**1. Introduction**

#### **1.1 Significance of vacuum and vacuum vessel**

 The word "vacuum*"* comes from the Latin word "vacuus" which means empty or void of stuff. The vacuum categories include natural vacuum and artificial vacuum. Space is a region with supreme zero pressure from a natural vacuum. A perfect or entire vacuum, which suggests an area that is entirely void of matter, is unachievable on earth, but it is possible to generate and hold the lower pressure region referred to as an artificial vacuum by means of a vacuum vessel and vacuum generation pump. The objective of artificial vacuum generation techniques is to simulate the vacuum pressure environment. A vacuum in space can have a pressure less than 760 Torr.

 A vacuum vessel is a compact, rigid, closed chamber in which air and other gas molecules have been evacuated by a vacuum pump. A vacuum vessel isolates the partial vacuum region from the outside atmosphere. The lowest pressures inside the vessel is reached by using modern vacuum pumping strategies where large amounts of molecules are evacuated. The core of the vacuum framework is the

#### *A Review on Development and Fabrication Methodology for Vacuum Vessel from Aluminum Alloy DOI: http://dx.doi.org/10.5772/intechopen.81083*

vacuum chamber, which is customized to each application. It encloses the application and reliably splits it from the outside or protects the surrounding from the processes inside. A vacuum vessel has numerous isolating ports that are covered with the flanges and this enables instruments or windows to be introduced in the dividers of the vessels.


**Application of a Gray-Based Decision Support Framework for Location Selection of a Temporary Hospital during COVID-19 Pandemic**

#### **Sarfaraz Hashemkhani Zolfani 1,\*, Morteza Yazdani 2 , Ali Ebadi Torkayesh 3 and Arman Derakhti 1**

Received: 3 May 2020; Accepted: 14 May 2020; Published: 30 May 2020

**Abstract:** The hospital location selection problem is one of the most important decisions in the healthcare sector in big cities due to population growth and the possibility of a high number of daily referred patients. A poor location selection process can lead to many issues for the health workforce and patients, and it can result in many unnecessary costs for the healthcare systems. The COVID-19 outbreak had a noticeable effect on people's lives and the service quality of hospitals during recent months. The hospital location selection problem for infected patients with COVID-19 turned out to be one of the most significant and complicated decisions with many uncertain involved parameters for healthcare sectors in countries with high cases. In this study, a gray-based decision support framework using criteria importance through inter-criteria correlation (CRITIC) and combined compromise solution (CoCoSo) methods is proposed for location selection of a temporary hospital for COVID-19 patients. A case study is performed for Istanbul using the proposed decision-making framework.

**Keywords:** COVID-19; criteria importance through inter-criteria correlation (CRITIC); combined compromise solution (CoCoSo); gray values; temporary hospital; location selection


2.2. Preoperative patient education

post-operative recovery [1, 2]. FT refers to a standardized, evidence-base and multimodal strategy to surgery [3]. It aims at early recovery, early discharge with better prognosis and less complications. Outpatient surgical pathways even go one step further and aim for same day of

This concept was pioneered by Professor Henrik Kehlet in the early 1990s and firstly applied in the colorectal surgery [4]. It is then expanded to many surgical fields, e.g., gynecologic, urologic, cardiovascular or orthopedic surgery. Over the past decades, it has been proven an effective and beneficial program for not only patients but also surgeons as well as the health insurance system. As increasing elderly population and increasing incidence of osteoarthritis, the growth of requirement of knee arthroplasty is to be expected [5]. However, the difficulty in economic health care along with increasing financial pressure has reinforced the necessity of putting FT into the focus. An increasing number of knee surgeons have already introduced fast track surgery for patients undergoing knee arthroplasty. It is carried out by the fast track team which composed of anesthetists, surgeons, pain specialist, physiotherapists, and nurses [6].The main strategy consists of five strands: careful patient selection, improving preoperative care, minimizing perioperative stress, decreasing postoperative discomfort, and improving postoperative recovery, thus leading to potentially lower mortality and morbidity as well as optimizing patient satisfaction. To make the FT program and related approaches easily-understood, we classified it into 3 phases according to the proceeding of operation: the pre-, intra- and post-

The challenge for the patients begins with the initial consultation and the decision to undergo total knee arthroplasty (TKA). It is important to select appropriate candidates and optimize

admission and discharge of the patient undergoing selective operations.

operative management. (Figure 1).

Figure 1. Algorithm of fast track TKA surgery.

114 Primary Total Knee Arthroplasty

2. Preoperative management

Adequate patient education is required to prevent patients or their family from holding unrealistic expectation. Surgeon should notify patients of what will happen during their inpatient stay, associated risks and the postoperative recovery plan. Some educational class illustrating the whole procedure and related impacts are therefore arranged for the candidates for TKA. In addition, the identification of assistive care companion at home is important to clarify the availability and ability of nursing after discharge, leading to the reduced anxiety of the patients for the coming surgical interventions. Through sufficient patient education and discussion, less anxiety with enhanced patient compliance could be expected [9].

#### 2.3. Preoperative preparation

#### 2.3.1. Infectious prophylaxis

One of the great challenge for orthopedic surgeon was avoiding the prosthetic joint-associated infections [10]. Several patients' preoperative conditions are considered as risk factors for increasing the rate of infection after TKA. Risk factors include old age, poor nutritional status, extreme body mass index, smoking, rheumatoid arthritis or diabetes mellitus [11–13]. Previous histories of trauma, steroid injection, or infection elsewhere in the body are also associated with increasing rates of infection. According to the Surgical Care Improvement Project initiated in 2004, rate of wound infections was reduced by the administration of prophylactic antibiotics. Current literature suggested that systemic administration of prophylactic antibiotics should be given within 60 min of surgical incision [14].

mixture of medications that include long-acting anesthetic, NSAIDs and epinephrine. Regimen varies from institution to institution. It is injected to the posterior capsule, collateral ligaments, capsular incision, quadriceps muscle tendon, and the adjacent subcutaneous tissues. Significant reduction in opioid consumption, improvement in pain VAS score and patient satisfaction are

Fast Track Surgery Program in Knee Replacement http://dx.doi.org/10.5772/intechopen.74026 117

The duration of the surgical procedure should be minimized as short as possible. However, the delay in operation duration is frequently reported with revision surgeries, the use of computer navigation and inexperienced surgeons. Prolonged operative time may be highly associated with the increase rate of surgical site infection, deep wound infection and other associated

Minimally invasive approach for total knee arthroplasty was introduced in 1990s, and popularized in recent 10 years. The minimally invasive approach allowed smaller wound incision, less soft tissue trauma, less invasion to muscle, especially vastus medialis obliquus (VMO). Minimally invasive approach has transformed from conventional parapatellar approach, and later converted to sub-vastus approach, which boasted no invasion to VMO, and had less soft tissue damage during the procedure. However, with VMO preserved the surgical field clearance decrease and may lead to difficulties in prosthesis sizing, and placement. Mini-midvastus approach was then introduced, and was shown to have similar outcomes comparing to sub-

Through minimal invasive approach, extensor muscles were maximally preserved. Large scale of RCTs showed the better short-term outcomes including better knee flexion/ extension torque, faster days to raise leg, greater range of motion, higher in knee society score (KSS), less total estimated blood loss and less postoperative pain [26, 27]. However, minimal invasive approach may also contribute to longer tourniquet time and operating time, as well as wound complication. In term of long term outcomes and longevity, there is still insufficient evidence to declare that minimal invasive approach had long term advantages over conventional total

Perioperative blood loss of TKA can be significant. It could lead to resultant anemia which leads to further need of blood transfusion associated morbidity and mortality [28]. Therefore, it is always an important issue to avoid massive blood loss during and after TKA procedure.

During TKA, a pneumatic tourniquet is commonly used to provide clear surgical field and significantly reduces blood loss and surgical time [29]. However, no significant difference was shown for postoperative knee-extension strength, hemoglobin level, pain, nausea, length of

observed in patients receiving LIA [21, 22].

3.2. Duration of the surgical procedure

3.3. Minimally invasive surgical approach

complications [23].

vastus approach [24, 25].

knee arthroplasty.

3.4. Blood management strategy

hospital stay, and local swelling.

#### 2.3.2. Preemptive analgesia

Analgesia given prior to surgery is assumed to prevent peripheral and central sensitization. Preemptive medications such as NSAIDs, COX-2 inhibitors, and the neuropathic agent gabapentin and pregabalin have all shown promising result in reducing the magnitude and duration of postoperative pain [15]. However, the optimal dose, timing of administration, and whether there is potential benefit of continuing the analgesics during operation remain debating issues.


**8. Accuracy**

10-bit resolution of FADC in the high-gain channels (responsible for a trigger generation) implies the ranges of *X*¯ *k* coefficients given in the 2nd column of Table 1. Multiplications of integer values N by real scaling factors sf give floating-point results. In order to keep possible high speed of calculation and not to utilize resources spendthrift the fixed-point algorithm of processing has been chosen. N×sf were approximated on each pipeline stage again to the integer value. For almost all scaling factors: sf ≤ 1, N×sf has a representation of the same or less amount of bits. For sf ≥ 1, N×sf extends the representation on 1 or 2 bits. This approximation introduces errors. However, the width of the data in the internal pipeline stages is extended from the N at the shift register *x*15,...,*x*0, to N+1, N+2, N+3, N+4, N+5, N+7, N+8 in routines A, B, C, D, E, F, G, respectively (Fig. 8). This reduces approximation errors mostly to the LSB, apart the *X*¯ 15. This coefficient will not be used for a trigger.

Table 1. Ranges of *X*¯ *k* coefficients and relative errors for least significant bits of *X*¯ *k*. For k ≤ 14 the errors appear practically only in the LSB.

According to above estimations, the configuration with 3 "engines" does not support all *ξk* sub-triggers due to limited amount of DSP blocks. However, for the next generation of the water Cherenkov detectors array, where probably only a single PMT will be used, 3 "engines" will be implemented to investigate and to detect 3 different shapes of FADC traces corresponding to i.e. different rise times of the rising edge.

#### **9. Preliminary tests**

Analysis of Auger ADC traces of very inclined showers shows that the maximum of the signal is mostly reach in a single time bin. The attenuation factor for a tail is in the range of *β* = (0.2 - 0.5). Fig. 12 shows shapes of signals with various attenuation factors with two first time bins on a pedestal level. For simplicity it has been set on zero. It does not reduce the generality of analysis, because the pedestal is irrelevant for DCT (k ≥ 1). The corresponding DCT coefficients are shown in upper Fig. 3 (Shape\_A). After a single clock cycle, when data is shifted in the registers chain, shifted signal with only one time bin on the pedestal level determines a new set of the DCT coefficients shown in lower Fig. 3 (Shape\_B). Pattern, which is going to be recognized, can be selected by a setting of DCT coefficient in the DCT engines.

18 Will-be-set-by-IN-TECH

of a signal requires a single clock cycle only. All routines are fast enough to work with 100 MHz sampling without an additional pipeline stages and they do not introduce an additional

10-bit resolution of FADC in the high-gain channels (responsible for a trigger generation) implies the ranges of *X*¯ *k* coefficients given in the 2nd column of Table 1. Multiplications of integer values N by real scaling factors sf give floating-point results. In order to keep possible high speed of calculation and not to utilize resources spendthrift the fixed-point algorithm of processing has been chosen. N×sf were approximated on each pipeline stage again to the integer value. For almost all scaling factors: sf ≤ 1, N×sf has a representation of the same or less amount of bits. For sf ≥ 1, N×sf extends the representation on 1 or 2 bits. This approximation introduces errors. However, the width of the data in the internal pipeline stages is extended from the N at the shift register *x*15,...,*x*0, to N+1, N+2, N+3, N+4, N+5, N+7, N+8 in routines A, B, C, D, E, F, G, respectively (Fig. 8). This reduces approximation errors

> k range of LSB 2*nd* 3*rd* and k range of LSB 2*nd* 3*rd* and *X*¯ *k* bit more *X*¯ *k* bit more 0...4092 0.0% 0.00% 0.00% 8 ± 2041 0.0% 0.00% 0.00% ± 2521 13.1% 0.00% 0.00% 9 ± 12224 23.8% 1.55% 0.00% ± 2581 8.7% 0.00% 0.00% 10 ± 4557 12.8% 0.00% 0.00% ± 2914 13.1% 0.00% 0.00% 11 ± 7519 17.7% 0.00% 0.00% ± 2348 4.8% 0.00% 0.00% 12 ± 5671 11.5% 0.00% 0.00% ± 4019 15.1% 0.00% 0.00% 13 ± 9605 24.3% 2.00% 0.00% ± 3045 8.6% 0.00% 0.00% 14 ± 12978 26.9% 2.86% 0.00% ± 10032 23.1% 1.10% 0.00% 15 ± 25597 30.9% 25.08% 6.83%

Table 1. Ranges of *X*¯ *k* coefficients and relative errors for least significant bits of *X*¯ *k*. For k ≤ 14

According to above estimations, the configuration with 3 "engines" does not support all *ξk* sub-triggers due to limited amount of DSP blocks. However, for the next generation of the water Cherenkov detectors array, where probably only a single PMT will be used, 3 "engines" will be implemented to investigate and to detect 3 different shapes of FADC traces

Analysis of Auger ADC traces of very inclined showers shows that the maximum of the signal is mostly reach in a single time bin. The attenuation factor for a tail is in the range of *β* = (0.2 - 0.5). Fig. 12 shows shapes of signals with various attenuation factors with two first time bins on a pedestal level. For simplicity it has been set on zero. It does not reduce the generality of analysis, because the pedestal is irrelevant for DCT (k ≥ 1). The corresponding DCT coefficients are shown in upper Fig. 3 (Shape\_A). After a single clock cycle, when data is shifted in the registers chain, shifted signal with only one time bin on the pedestal level determines a new set of the DCT coefficients shown in lower Fig. 3 (Shape\_B). Pattern, which is going to be recognized, can be selected by a setting of DCT coefficient in the DCT engines.

mostly to the LSB, apart the *X*¯ 15. This coefficient will not be used for a trigger.

the errors appear practically only in the LSB.

**9. Preliminary tests**

corresponding to i.e. different rise times of the rising edge.

latency.

**8. Accuracy**

Fig. 12. Shapes of signals with various attenuation factors and two first time bins on the pedestal level

All signals with first two time bins on the pedestal level for sure will be with only one time bin on the pedestal level in the next clock cycle. But, not vice versa. A signal with only a single time bin on the pedestal level before sharp rising edge can have significant contribution in the 2nd time bin before rising edge and it will not be recognized by a pattern recognition procedure tuned on the Shape\_A. A procedure recognizing Shape\_A is more restrictive and gives lower trigger rate than for the Shape\_B. Due to limited amount of the DSP blocks only 11 DCT coefficients can be analyzed simultaneously. For the Shape\_A the *X*¯ 4 and *X*¯ 10 are ignored and for the Shape\_B : *X*¯ 6 and *X*¯ 14, respectively, as weakly sensitive on changes of signal shapes. The trigger based only on the DCT pattern recognition gives too high rate, due to a contribution of very week signals with also appropriate shape, but usually treated as noise. In order to reduce and control the trigger rate, the veto threshold has been introduced. The calculation of the DCT coefficients in the pipeline chain and next the calculation of sub-triggers in multipliers and comparators block takes 12 clock cycles. The signal is synchronized with the DCT sub-triggers delayed the same time to be compared with the veto threshold, simultaneously with a generated DCT sub-triggers. If the signal is above the sum of the veto threshold and the pedestal, the sub-triggers are enabled to generated a final spectral trigger. The condition that all 11 DCT coefficients were inside the acceptance lane is too strong. The shapes are not ideal, noise introduces additional shape distortions. Similarly as in the ToT trigger only a part of "fired" sub-triggers (Occupancy ≤ 11 = max. number of sub-triggers) is enough to generate the final spectral trigger.

Fig. 13. Coefficients for signals with various attenuation factors and two first time bins (left) and only one time bin (right) on the pedestal level

Experiments 21

399 An Optimization of 16-Point Discrete Cosine Transform Implemented into a FPGA as a Design for a Spectral First Level Surface Detector Trigger in Extensive Air Shower Experiments

> 0 10 20

Rates vs. time for 25-35 requirement

0 96 192 288 384 480 576 672 768 864 960

Contribution of the DCT sub-triggers to the final trigger for the Occ = 7

2 3 4 5 6 7 8 9 10 11 12 13 14

rate veto

time (s)

PMT1 : 15-25 PMT2 : 15-25 PMT3 : 15-25 PMT1 : 25-35 PMT2 : 25-35 PMT3 : 25 35

PMT3 : 25-35PMT1 : 35-45 PMT2 : 35-45 PMT3 : 35-45

B

Rates (Hz) &

to (ADC-counts)

veto

> 0 2

A

rate - 0.26 rate - 0.27

veto

time (s)

C

rate - 0.28 rate - 0.27 rate - 0.26 t 0 25

rate - 0.25veto

time (s)

and C). A comparison of a contribution of "fired" DCT coefficients generating the sub-triggers for all three PMT channels and for various trigger rates requirements

different PMT and various configuration of trigger rate requirements

surface detector, when the coincidence technique cannot be longer used.

other than the present Pierre Auger Observatory - only one PMT per station.

for arbitrarily selected acceptance lane of the spectral trigger rate.

Fig. 15. Rates obtained in the test detector during an auto-calibration process (graphs A, B

(right-down). There are no significant differences in a contribution of a fixed coefficients for

high and the attenuation factor has to be increased (the acceptance lane is narrowed down). The graph B shows the process, when the initial parameters are optimal and the acceptance lane is not modified (only the veto threshold is tuned). The graph C shows the process when the initial parameters give too low trigger rate and the acceptance lane is changed three times.

The pattern recognition technique implemented parallel with the standard threshold detection may improve an efficency of a registration of rare events, especially for a single PMT in the

The optimized algorithm of the spectral trigger based on the Discrete Cosine Transform with veto and auto-calibration procedure has been successfully implemented into the FPGA and showed the perfect stability in the real detector. Measurements in the test detector confirmed assumption for a selection of limited amount of DCT coefficients and a stability of algorithm

Although 6 surface detectors from the Pierre Auger Observatory have been used for the tests, the spectral trigger is being developed more generally for future ground EAS arrays using -

The author would like to thank the Pierre Auger Collaboration for being allowed to use a PAO infrastructure and a test-hexagon and for getting the data made available. The successful installation and preliminary tests of the new Front End Boards with the DCT trigger would

0 10 20

0 10 20

**10. Conclusion**

**11. Acknowledgement**

es (Hz) &

Rates veto (ADC-counts)

ADC-counts)

Rates vs. time for 15-25 Hz requirement

0 0 103 206 309 412 515 618 721 824 927

Rates vs. time for 35-45 Hz requirement

0 0 96 192 288 384 480 576 672 768 864 960

Rates (Hz) &

& veto

(ADC-counts)

Although the spectral trigger is being developed for the future and for a single detection channel (a single PMT), the DCT trigger in the Auger surface detector has been tested in a 2-fold coincidences of any 3 PMTs, to be close as possible for a comparison of the results with the standard Auger data.

Fig. 14. Rates for various acceptance lanes (from 0.20 - 1.00 till 0.30 - 1.50) and the veto threshold (0 - 80 ADC-counts) for the Occupancy = 6 (left) and 7 (right), respectively. Due to a possible saturation of a transmission channel, the rate of the spectral trigger should not exceed ca. 40 Hz.

Fig. 14 show the trigger rate for the Occupancy = 6 and 7, respectively. The T1 trigger rate is calibrated to ca. 100 Hz. Generally, the trigger rate for Occ = 6 is too high. In order not to saturate the microcontroller and the transmission chain the total (standard Auger + spectral) trigger rate should not exceed 150 Hz. This gives max. 40 - 50 Hz for the spectral trigger only. The Occupancy = 7 with a range of attenuation factors limited to *β* = (0.20 - 1.14) gives a trigger range on the reasonable level. The Occupancy = 8 reduces the trigger rate below 1 Hz and seems to be too restrictive. The FPGA contains internal counters counting the trigger rate and a contribution of DCT sub-triggers to the final trigger. The required trigger rate range can be set remotely from the Central Data Acquisition System (CDAS). The FPGA automatically tunes the veto threshold to get the required trigger rate. If the veto threshold is above 60 ADC-counts (ca. 1.2 VEM) the acceptance lane is modified. The attenuation factor *β* from the left side of the range is increased/decreased in the range of (0.20 - 0.40) by the fixed right boundary *β* = 1.3.

Fig. 15 show three calibration processes, when either initial parameters has been set ideally (B) or they have to be tuned to get required trigger rate (A and C). The tuning process typically does not exceed 3 minutes. In contrary to the standard Auger tuning procedure, when the thresholds for the Threshold trigger are calculated by the external microcontroller located on the Unified Board (UB), the thresholds for the acceptance DCT lane are initially calculated and next stored in the ROM inside the FPGA and they are only multiplexed. This allows a full autonomous FPGA calibration process without a support by any external microcontroller. The new Front-End Board samples analog signals with 80 MHz. Data is written via a left port in the dual-port RAM. Stored data are next read via the right port with 40 MHz. The new board is seen by the rest of electronics as the standard one. Only a additional flag informs the system on the type of the trigger. Internal FPGA counters allow counting a contribution of the DCT coefficients to the final spectral trigger. Fig. 15d shows a relative contribution of the DCT coefficients for the Shape\_A. The contribution of the *X*¯ 5 and *X*¯ 9 is a little bit lower than the rest ones. For the *X*¯ 9 the acceptance lane (compare Fig. 13A) is relatively narrow, so the lower contribution is not strange. *X*¯ 5 is probably more sensitive on signal noise and possible signal distortions. The graph A (Fig. 15) shows a process when the trigger rate is initially too

Fig. 15. Rates obtained in the test detector during an auto-calibration process (graphs A, B and C). A comparison of a contribution of "fired" DCT coefficients generating the sub-triggers for all three PMT channels and for various trigger rates requirements (right-down). There are no significant differences in a contribution of a fixed coefficients for different PMT and various configuration of trigger rate requirements

high and the attenuation factor has to be increased (the acceptance lane is narrowed down). The graph B shows the process, when the initial parameters are optimal and the acceptance lane is not modified (only the veto threshold is tuned). The graph C shows the process when the initial parameters give too low trigger rate and the acceptance lane is changed three times.


**3.2 Acetic acid-induced writhing in mice**

The aqueous extract of *V. amygdalina* (50, 100, 200mg/kg i.p) exhibited a significant (P<0.05) antinocicetive activity against acetic acid-induced writhing in mice. 50 and 100 mg/kg oral doses exhibited a dose-dependent anti-nociception that progressively reduced over a period of 90 min post-treatment. However, at 120min the reduced anti-nociceptive activity increases again at these doses. The dose of 200 mg/kg on the other hand caused a total antinociception up to 120min. These results compared favorably with those of aspirin (100mg/kg i.p; Fig 1).

Fig. 1. Effect of aqueous extract of *V.amygdalina* leaves on acetic acid-induced writhing in mice for 5 min (NS, normal saline; Va, *V. Amygdalina*; ASA acetysalicylic acid). All data are presented as means ± S.E.M., n=5. The asterisk (\*) denotes, significance (p<0.05) between treated group and NS control


Fujino et al., 2001, 2002; Nanni et al., 2003). The proband described here, however, was followed closely since birth given her brother's diagnosis, and she never had any echocardiographic or other evidence of HCM, suggesting that this combination of cTnC alleles results purely in DCM. Of significance is another example of compound heterozygosity where the D145E variant was again associated with inherited cardiomyopathy (Ploski et al., 2016). In this case, two pediatric patients were diagnosed with autosomal recessive restrictive cardiomyopathy which was fatal during infancy. Sanger sequencing revealed that both of the affected individuals were compound heterozygous for genetic variants in TNNC1 corresponding to D145E and A8V in the cTnC protein.

Variants in TNNC1 can affect skeletal muscle function because cTnC is also expressed in slow skeletal muscle. The D145E mutant was reported to increase slow skeletal muscle ATPase activity at low and high calcium concentrations in reconstituted myofibrils (Veltri et al., 2017b). Although it is known that D145E does not affect myofilament calcium sensitivity in slow skeletal skinned fibers (Veltri et al., 2017b), it is unclear whether D132N has any effect on slow skeletal muscle function. Importantly, no skeletal muscle weakness was reported in the patient's history.

### Implications for cTnC Function in the Cardiac Thin Filament

Both variants are located in the C-domain of cTnC (**Figure 3**). The missense D132N variant is located within the G-helix that is part of EF-hand site IV, and D145E is one of the divalent cation (Ca2+ or Mg2+) coordinating residues (+Z position) in site IV. While the regulatory site for Ca2+ binding is site II in the N-domain of cTnC, our prior results indicate that there is communication between the C- and N-domains (Badr et al., 2016) such that changes in the C-domain can influence the regulatory N-domain (Marques et al., 2017; Veltri et al., 2017a). The results of this study demonstrated that the D145E variant (WT/D145E) increases myofilament Ca2+ sensitivity even when WT is also present, while the D132N mutation (WT/D132N) had the opposite effect of decreasing Ca2+ sensitivity (**Table 2** and

TABLE 4 | Optimized parameter estimates and predictions from the 3-state model for force-kTR data depicted in Figure 7.

Best fit parameter estimates for three parameters (f, g, and kOFF) were obtained in MatLab using the Simplex method for the force-kTR data in Figure 7, taking into account the Ca2+ sensitivity (pCa50) from steady-state isometric force measurements (section "Experimental Procedures") as reported in Table 2; values for kON were derived from measurements reported by Pinto et al. (2011a) and maximal isometric force was assumed to be 1.0 for each dataset. These parameter sets were used to illustrate relations between kTR and steady-state, isometric force in Figure 7 (dashed lines in Figure 7B).

**Figures 5A,B**) even though the two affected residues are relatively close in the primary sequence and the 3D structure (**Figure 3**). In general, myofilament incorporation of troponin variants that cause opposing effects on thin filament Ca2+ responsiveness are expected to normalize the myofilament Ca2+ sensitivity (Li et al., 2010; Alves et al., 2014, 2017; Dieseldorff Jones et al., 2018), and that was what was found in this study. Compared to WT, the 50%:50% mix of D132N/D145E showed similar myofilament Ca2+ sensitivity and Hill coefficient (nHill) values, the latter generally reflecting cooperative processes associated with isometric force generation. While incorporation of the two mutants appeared to normalize opposite effects of the two variants on Ca2+ dependence of isometric force, what is difficult to reconcile is the severe DCM pathology associated with the presence of the two mutations.

A paradigm associating changes in myofilament Ca2+ sensitivity commonly found in HCM (increased) and DCM (decreased) is generally well-accepted (Willott et al., 2010; van der Velden and Stienen, 2019). This study, however, presents a unique clinical case in which we observed no significant changes in myofilament Ca2+ sensitivity due to a combination of variants in patients with DCM. How can this finding be explained? After depleting native cTnC from CMPs, each available unfilled cTnC site was presumably reconstituted, depending on the experiment, with either exogenous WT, WT/D132N mix, WT/D145E mix, or D132N/D145E mix in an attempt to mimic what is expected to be found in the patients (D132N/D145E) or the parents (WT/D132N or WT/D145E), along with the control condition (100% WT). Previous studies showed that D145E cTnC dissociates more slowly from the thin filament compared to WT cTnC indicating that D145E has a higher binding affinity for the thin filament (Marques et al., 2017), although the functional data (**Table 2**) do not suggest that this difference in affinity influenced the results.

As with our finding, there has been some variation in prior reports on variants in TNNC1 with regard to the hypothesis that HCM is always associated with increased Ca2+ sensitivity and DCM with decreased Ca2+ sensitivity. The first DCM-associated variant identified in TNNC1, G159D (Mogensen et al., 2004), has been explored by several groups with conflicting findings. It was initially reported that G159D did not alter myofilament Ca2+ sensitivity in Tn-exchanged skinned rat trabeculae (Biesiadecki et al., 2007). In contrast, both cTnC-depleted skinned porcine papillary muscle and in vitro regulated thin filaments reconstituted with cTnC-G159D exhibited reduced myofilament Ca2+ sensitivity (Robinson et al., 2007; Dweck et al., 2008). Conversely, G159D exhibited increased myofilament Ca2+ sensitivity in both human skinned ventricular myocytes obtained from a patient bearing G159D, and thin filaments reconstituted with skeletal muscle actin, human cardiac tropomyosin, and troponin complex containing cTnC G159D (Dyer et al., 2009).

Loss of the lusitropic response (cardiac muscle relaxation) is considered a common feature of DCM and HCM (Wang et al., 2012; Dweck et al., 2014; Messer and Marston, 2014). In the case of thin filament mutants, this can be partially attributed to uncoupling of cTnI phosphorylation from modulation of Ca2+ sensitivity (Memo et al., 2013; Messer and Marston, 2014). Studies using TnC-extracted porcine papillary muscle fibers reconstituted with cTnC DCM-associated mutants (Y5H, M103I or I148V) exhibited diminished or abolished PKA-mediated Ca2+ desensitization (Pinto et al., 2011b). Although we did not test for PKA-mediated myofilament Ca2+ desensitization in our current study, we speculate that the presence of both mutant proteins could interfere with cTnI phosphorylation levels and functional consequences. This could cause a loss of downstream β-adrenergic stimulation which is crucial to effect the lusitropic response and diastolic ventricular filling.

Alternative possibilities exist because the two mutations are not present in the same protein. In the affected siblings, the two variant proteins are expected to be distributed randomly in thin filaments, and differences in function might affect communication between adjacent regulatory units when they contain different cTnCs. The lack of change in nHill for D132N/D145E (**Table 2** and **Figures 5A,B**) suggests this might not be a plausible explanation, although the Hill coefficient represents the overall response and does not inform about highly localized effects when two variant cTnCs are present. Mechanical kinetics provides information beyond steady-state force measurement, and cooperative interactions among regulatory units do not appear to be required for fast rates of tension redevelopment, kTR (Chase et al., 1994). Both the force-kTR data (**Figure 7B**) and the

3-state model analyses of those force-kTR relations (**Table 4**) suggest that the presence of either variant (WT/D132N or WT/D145E) shifts the force-kTR relation, and alters contractile kinetics (kTR in **Figure 7B**, and f and g in **Table 4**) in the same direction. However, both the kinetic data (**Figure 7B**) and modeling (**Table 4**) support the conclusion from analysis of the steady-state force-pCa data (**Table 2** and **Figures 5A,B**) that the combination of the two variants, D132N/D145E, in thin filaments may partially ameliorate the effects of either mutant on contractile function. This is because D132N/D145E yields results more similar to 100% WT than either WT/D132N or WT/D145E (**Table 4** and **Figure 7B**). Thus it appears to be necessary to seek alternative explanations, beyond changes in biomechanical function, to explain the deleterious effects of the two variants in combination in the proband and her brother.

### cTnC Function Beyond the Sarcomere

We and others have previously suggested that troponin subunits may participate in cellular functions that extend beyond contractile regulation in the sarcomere (Asumda and Chase, 2012; Wu et al., 2015; Zhang et al., 2016; Pinto et al., 2017). In fact, cTnC has been previously detected in cardiomyocyte nuclei, as well as the nuclei of certain cancer cells (Johnston et al., 2018). In addition, cTnC has been identified in the mitochondrial compartment of some cancer cell lines. Therefore, a potential mechanistic explanation for the severe disease phenotype associated with troponin mutants may be attributed to perturbations in its putative nuclear/mitochondrial function, whereby gene expression or bioenergetics could be altered by cTnC variants such as those reported here.

In summary, we report the association of compound heterozygous variants in TNNC1 in siblings with early onset familial DCM. We provide functional evidence that both variants separately lead to abnormal myofilament Ca2+ sensitivity in which the variants D132N and D145E are, respectively, associated with reduced or increased myofilament Ca2+ sensitivity relative to WT, while these effects compensated for no change relative to WT when the two variants were combined. Analyses of activation-dependence of kTR suggest that both variants have similar effects on contractile kinetics, although there was no significant change in maximal kTR, and that the combination of the two variants again partially ameliorated the effects of the individual variants. None of the observed changes in mechanics, however, can explain the clinical outcomes in this family. Further studies are needed to understand precisely how these variants combine to result in a severe DCM phenotype.

### DISCLOSURE

SL is part owner of Qiyas Higher Health, a startup company unrelated to this work.

### DATA AVAILABILITY STATEMENT

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher.

### ETHICS STATEMENT

The studies involving human participants were reviewed and approved by the Institutional Review Board of Yale University School of Medicine. Written informed consent to participate in this study was provided by the participants' legal guardian/next of kin. The animal study was reviewed and approved by the Yale University Institutional Animal Care and Use Committee.

### AUTHOR CONTRIBUTIONS

JP and SL conceptualized the study and oversaw project administration. ML-V, JJ, WJ, EM, JT, MS-M, LJ, EH, DP-M, MK, and ED contributed to the data curation. PC contributed to computational data analysis and modeling. ML-V, JJ, JT, EM, PC, SL, and JP wrote the manuscript. All authors reviewed and approved the final version of the manuscript.

### FUNDING

JP was supported by the National Institutes of Health (grant no. HL128683). The Yale PGDP was supported by the Yale New Haven Hospital and Sara and Jeffery Buell.

### ACKNOWLEDGMENTS

The authors are grateful to this family for sharing their story. The authors also thank Bradley's Country Store, Tallahassee, FL, United States, for generously supplying porcine hearts, and Monica Konstantino for patient recruitment.

### SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphys.2019. 01612/full#supplementary-material

VIDEO S1 | Optical Coherence Tomography imaging showing contractility of tadpole hearts. Control heart (top panel) with good systolic contractility. Two views of TNNC1-F0 mutant hearts (middle and lower panels) showing minimal motion and lack of effective contractility.

### REFERENCES

fphys-10-01612 January 17, 2020 Time: 16:21 # 12

that causes familial dilated cardiomyopathy, studied in explanted heart muscle. Circ. Heart Fail 2, 456–464. doi: 10.1161/CIRCHEARTFAILURE.108.818237

**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Landim-Vieira, Johnston, Ji, Mis, Tijerino, Spencer-Manzon, Jeffries, Hall, Panisello-Manterola, Khokha, Deniz, Chase, Lakhani and Pinto. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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Recently, researchers specified their attention toward the complex and interesting properties of non-Newtonian meterials due to their miscellaneous application in many industries and technologies. The non-Newtonian materials due to convoluted features attracted special attention especially in the current century. The novel physical importance of such non-Newtonian liquids in various engineering and physical processes, biological sciences, physiology, and manufacturing industries is associated due to complex rheological features. Some useful applications associated with the non-Newtonian fluids include polymer solutions, certain oil, petroleum industries, blood, honey, lubricants, and many more. It is commonly observed that distinctive features of such non-Newtonian fluid cannot be pointed out via single relation. Therefore, different non-Newtonian fluid models are suggested by investigators according to their rheology. Among these, Eyring–Powell fluid is inferred from kinetic laws of gases instead of any empirical formulas. This model reduces the viscous fluid at both low and high shear rates (see Powell and Eyring [19]). Gholinia et al. [20] carried out the homogenous and heterogeneous impact in flow of Powell–Eyring liquid due to rotating. Khan et al. [21] focused on viscositydependent mixed convection flow of Eyring–Powell nanofluid encountered by inclined surface. Salawu and Ogunseye [22] reported the entropy generation prospective in Eyring–Powell nanofluid featuring variable thermal consequences and electric field. Another useful continuation performed by Abegunrin et al. [23] examined the change in the boundary layer for the flow of Eyring–Powell fluid subjected by the catalytic surface reaction. Rahimi et al. [24] adopted a numerical technique to compute the numerical solution of a boundary value problem modeled due to the flow of Eyring–Powell fluid. Reddy et al. [25] involved some interesting thermal features, like activation energy, chemical reaction, and non-linear thermal radiation in the 3D flow of Eyring–Powell nanofluid induced via slandering surface. Hayat and Nadeem [26] examined the flow of Eyring–Powell fluid and suggested modification in energy and concentration expressions by using generalized Fourier's law. Ghadikolaei et al. [27] reported the Joule heating and thermal radiation features inflow of Eyring–Powell non-Newtonian fluid in a stretching walls channel.

The double diffusion convection is a natural phenomenon that encountered multiple novel applications in area soil sciences, groundwater, oceanography, petroleum engineering, food processing, etc. The double-diffusive convection refers to the intermixing of components of two fluid having different diffuse rates. However, the situation becomes quite interesting when double-diffusive convection depends upon more than two components of fluids. Examples of such multiple diffusive phenomenons include seawater, molten alloy solidification, and geothermally heated lakes. The triple diffusion flow appears in diverse engineering and scientific fields like geology, astrophysics, disposals of nuclear waste, deoxyribonucleic acid (DNA), chemical engineering, etc. [28–30].

After careful observation of the previously cited work, it is claimed that no efforts have been made to report the triple diffusion flow of Eyring–Powell nanofluid induced by an oscillatory stretching surface with variable thermal features. Although some investigations on flow that is due to periodical acceleration have been available in the literature, thermodiffusion features for Eyring–Powell nanofluid are not studied yet. Therefore, our prime objective of this contribution is to report the triple diffusion aspects of Eyring–Powell nanofluid flow by using variable thermal properties. The most interesting convergent technique homotopy analysis procedure is followed to simulate the solution [31–35]. The graphs are prepared to see the impact of different flow parameters with relevant physical consequence.

### FLOW PROBLEM

To develop governing equations for unsteady flow of Eyring– Powell nanofluid, we have considered a periodically stretching surface where x-axis is assumed along with the stretched configuration, whereas y-axis is taken normally. The source of induced flow is based on the periodically moving surface where amplitude of oscillations are assumed to be small. Let velocity of the moving surface as u = uω = bx sin ωt, b as stretching rate, ω being angular frequency, whereas t represent time. The uniform features of the magnetic field are reported by implementing it vertically. Let T represent the temperature, C solutal concentration, whereas 8 report the nanoparticle volume fraction. Furthermore, T∞, C∞, and 8∞ denote free stream nanoparticle temperature, free stream solutal concentration, and volume fraction of nanofluid, respectively. After using such assumptions, the flow problem is modeled through the following equations [22, 33]:

$$
\frac{\partial \mu}{\partial x} + \frac{\partial \nu}{\partial y} = 0,\tag{1}
$$

$$\begin{split} u\left(\frac{\partial u}{\partial \mathbf{x}}\right) &+ \nu \left(\frac{\partial u}{\partial \mathbf{y}}\right) + \frac{\partial u}{\partial t} = \left(\nu + \frac{1}{\rho\_f \beta^\* \mathbf{C}}\right) \left(\frac{\partial^2 u}{\partial \mathbf{y}^2}\right) \\ &- \frac{1}{2\rho \beta^\* \mathbf{C}^3} \left[ \left(\frac{\partial u}{\partial \mathbf{y}}\right)^2 \frac{\partial^2 u}{\partial \mathbf{y}^2} \right] - \left(\frac{\sigma B\_0^2}{\rho\_f} + \frac{\nu \vartheta}{k'}\right) u, \end{split} \tag{21}$$

$$\begin{split} \mu \left( \frac{\partial T}{\partial x} \right) &+ \left( \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} \right) + \frac{\partial T}{\partial t} = \frac{1}{\left( \rho \boldsymbol{c} \right)\_p} \frac{\partial}{\partial \boldsymbol{\mathcal{Y}}} \left( K \left( T \right) \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} \right) \\ &+ \tau\_T \left[ D\_T \frac{\partial \boldsymbol{\phi}}{\partial \boldsymbol{\mathcal{Y}}} \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} + \frac{D\_T}{T\_\infty} \left( \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} \right)^2 \right] + DK\_{TC} \left( \frac{\partial^2 C}{\partial \boldsymbol{\mathcal{Y}}^2} \right), \end{split} \tag{3}$$

$$u\left(\frac{\partial C}{\partial \mathbf{x}}\right) + \nu \left(\frac{\partial C}{\partial \mathbf{y}}\right) + \frac{\partial C}{\partial t} = D\_s \left(\frac{\partial^2 C}{\partial \boldsymbol{\eta}^2}\right) + D \mathcal{K}\_{\text{CT}} \left(\frac{\partial^2 T}{\partial \boldsymbol{\eta}^2}\right), \text{(4)}$$

$$u\left(\frac{\partial \Phi}{\partial \mathbf{x}}\right) + \nu \left(\frac{\partial \Phi}{\partial \boldsymbol{\eta}}\right) + \frac{\partial \Phi}{\partial t} = D\_B \left(\frac{\partial^2 \Phi}{\partial \boldsymbol{\eta}^2}\right) + \frac{D\_T}{T\_{\infty}} \left(\frac{\partial^2 T}{\partial \boldsymbol{\eta}^2}\right), \text{(5)}$$

where ν is viscosity, ρf fluid density, (β ∗ , C) fluid parameters, ϑ permeability of porous medium, σe electrical conductivity, α1 thermal diffusivity, DKTC Dufour diffusivity, τT = (ρc)p/(ρc)f ratio of heat capacity of nanoparticles to heat capacity of fluid, DB Brownian diffusion coefficients, Ds solutal diffusivity, DT thermophoretic diffusion coefficient, whereas DKCT Soret diffusivity.

Following boundary assumptions are articulated for current flow problem

$$\begin{aligned} \mu &= \mu(\mathbf{x}, t) = u\_{\text{w}} \sin \omega t = b\mathbf{x} \sin \omega t, & \nu &= 0, \\ T &= T\_{\text{w}}, \ C = C\_{\text{w}}, \Phi = \Phi\_{\text{w}} \text{ at } \ \mathcal{Y} = 0, & \text{(6)} \\ \mu &\to 0, \ \nu &\to 0, \ T \to T\_{\infty}, \ C \to C\_{\infty}, \Phi \to \Phi\_{\infty} \text{ at } \ \mathcal{Y} \to \infty. \end{aligned} \tag{7}$$

In order to suggest modification in energy equation (3), we used the following relations for variable thermal conductivity [33, 34]

$$K(T) = K\_{\infty} \left[ 1 + \varepsilon \frac{(T - T\_{\infty})}{\Delta T} \right],\tag{8}$$

where K∞ ambient fluid conductivity and ε thermal dependence conductivity constant. Now, before perfume analytical simulations, first, we reduce the number of independent variables in the governing equations by using the following variables:

$$\begin{aligned} \xi &= \left(\frac{b}{\upsilon}\right)^{1/2} \wp, \ \mathfrak{r} = t\wp, \ u = u\_{\mathfrak{w}} f\_{\mathfrak{f}}\left(\xi, \ \mathfrak{r}\right), \\\ \upsilon &= -\sqrt{\upsilon b f}\left(\xi, \ \mathfrak{r}\right), \\\ \theta\left(\xi, \mathfrak{r}\right) &= \frac{\left(T - T\_{\infty}\right)}{\left(T\_{\mathfrak{w}} - T\_{\infty}\right)}, \varphi\left(\xi, \mathfrak{r}\right) = \left(\frac{\left(C - C\_{\infty}\right)}{C\_{\mathfrak{w}} - C\_{\infty}}\right), \phi\left(\xi, \mathfrak{r}\right) \end{aligned} \tag{9}$$

$$=\frac{(\Phi-\Phi\_{\infty})}{(\Phi\_{\text{w}}-\Phi\_{\infty})},\tag{10}$$

The dimensionless set of equations in view of the previously mentioned transformations is

$$(1+K)f\_{\xi\xi\xi} - \S f\_{\xi\tau} - f\_{\xi}^2 + f f\_{\xi\xi} - \Omega f\_{\xi} - \Gamma K f\_{\xi\xi}^2 f\_{\xi\xi\xi} = 0,\tag{11}$$

$$\begin{aligned} \left( \left( 1 + \delta \theta \right) \theta\_{\xi} \xi \right) + \delta \left( \theta\_{\xi} \right)^{2} + \Pr \left[ f \phi\_{\xi} - \mathbb{S} \phi\_{\xi} + \mathrm{Nb} \theta\_{\xi} \phi\_{\xi} + \mathrm{Nt} \left( \theta\_{\xi} \right)^{2} \right] \\ + \left( \mathrm{Nd} \right) \varphi\_{\xi} \xi \right] = 0, \end{aligned} \tag{12}$$

$$\begin{aligned} \text{2. } \begin{pmatrix} \text{Nd} \end{pmatrix} \begin{aligned} \varphi\_{\xi\xi} \end{aligned} \Big| \begin{aligned} \text{2. } \begin{cases} \text{2. } \xi \end{cases} \end{aligned} \Big| \begin{aligned} \text{2. } \begin{cases} \text{2. } \xi \end{cases} \end{aligned} \Big| \begin{aligned} \text{2. } \begin{cases} \text{2. } \xi \end{cases} \end{aligned} \Big| \begin{aligned} \text{2. } \xi \end{aligned} \Big| \begin{aligned} \text{2. } \xi \end{aligned}$$

$$
\varphi\_{\xi\xi} - \mathcal{S}\varphi\_{\mathfrak{t}} + L e\left(f\varphi\_{\xi}\right) + L d\theta\_{\xi\xi} = 0,\tag{13}
$$

$$
\phi\_{\xi}\xi \, - \, \mathcal{S}\phi\_{\mathsf{T}} + \ln \left( f \phi\_{\xi} \right) + \frac{Nt}{Nb} \theta\_{\xi}\xi \, = 0,\tag{14}
$$

The boundary constraints in the non-dimensional form are

$$\begin{aligned} f\_{\xi} \left( 0, \tau \right) &= \sin \tau, \ f \left( 0, \tau \right) = 0, \\ \theta \left( 0, \tau \right) &= 1, \; \phi(0, \tau) = 1, \phi(0, \tau) = 1, \\ \left. f\_{\mathcal{Y}} \left( \infty, \tau \right) &\to 0, \theta \left( \infty, \tau \right) \to 0, \phi \left( \infty, \tau \right) \to 0, \phi \left( \infty, \tau \right) \to 0, \end{aligned} \tag{15}$$

where K = 1/µβ∗C and Ŵ = uw 2b/2νC 2 denote the material parameters, = σB 2 0 /ρf b + νϑ/k ′b is magneto-porosity constant, S = ω/b oscillating frequencyto-stretching rate ratio, Nt = (ρc)pDT (Tw − T∞) /(ρc)f T∞ν thermophoresis parameter, Pr = ν/αm is Prandtl number, Nb = (ρc)pDB (Cw − C∞) /(ρc)f ν Brownian motion constant, Nd = DTC (Cw − C∞) /αm (Tw − T∞) modified Dufour number, Ld = DCT (Tw − T∞) /αm (Cw − C∞) Dufour Lewis number, Le = ν/Ds regular Lewis number, whereas Ln = ν/DB nano-Lewis number.

We define the following relations associated with the definitions of wall shear stress, local Nusselt number, Sherwood

number, and nano-Sherwood number:

$$\begin{split} \mathcal{C}\_{f} &= \frac{\mathsf{r}\_{\text{w}}}{\rho u\_{\text{w}}^{2}}, \mathsf{N}u\_{\text{x}} = \frac{\mathsf{x}q\_{\text{s}}}{k \,(T\_{\text{w}} - T\_{\text{\infty}})}, \mathsf{Sh}\_{\text{x}} = \frac{\mathsf{x}j\_{\text{s}}}{D\_{\text{B}} \,(\mathsf{C}\_{\text{w}} - \mathsf{C}\_{\text{\infty}})},\\ \mathcal{S}\mathsf{h}\_{\text{xn}} &= \frac{\mathsf{x}q\_{\text{m}}}{D\_{\text{s}} \,(\varphi\_{\text{w}} - \varphi\_{\text{\infty}})}, \end{split} \tag{17}$$

where qs , js , and gs stand for surface heat flux, surface mass flux, and motile microorganisms flux, respectively. The dimensionless forms of the previously mentioned physical quantities are

$$\begin{array}{c} \text{Re}\_{\mathbf{x}}^{1/2}C\_{\mathbf{f}} = (1+K)f\_{\xi\xi} - \frac{K}{3}\theta\left(f\_{\xi\xi}\right)\_{\xi=0}, \\\quad \text{Nu}\_{\mathbf{x}}\,\text{Re}\_{\mathbf{x}}^{-1/2} = -\theta\_{\xi}\left(0,\tau\right), \\\quad \text{Sh}\_{\mathbf{x}}\,\text{Re}\_{\mathbf{x}}^{-1/2} = -\varphi\_{\xi}\left(0,\tau\right), \\\quad \text{Sh}\_{\mathbf{n}}\,\text{Re}\_{\mathbf{x}}^{-1/2} = -\phi\_{\xi}\left(0,\tau\right). \end{array} \tag{18}$$

TABLE 1 | Comparison of *f*ξ ξ (0, τ ) for τ with Abbas et al. [35] when *S* = 1, = 12, Ŵ = 0, and *K* = 0.

where Rex = uwx¯/ν is mentioned for local Reynolds number.

### SOLUTION METHODOLOGY

The structured set of non-linear partial differential equations (12–16) with boundary conditions (17–18) are simulated analytically via homotopy analysis technique. Due to efficient and convincing accuracy, various physical problems in recent years have been solved by following this procedure. The initial guesses for the present flow problem are

$$f\_0\left(\xi,\tau\right) = \sin\tau \left(1 - e^{-\xi}\right), \ \theta\_0\left(\xi\right) = e^{-\xi}, \ \varphi\_0\left(\xi\right) = e^{-\xi},$$

$$\phi\_0\left(\xi\right) = e^{-\xi},\tag{19}$$

Following auxiliary linear operators that are followed to precede the solution

$$\mathfrak{E}\_f = \frac{\partial^3}{\partial \xi^3} - \frac{\partial}{\partial \xi}, \mathfrak{E}\_\theta = \frac{\partial^2}{\partial \xi^2} - 1, \mathfrak{E}\_\varphi = \frac{\partial^2}{\partial \xi^2} - 1,$$

$$\mathfrak{E}\_\phi = \frac{\partial^2}{\partial \xi^2} - 1,\tag{20}$$

satisfying

$$\left[\mathfrak{E}\_f\left[a\_1 + a\_2e^{\xi} + a\_3e^{-\xi}\right] = 0,\tag{21}$$

FIGURE 2 | Impact of (A) *K* (B) *S* on skin friction coefficient.

$$
\varepsilon\_{\theta} \left[ a\_4 e^{\xi} + a\_5 e^{-\xi} \right] = 0,\tag{22}
$$

$$
\xi\_{\varphi} \left[ a\_6 e^{\xi} + a\_7 e^{-\xi} \right] = 0,\tag{23}
$$

$$
\varepsilon\_{\phi} \left[ a s e^{\xi} + a \wp e^{-\xi} \right] = 0,\tag{24}
$$

where a1, . . . a9 are arbitrary constants.

### CONVERGENT REGION

The convergence procedure in homotopic solution is regulated with auxiliary parameters hf , hθ , hϕ, and hφ. On this end, we prepare h−curves to report the convincible range of such parameters. It is obvious from **Figure 1** that the preferable range of these such parameters can be utilized from −2 ≤ hf ≤ 0, −1.2 ≤ hθ ≤ −0.2, −1.4 ≤ hϕ ≤ 0 and −1.2 ≤ hφ ≤ −0.2.

### VALIDATION OF RESULTS

**Table 1** shows the comparison of present results with Abbas et al. [35] as a limiting case. An excellent accuracy of results is noted with these reported investigations.

### DISCUSSION

This section aims to manifest the features of some interesting flow parameters that appeared in the dimensionless equations, where material parameter K, magneto-porosity constant , oscillating frequency-to-rate of stretching ratio S, Brownian motion Nb, thermophoresis constant Nt, variable thermal conductivity δ, Prandtl number Pr, Dufour Lewis number Ld, modified Dufour constant Nd, regular Lewis number Le, and nano-Lewis number Ln. During variation of each flow parameter, we fixed some numerical values to remaining parameters, like K=0.2, = 0.4, S = 0.2, Nt = 0.3, Pr = 0.7, Nb = 0.4, Nd, = 0.5, Ld = 0.3, Le = 0.2, and Ln = 0.3.

#### Skin Friction Coefficient

The impact of the skin friction coefficient against time τ for diverse variation of K and S is evaluated in **Figures 2A,B**. An interesting periodic oscillation in the wall shear stress is evaluated by both figures. Furthermore, the growing values of both parameters increase the amplitude of oscillation sufficiently. Due to no-slip conditions at the surface, the fluid particles accelerated together with surface in same amplitude and phase. However, the occurrence of a phase shift in both curves is almost negligible.

### Velocity Profile

The results reported in **Figures 3A,B** show the change in velocity fξ , verse time τ and leading values of material constant K, and magneto-porosity parameter . **Figure 3A** characterized the influence of K on fξ , which shows that an increment in K leads to higher velocity amplitude. The physical justification of such enhancing velocity distribution is attributed to the lower viscosity of fluid associated with the higher values of K. However, reverse observations are predicated for . In fact, magneto-porosity

constant is the combination of magnetic field and porous space. The existence of a magnetic force encountered the effects of Lorentz force, in which a declining oscillation behavior is noticed. Similarly, the permeability of a porous medium also retards the velocity amplitude due to the loss of fluid. Moreover, the utilization of magnetic force enhances the apparent viscosity of fluid up to a certain point of becoming an elastic solid, and subsequently, the fluid stress can be managed upon changing the magnetic force. These interesting observations can be used in various processes, like magnetohydrodynamic drive ion propulsion, magnetohydrodynamic drive power generators, electromagnetic material casting, etc.

#### Temperature Distribution

To visualize the alter profile of nanoparticle temperature θ due to δ, K, and Nd, **Figures 4A–C** are prepared. **Figure 4A** reveals that temperature distribution θ increases with variable thermal conductivity constant δ. **Figure 4B** is constituted to observe the change in θ due to material parameter K. A fall in θ is associated with leading variation of K. An increment in viscosity would yield for arising values of K that increases the fluid velocity but a decline in the temperature of fluid. The change in θ with effect of modified Dufour number Nd has been reported in **Figure 4C**. A slightly dominant variation in θ is seen with larger values of Nd.

### Solutal Concentration Profile

Now, we observe the variation in solutal concentration profile ϕ by varying regular Lewis number (Le), Dufour Lewis constant Ld , and material constant(K). **Figure 5A** is designed to observe the impact of Le on ϕ. A decreasing solutal concentration profile ϕ is notified due to Le. Physical explanation of such decling variation of ϕ can be justified on the fact that Le captures reverse relation with species diffusion, which means that when Le is maximum, species diffusion is lower, which leads to the decrement of the resulting solutal concentration. From **Figure 5B**, ϕ increases with the growth of Ld. Physically, Ld depends upon the Lewis number due to lower mass diffusivity. **Figure 5C** presents change in ϕ due to material constant K. Again, an enhanced distribution of solutal concentration profile ϕ has resulted for maximum values of K.

### Nanoparticle Concentration

The physical consequences of Ln, Nt, and Nb on concentration distribution φ are deliberated in **Figures 6A–C**. **Figure 6A** specified the input of Ln on φ. A declining concentration distribution φ is examined in the peak values of Ln. This decreasing behavior of φ is attributed to the fact that Ln is associated with the Brownian diffusion coefficient because Ln expresses the thermal diffusivity-to-mass diffusivity ratio. This parameter that is referred to the fluid flow in a

phenomenon of heat and mass transfer occurs due to convection. The consequences for another important parameter, namely thermophoresis constant Nt, are analyzed in **Figure 6B**. As expected, a larger concentration distribution φ is reported due to involvement of Nt. The larger variation of Nt helps to improve the thermal conductivity of fluid. Physically, the thermophoretic process is based on collective migrated heat particles in the region of low temperature and plays a momentous role in many physical

phenomenons. The curve of φ attained maximum level due to Nt. However, a reduced concentration distribution φ is associated with Nb, as shown in **Figure 6C**. The Brownian movement is based on random pattern moving fluid particles in flow surface. It is further justified from Equation (16), which clearly shows that reverse relation is developed between Nb and φ. In fact, the specified numerical values of Nb are associated to the more prominent nanoparticle moments that are being pushed back from accelerated plate to quiescent, which resulted in a retarded concentration distribution.

#### Physical Quantities

To perform the numerical simulations for local Nusselt number −θξ (0, τ ), local Sherwood Number −ϕξ (0, τ ) and nanofluid Sherwood number −φξ (0, τ ), **Table 2** is designed. It is observed that when , ε, and K assigned larger numerical values, a decreasing trend in −θξ (0, τ ), −ϕξ (0, τ ), and −φξ (0, τ ) is reported. However, these physical quantities increase with the variation of Pr.

### CONCLUSIONS

We have focused on periodically accelerated unsteady flow of Eyring–Powell nanofluid with utilization of thermal diffusive features. The variable impact of thermal conductivity, porous medium, and magnetic field consequences are also utilized. The important observations from current flow problem are summarized as:

➢ The magneto-porosity parameter declined the periodic oscillation in the velocity, and subsequently, the magnitude of velocity declined.

### REFERENCES

The observation based on the reported results can be used to improve thermal extrusion processes, heat exchangers, solar technology, energy production, cooling processes, etc.

### DATA AVAILABILITY STATEMENT

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

### ACKNOWLEDGMENTS

The authors extend their appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under project number: RGP-2019-1.

presence of radiation heat source, Int J Heat Mass Transfer. (2019) **139**:87–94. doi: 10.1016/j.ijheatmasstransfer.2019.05.006

**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Khan, Vaidya, Chammam, Musmar, Prasad and Tlili. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Entropy Optimization of Third-Grade Nanofluid Slip Flow Embedded in a Porous Sheet With Zero Mass Flux and a Non-Fourier Heat Flux Model

K. Loganathan1 \*, G. Muhiuddin2 \*, A. M. Alanazi 2 , Fehaid S. Alshammari 3 , Bader M. Alqurashi 4 and S. Rajan1

*1 Department of Mathematics, Erode Arts & Science College, Erode, India, 2 Department of Mathematics, University of Tabuk, Tabuk, Saudi Arabia, 3 Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia, 4 Department of Mathematics, Faculty of Arts and Science in Almandaq, Albaha University, Albaha, Saudi Arabia*

#### Edited by:

*Ahmed Zeeshan, International Islamic University, Islamabad, Pakistan*

#### Reviewed by:

*Ali Chamkha, Prince Mohammad bin Fahd University, Saudi Arabia Precious Sibanda, University of KwaZulu-Natal, South Africa*

#### \*Correspondence:

*G. Muhiuddin<EMAIL_ADDRESS>K. Loganathan loganathankaruppusamy304@ gmail.com*

#### Specialty section:

*This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics*

> Received: *21 April 2020* Accepted: *08 June 2020* Published: *24 November 2020*

#### Citation:

*Loganathan K, Muhiuddin G, Alanazi AM, Alshammari FS, Alqurashi BM and Rajan S (2020) Entropy Optimization of Third-Grade Nanofluid Slip Flow Embedded in a Porous Sheet With Zero Mass Flux and a Non-Fourier Heat Flux Model. Front. Phys. 8:250. doi: 10.3389/fphy.2020.00250*

The prime objective of this article is to explore the entropy analysis of third-order nanofluid fluid slip flow caused by a stretchable sheet implanted in a porous plate along with thermal radiation, convective surface boundary, non-Fourier heat flux applications, and nanoparticle concentration on zero mass flux conditions. The governing physical systems are modified into non-linear ordinary systems with the aid of similarity variables, and the outcomes are solved by a homotopy analysis scheme. The impression of certain governing flow parameters on the nanoparticle concentration, temperature, and velocity is illustrated through graphs, while the alteration of many valuable engineering parameters viz. the Nusselt number and Sherwood number are depicted in graphs. Entropy generation with various parameters is obtained and discussed in detail. The estimation of entropy generation using the Bejan number find robust application in power engineering and aeronautical propulsion to forecast the smartness of entire system.

Keywords: entropy generation, Christov-Cattaneo heat flux, third-grade nanofluid, porous medium, homotopy analytic technique

### INTRODUCTION

Nanoliquids are the type of liquids that have small volumetric quantities of nanoscale 10−9 − 10−7m metallic Cu, Ti, Hg, Fe, Ag, Au, etc. or non-metallic particles TiO2, SiO2, CuO, Al2O3, etc taken as nano particles. Usually nanoliquids have a colloidal suspension of nanoparticles inside a base liquid for example water, oil, ethylene glycol, etc. Initially, Choi [1] proposed the "nanofluid" term. In general, the effective heat transfer enhancement has the reason of nanoliquids generally restrict up to volume fraction of nanoparticles. Therefore, in the latest technologies and engineering areas, nanoliquids receiving a phenomenal impact. Mushtaq et al. [2] attempted the numerical study of the nanoliquids induced by an exponentially stretchable sheet with rotating flow model. MHD nanoliquid flow toward a porous plate with internal heat generation effects was presented by Reddy and Chamka [3].  Recently, Gireesha et al. [5] explained a Hall current effects of two-phase transient nanoliquid flow induced by a stretchable sheet. Reddy et al. [6] performed the combined convection flow of nanoliquid toward a semi-infinite vertical flat sheet with convectively heated boundary and Soret effect. Few more significant studies in this research area are seen in ref's [7–13].

Due to various applications in different technical and industrial areas, the fluid flow problems toward a stretching surface have developed. It finds application in rubber and plastic sheets production, melt-spinning, production of glass-fiber, and metallic plate cooling systems. Sakiadis [14] studied the uniform velocity of a magnetohydrodynamic flow past a solid medium. Magyari et al. [15] examined the first order chemical reaction and heat generation combination on micropolar fluid flow induced by a permeable stretching surface. Gupta [16] analyzed the heat and mass transfer effects of a boundary layer flow induced by a stretchy sheet with suction or blowing impact. Magyari and Keller [17] exposed the fluid flow over exponentially extending sheet with heat and mass transfer impacts. Significances of thermal boundary layer flow for a linearly stretchy sheet with viscous dissipation were examined by Cortell [18]. The cutting-edge research reports on stretching sheet flow are highlighted in these works of literature [19–24].

Fluid flow saturated in the porous surface have numerous applications in various fields like geothermal energy, fuel cell technologies, material processing, etc. Chamkha et al. [25] described the free convective flow past an inclined plate fitted in a porous medium of variable porosity with solar radiation. Khan and Aziz [26] studied the natural convective flow with double diffusion caused by a vertical porous sheet. Oyelakin et al. [27] analyzed the slip flow of unsteady radiative Casson nanofluid toward a stretchy surface. Gorla and Chamkha [28] studied the nanofluid flow toward a non-isothermal vertical plat entrenched in a porous sheet. The same research group extended their work for different models for various applications [29–32].

Heat transport problems in the flow of liquids have been examined by several researchers for the last decade. In 1822, Fourier [33] constructed the heat conduction law. This states that "the heat transfer in a medium with inertial rate." A parabolictype equation was used to state the heat conduction equation. The problem rising at this time is that there exists no such object or material that satisfies Fourier's law, as argued by Cattaneo [34] when using thermal relaxation time to customize Fourier's law. Later, Christov [35] developed and joined the upper convected Maxwell fluid. This developed model is called a Christov-Cattaneo heat flux model. Loganathan et al. [36] studied the thermal relaxation time effects on Oldroyd-B liquid with second-order slip and cross-diffusion impacts. The Christov-Cattaneo heat flux model for a third-grade liquid with chemical reaction effects was examined by Imtiaz et al. [37].

In the last decade, several scientists have researched entropy generation in the flow of fluids and heat transfer over a stretching surface. In various engineering and industrial divisions, the performance of heating and cooling are of massive importance in different electronic and energy issues. Aiboud and Saouli [38] examined the MHD viscoelastic fluid flow with the application of entropy analysis using Kummer's function. Makinde [39] presented the thermal radiation and Newtonian heating impacts of variable viscosity fluid caused by a semi-infinite plate using shooting quadrature and obtained the entropy generation number. Loganathan et al. [40, 41] verified the entropy analysis for the third grade and Williamson nanoliquid flow caused by a stretchable sheet with various effects. They employed HAM to solve the non-linear governing systems.

Based on our research in previously published works, entropy generation of third-grade nanofluid flow caused by a stretching sheet with a modified Fourier law has not been discussed with a high standard of scientific attention. As far as we noticed in literature, the studies taken for entropy generation are limited with some parameters viz the Brinkmann number, Reynolds number, Temperature difference parameter, and Hartmann number. We have extended our investigations to include thermal relaxation time, the Biot number, thermal radiation, slip parameter, porous parameter, etc. The effective collection and analysis of these results will open new gateways for diverse engineering application in various streams.

#### PROBLEM DEVELOPMENT

We were interested in analyzing the entropy and Bejan number of third-grade nanofluid saturated in a porous medium with Christov-Cattaneo heat flux. In **Figure 1**, the stretching parameter is taken along x direction where TW and CW are represented the wall temperature and concentration, respectively. T∞ and C∞ are used to index the ambient temperature and concentration, respectively. A convective heating temperature Tf is stimulated at the bottom of the sheet surface. The Buongiorno nanofluid [42] is used for the present case.

Based on his consideration, the slip mechanisms, namely, the Brownian diffusion, inertia, Magnus effect, diffusiophoresis, thermophoresis, gravity, and fluid drainage, were analyzed. He recommended that the thermophoresis and Brownian diffusion are essential slip mechanism in low dimensional materials.

The following is an incompressible fluid model containing body forces with the equation of continuity and motion:

$$d\dot{\nu}\,\nu = 0\tag{1}$$

$$
\rho \frac{d\nu}{dt} = di\nu T + \rho b + J + B \tag{2}
$$

Here, ρ is the fluid density, which is taken as a constant, v is the velocity field, b indicates the body forces, J denotes the electric current, and T states the third-grade incompressible fluids Cauchy stress tensor [43]

$$T = -pI + \mu E\_1 + A\_1^\* E\_2 + A\_2^\* E\_1^2 + \beta\_1 E\_3$$

$$+ \beta\_2 \left(E\_1, E\_2 + E\_2 E\_1\right) + \beta\_3 \left(tr E\_1^2\right) E\_1 \tag{3}$$

where µ,(E1, E2, E3) and A ∗ 1 , βi indicate the viscosity coefficient, kinematics tensors, and material modulis as in

$$E\_1 = L + \left(L\right)^T \tag{4}$$

$$E\_n = \frac{d}{dt}E\_{n-1} + E\_{n-1}L + (L)^T E\_{n-1}, \qquad n = 2, 3, \text{and} \tag{5}$$

$$L = \nabla \mathbf{v}.\tag{6}$$

d dt is expressed as the material time derivative

$$\frac{d\langle\rangle}{dt} = \frac{\partial\langle\rangle}{\partial t} + \nu.\nabla\langle\rangle. \tag{7}$$

The relationship between Clausius-Duhem inequality and thermodynamically compatible fluid is stated by Fosdick and Rajagopal [44]:

$$
\mu \ge 0, \ A\_1^\* \ge 0, \ \beta\_1 = \beta\_2 = 0, \ \beta\_3 \ge 0 \tag{8}
$$

$$\left|A\_1^\* + A\_2^\*\right| \le 2\sqrt{6\mu\beta\_3} \tag{9}$$

$$T = -pI + \mu E\_1 + A\_1^\* E\_2 + A\_2^\* E\_1^2 + \beta\_3 \left( trE\_1^2 \right) E\_1 \tag{10}$$

Boussinesq and normal boundary layer approximations were considered by Pakdemirli [45]. We made the following assumptions:

$$\frac{\partial \mu}{\partial \mathbf{x}} + \frac{\partial \nu}{\partial y} = \mathbf{0} \tag{11}$$

$$\begin{split} \frac{\partial u}{\partial x} &+ \nu \frac{\partial u}{\partial \boldsymbol{\nu}} \\ &= \nu \frac{\partial^2 u}{\partial \boldsymbol{\nu}^2} + \frac{A\_1^\*}{\rho} \left( u \frac{\partial^3 u}{\partial \boldsymbol{\nu}^2 \partial \boldsymbol{x}} + \nu \frac{\partial^3 u}{\partial \boldsymbol{\nu}^3} + \frac{\partial u}{\partial \boldsymbol{x}} \frac{\partial^2 u}{\partial \boldsymbol{\nu}^2} + 3 \frac{\partial u}{\partial \boldsymbol{\nu}} \frac{\partial^2 u}{\partial \boldsymbol{x} \partial \boldsymbol{\nu}} \right) \\ &+ 2 \frac{A\_2^\*}{\rho} \frac{\partial u}{\partial \boldsymbol{\nu}} \frac{\partial^2 u}{\partial \boldsymbol{x} \partial \boldsymbol{\nu}} + 6 \frac{\theta\_1^\*}{\rho} \left( \frac{\partial u}{\partial \boldsymbol{\nu}} \right)^2 \frac{\partial^2 u}{\partial \boldsymbol{\nu}^2} - \frac{\sigma B\_0^2}{\rho} u - \frac{\nu}{k\_p} u \end{split} \tag{12}$$

u

$$\begin{aligned} \mu \frac{\partial T}{\partial \mathbf{x}} + \nu \frac{\partial T}{\partial \mathbf{y}} &= \frac{k}{\rho c\_p} \frac{\partial^2 T}{\partial \mathbf{y}^2} - \frac{1}{\rho c\_p} \frac{\partial q\_r}{\partial \mathbf{y}} + \frac{Q\_0}{\rho c\_p} \left(T - T\_\infty\right) \\ &+ \pi \left[D\_B \frac{\partial C}{\partial \mathbf{y}} \frac{\partial T}{\partial \mathbf{y}} + \frac{D\_T}{T\_\infty} \left(\frac{\partial T}{\partial \mathbf{y}}\right)^2\right] \end{aligned} \tag{13}$$

$$\begin{aligned} \mu \frac{\partial C}{\partial x} + \nu \frac{\partial C}{\partial \mathcal{Y}} &= D\_B \frac{\partial^2 C}{\partial \mathcal{Y}^2} + \frac{D\_T}{T\_{\infty}} \frac{\partial^2 T}{\partial \mathcal{Y}^2} \\ \mu = u\_w \left( \mathbf{x} \right) &= a\mathbf{x} + L \frac{\partial u}{\partial \mathcal{Y}}, \quad -k \frac{\partial T}{\partial \mathcal{Y}} = h\_f \left( T\_f - T\_{\infty} \right), \\ D\_B \frac{\partial C}{\partial \mathcal{Y}} + \frac{D\_T}{T\_{\infty}} \frac{\partial T}{\partial \mathcal{Y}} &= 0 \text{ at } \mathcal{y} = 0 \\ \mu \to 0, \ T \to T\_{\infty}, \ c \to c\_{\infty} \text{ as } \mathcal{y} \to \infty \end{aligned} \tag{15}$$

The energy equation with a Cattaneo-Christov heat flux model is stated as

$$\begin{split} \mu \frac{\partial T}{\partial \boldsymbol{x}} + \nu \frac{\partial T}{\partial \boldsymbol{\mathcal{Y}}} &+ \lambda \boldsymbol{\tau} \left( \mu^{2} \frac{\partial^{2} T}{\partial \boldsymbol{x}^{2}} + \nu^{2} \frac{\partial^{2} T}{\partial \boldsymbol{y}^{2}} + \left( \mu \frac{\partial \mu}{\partial \boldsymbol{x}} \frac{\partial T}{\partial \boldsymbol{x}} + \nu \frac{\partial \mu}{\partial \boldsymbol{y}} \frac{\partial T}{\partial \boldsymbol{x}} \right) \right) \\ &+ 2\mu \nu \frac{\partial T^{2}}{\partial \boldsymbol{x} \partial \boldsymbol{y}} \bigg) + \left( \mu \frac{\partial \nu}{\partial \boldsymbol{x}} \frac{\partial T}{\partial \boldsymbol{y}} + \nu \frac{\partial \nu}{\partial \boldsymbol{y}} \frac{\partial T}{\partial \boldsymbol{y}} \right) = \frac{k}{\rho c\_{\rho}} \frac{\partial^{2} T}{\partial \boldsymbol{y}^{2}} \\ &- \frac{1}{\rho c\_{p}} \frac{\partial q\_{r}}{\partial \boldsymbol{y}} + \frac{Q\_{0}}{\rho c\_{p}} \left( T - T\_{\infty} \right) \\ &+ \pi \left[ D\_{B} \frac{\partial C}{\partial \boldsymbol{y}} \frac{\partial T}{\partial \boldsymbol{y}} + \frac{D\_{T}}{T\_{\infty}} \left( \frac{\partial T}{\partial \boldsymbol{y}} \right)^{2} \right]. \tag{16} \end{split}$$

Consider the transformation given below:

$$\psi = \sqrt{a\nu} \text{xf}\left(\eta\right), \ u = \frac{\partial \psi}{\partial \boldsymbol{\eta}}, \ \boldsymbol{\nu} = \ -\frac{\partial \boldsymbol{\psi}}{\partial \boldsymbol{x}}, \ \boldsymbol{\eta} = \sqrt{\frac{a}{\nu}} \boldsymbol{\gamma},$$

$$\boldsymbol{\nu} = -\sqrt{a\boldsymbol{\nu}} \boldsymbol{f}\left(\boldsymbol{\eta}\right), \ \boldsymbol{u} = a \boldsymbol{\text{xf}}'\left(\boldsymbol{\eta}\right), \ \boldsymbol{\theta}\left(\boldsymbol{\eta}\right) = \frac{T - T\_{\infty}}{T\_{f} - T\_{\infty}},$$

$$\boldsymbol{\phi}\left(\boldsymbol{\eta}\right) = \frac{C - C\_{\infty}}{C\_{\infty}}.\tag{17}$$

The non-linear governing equations are:

$$\left(f'' + f f'' - f'^2 + \alpha\_1 \left(2f'f'' - ff^{i\nu}\right) + \left(3\alpha\_1 + 2\alpha\_2\right)f'^2\right.$$

$$\left. + 6\beta \text{Ref}^{\prime\prime} f^{\prime\prime2} - \left(Mf' + Kf'\right) = 0\tag{18}$$

$$\left(1+\frac{4}{3}Rd\right)\theta''+Pr\theta'+Pr\Theta-Pr\gamma f^2\theta''-Pr\gamma f f'\theta'$$

$$+Pr\text{N}b\,\,\theta'\phi'+Pr\text{N}t\theta'^2=0\tag{19}$$

Frontiers in Physics | www.frontiersin.org

$$
\phi'' + Lef\phi' + \frac{Nt}{Nb}\theta'' = 0\tag{20}
$$

with the end points

$$\begin{aligned} f\ (0) &= f\_{\mathcal{W}} f'\ (0) = 1 + \Gamma \overline{f}'\ (0), \boldsymbol{\theta}'\ (0) = -Bi(1 - \boldsymbol{\theta}\ (0)),\\ \mathcal{N}b\ \boldsymbol{\phi}'\ (0) + \mathcal{N}t\ \boldsymbol{\theta}'\ (0) &= 1\\ f'\ (\infty) &= 0, \ \boldsymbol{\theta}\ (\infty) = 0, \boldsymbol{\phi}\ (\infty) = 0\end{aligned} \tag{21}$$

The non-dimensional variables are

$$\alpha\_1 = \frac{aA\_1^\*}{\upsilon}, \alpha\_2 = \frac{aA\_2^\*}{\upsilon}, \beta = \frac{a\beta\_1^\*}{\upsilon}, Re = \frac{u\_\upsilon \chi}{\upsilon}, Pr = \rho C\_p/k,$$

$$M = \sigma B\_0^2/\rho a, Rd = \left(4\sigma^\* T\_\infty^3\right)/\left(kk^\*\right), S = \frac{Q\_0}{\rho c\_p}, \gamma = \lambda\_T a,$$

$$Bi = \frac{h\_f}{k}\sqrt{\upsilon\_a^\prime}, Nb = \frac{\tau D\_B}{\upsilon}\left(C\_\infty\right), Nt = \frac{\tau D\_T}{\upsilon}\left(T\_f - T\_\infty\right).$$

The application of physical entitles is such that

$$\operatorname{Re}^{\frac{1}{2}}C\_f = f''(0) + \alpha\_1 f'(0)f''''(0) + \beta \operatorname{Re}[f''(0)]^{\frac{3}{2}} \tag{22}$$

$$Re^{-\frac{1}{2}}Nu\_{\mathbf{x}} = -(1 + \frac{4}{3}Rd)\theta'(0) \,. \tag{23}$$

The local mass transfer rate becomes identically zero due to the zero mass flux state [46]

$$Re^{-\frac{1}{2}}Sh = \frac{Nt}{Nb}\theta'(0) \,. \tag{24}$$

#### ENTROPY OPTIMIZATION

The entropy minimization optimization for fluid friction, heat, and the irreversibility of mass transfer are given below:

$$\begin{split} S\_{\text{gen}}^{'''} &= \frac{K\_1}{T\_{\infty}^2} \left[ \left( \frac{\partial T}{\partial \boldsymbol{x}} \right)^2 + \left( \frac{\partial T}{\partial \boldsymbol{y}} \right)^2 + \frac{16 \boldsymbol{\sigma}^\* T\_{\infty}^3}{3kk^\*} \left( \frac{\partial T}{\partial \boldsymbol{y}} \right)^2 \right] \\ &+ \frac{\mu}{T\_{\infty}} \Bigg[ 2 \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{x}} \right)^2 + \left( \frac{\partial \boldsymbol{\nu}}{\partial \boldsymbol{\eta}} \right)^2 \Bigg] + \left[ \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\eta}} + \frac{\partial \boldsymbol{\nu}}{\partial \boldsymbol{x}} \right]^2 \\ &+ \frac{RD}{C\_{\infty}} \Bigg[ \left( \frac{\partial \boldsymbol{C}}{\partial \boldsymbol{x}} \right)^2 + \left( \frac{\partial \boldsymbol{C}}{\partial \boldsymbol{\eta}} \right)^2 \Bigg] + \frac{RD}{T\_{\infty}} \Bigg[ \left( \frac{\partial T}{\partial \boldsymbol{x}} \right) \left( \frac{\partial \boldsymbol{C}}{\partial \boldsymbol{x}} \right) \\ &+ \left( \frac{\partial T}{\partial \boldsymbol{\eta}} \right) \left( \frac{\partial \boldsymbol{C}}{\partial \boldsymbol{\eta}} \right) \Bigg] + \frac{\sigma B\_0^2}{T\_{\infty}} u^2 + \frac{\nu}{k\_p} u^2. \tag{25} \end{split} \tag{25}$$

Using Equation (25) modified with the help of Equation (17),

$$\begin{split} S\_{\text{gen}}^{\prime\prime} &= \frac{K\_1}{T\_{\infty}^2} \left[ \left( \frac{\partial T}{\partial \boldsymbol{\chi}} \right)^2 + \frac{16 \boldsymbol{\sigma}^\* T\_{\infty}^3}{3 k k^\*} \left( \frac{\partial T}{\partial \boldsymbol{\chi}} \right)^2 \right] + \frac{\mu}{T\_{\infty}} \left( \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{\chi}} \right)^2 \\ &+ \frac{RD}{C\_{\infty}} \left( \frac{\partial C}{\partial \boldsymbol{\chi}} \right)^2 + \frac{RD}{T\_{\infty}} \left( \frac{\partial T}{\partial \boldsymbol{\chi}} \right) \left( \frac{\partial C}{\partial \boldsymbol{\chi}} \right) + \frac{\sigma B\_0^2}{T\_{\infty}} \boldsymbol{u}^2 + \frac{\nu}{k\_{\text{p}}} \boldsymbol{u}^2. \end{split} \tag{26}$$

Dimensionless system of entropy generation is defined as:

$$E\_G = \operatorname{Re}\left(1 + \frac{4}{3}Rd\right)\left{\boldsymbol{\theta}^{\prime 2}} + \operatorname{Re}\frac{\operatorname{Br}}{\Omega}{\Omega}\boldsymbol{\xi}^{\prime 2} + \operatorname{Re}\left(\frac{\boldsymbol{\xi}}{\Omega}\right)^2\lambda\boldsymbol{\phi}^{\prime 2}$$

$$+\operatorname{Re}\frac{\zeta}{\Omega}\lambda\phi^{\prime}\theta^{\prime}+\frac{Br}{\Omega}(\!M+K)\text{f}^{\prime}.\tag{27}$$

The Bejan number states

Be = Entropy genration due to irrevesablity of heat and mass transfer Total entropy generated

Be =

$$\frac{\left(\text{Re}\left(1+\frac{4}{3}\text{Rd}\right)\boldsymbol{\theta}^{\text{2}}+\text{Re}\left(\frac{\zeta}{\Omega}\right)^{2}\lambda\boldsymbol{\phi}'^{2}+\text{Re}\frac{\zeta}{\Omega}\lambda\boldsymbol{\phi}'\boldsymbol{\theta}'\right)}{\text{Re}\left(1+\frac{4}{3}\text{Rd}\right)\boldsymbol{\theta}'^{\text{2}}+\text{Re}\frac{\text{Re}}{\Omega^{2}}\boldsymbol{f}'^{'1}+\text{Re}\left(\frac{\zeta}{\Omega}\right)^{2}\lambda\boldsymbol{\phi}'^{2}+\text{Re}\frac{\zeta}{\Omega}\lambda\boldsymbol{\phi}'\boldsymbol{\theta}'+\frac{\text{Re}}{\Omega^{2}}(M+K)\boldsymbol{f}'^{2}}.\tag{28}$$

#### HOMOTOPY SOLUTIONS

There are several techniques available to solve non-linear problems. The homotopy analysis method (HAM) is initially constructed by Liao [47]. Moreover, he altered a non-zero auxiliary parameter [48]. This parameter shows the way to calculate the convergence rate. It also offers great independence with which to make the initial guesses of the solutions.

 \Gamma \right|\_m^{\prime\prime} \left( 0 \right) = 0, \left. \theta\_m^{\prime\prime} \left( 0 \right) - Bi \theta\_m \left( 0 \right) = 0, \\ \phi\_m^{\prime\prime}(0) + \frac{Nt}{Nb} \theta\_m^{\prime\prime}(0) &= 0 \\ \left. f\_m^{\prime\prime}(\eta) \right| \to 0, \left. \theta\_m^{\prime\prime} \left( \eta \right) \to 0, \left. \phi\_m^{\prime\prime} \left( \eta \right) \to 0 \text{as } \eta \to \infty. \end{aligned}$$

with boundary conditions

$$\begin{aligned} f\_m' \left( 0 \right) &= f\_m \left( 0 \right) = f\_m' \left( \infty \right) = \theta\_m \left( 0 \right) = \theta\_m \left( \infty \right) = \phi\_m \left( 0 \right) \\ &= \phi\_m \left( \infty \right) = 0. \end{aligned}$$

The appropriate solutions - f ∗ m, θ ∗ m, φ ∗ m are

$$\begin{aligned} f\_m(\eta) &= f\_m^\*(\eta) + E\_1 + E\_2 e^{\eta} + E\_3 e^{-\eta}, \\ \eta\_m(\eta) &= \eta\_m^\*(\eta) + E\_4 e^{\eta} + E\_5 e^{-\eta}, \\ \eta\_m(\eta) &= \eta\_m^\*(\eta) + E\_6 e^{\eta} + E\_7 e^{-\eta}. \end{aligned}$$

#### CONVERGENCE ANALYSIS

The auxiliary parameters hf , hθ , and hφ act as a vital part of convergence series solutions. The h-charts of f ′′ (0), θ ′ (0), and φ ′ (0) for Re,γ, and Nb are shown in **Figure 2**. From these curves, the straight line is referred as the h-curve. The convergent approximation is selected from this straight line of the curves. We note that h-curve of f ′′ (0), θ ′ (0), and φ ′ (0) shrinks as we enhance the range of Re,γ, and Nb, which shows the larger order approximation will be needed if the larger value of Re,γ, and Nb is employed. Approximations values of HAM with CPU time is denoted in **Table 1**.

### COMPUTATIONAL RESULTS AND DISCUSSION

The numerical calculations of velocity, concentration, temperature, entropy generation, and the Bejan number are discussed in this section. The homotopy technique is used for solving the non-linear governing Equations (18)–(20) with boundary conditions (21). The graphical results of entropy, Bejan number, temperature, Nusselt number, nanoparticle concentration, Sherwood number, and velocity profiles are computed via different flow parameters included in this study with the fixed values of α1 = α2 = β = Bi = fw = Nt = 0.2, Re = Pr = Le = Ŵ = 1.0, M = γ = 0.5, Rd = 0.3, Hg = −0.1, Nb = 0.4, Br = 5.0, and λ = = ζ = 1.0.

#### Impact on Velocity

It is observed from **Figure 3A** that the velocity is reduced to increase the values of the velocity slip parameter due to the ratio of stretching velocity and the viscosity of the fluid. The effect of the slip parameter in velocity has had more impact in the absence of a porous medium. **Figure 3B** indicates that the velocity diminishes when the magnetic field (M) raises. Improved Lorentz force is observed because of the increasing the values of M that opposes the fluid motion. Thus, we conclude that the velocity profile diminishes.

### Impact on Temperature

The impact of the radiation parameter Rd on the temperature profile is examined in **Figure 4A**. The temperature in the radiation parameter is high. Comparing the radiation effects with Christov-Cattaneo and normal heat flux, we observed that the radiation effect is quite low for Christov-Cattaneo heat flux. **Figure 4B** displays the influence of the Biot number on the temperature profile. From this figure, we observed that temperature is a rising function of Bi close to the sheet. Since Bi affects more temperature near the surface. Heat transfer

resistance is higher within the body compared to the surface of the sheet for rising values Bi.

### Impact on Nanoparticle Volume Concentration

In **Figure 5A**, the effect of concentration profile with growing values of the Brownian motion parameter Nb is depicted. In suction cases, higher values of Nb increase the concentration profile close to that of the surface of the sheet, and, suddenly, the concentration begins to fall, stabilizing far away from the surface of the sheet. This is due to the appearance of passive surface conditions for the concentration profile. Moreover, the injection at the sheet shows the concentration is rising near the sheet, and it diminishes far away from the surface of the sheet. The influence of the thermophoresis parameter Nt in the concentration profile φ (η). is highlighted in **Figure 5B**. When

there is suction in the sheet, with a rise in Nt, the concentration of the liquid decreases near to thsheet at certain stage starts to fall and stabilize away from the endpoints of the surface of the sheet. When there is an injection at the sheet, higher values of Nt decreases the concentration profile near the sheet, and it increases far away from the surface of the sheet.

#### Impact on Entropy Generation

The Effects of distinct fluid parameters on the entropy generation profile are highlighted in **Figures 6A–G**. **Figure 6A** depicts the influence of porous parameter in entropy profile. Initially, the entropy rate increases for the porous parameter at a certain stage (η = 0.6) before it becomes to fall. The responses of the slip parameter on entropy generation were succinctly depicted in **Figure 6B**. From this figure, it was obviously noted that entropy generation was inversely proportional to slip parameter. This causes a decrease in large values of slip parameter and temperature gradients in the boundary layer when retaining the fluid friction as we proceed. This occurrence induces a suppression in entropy generation since heat transfer was committed. **Figures 6C–E** displayed the impact of the suction\ injection parameter (fw.), material parameter (β) and Biot number (Bi) on the entropy profile (EG). Our examination obtained that higher range of fw, reduces the entropy generation profile, and the entropy generation rate is enhanced for higher β and Bi.

The effect of thermal relaxation time (γ ) on the entropy genation profile is sketched in **Figure 6F**. It is obvious that thermal relaxation time is small for temperature and heat transfer rates. In addition, domination of the irreversibility in heat transfer affected the heat flux. Thus, we have seen a small increase in the entropy of the system. Performance of Nt and Nb on entropy generation profile (EG) is shown in **Figures 6G,H**, which shows that entropy is increased with an increase of Nt, whereas EG is inversely proportional to Nb. The Brownian motion induces the nanoparticles temperature, but it reduces the temperature gradient on wall. As a result, entropy generation parameter reduces where Nt is directly proportional to temperature gradient and creates ambient atmosphere for higher values of EG.

### Impact on the Bejan Number

The Bejan number (Be) is a dimensionless quantity that specifies the ratio of entropy generation between heat transfer and the total entropy generation where Bejan numbers take values from 0 to 1. If Be is nearly equal to 1, the entropy generation will become more due to heat transfer. It is clear from **Figure 7A** that, with an escalation in the applied magnetic field, there is an augmentation in the Bejan number. The consequence of heat transfer entropy develops as we move up from the surface. The entropy effect has full domination because of the heat transfer while it is also outlying from the region. This is the reason behind how the augmenting value of M brings a stronger frictional effect, which leads to an increase in the liquid temperature. There is also a consequent development in the Be, as shown in **Figure 7A**. From **Figure 7B**, it is observed that the Be is increased for the increase in the slip parameter (Ŵ). Physically, larger values of (Ŵ) enhance the temperature gradient inside the regime, which induces the Bejan number and irreversibility of heat transfer.

**Figure 7C** states that, with the rise of the radiation constant Rd, the Bejan number is boosted. This is due to the total entropy generation dominated by thermal irreversibility. **Figure 7D** shows the effect of the thermal relaxation time (γ ) on the Be. At first, the Be is augmented for higher values of γ at a sudden point (η = 2.2). Consequently, the Bejan number profile reduces for the values of γ . **Figure 7E** shows that the Bi displays a trend of raising the Be. The demonstration of such an increasing trend of the Bejan number explains how the entropy production near the surface is large due to the liquid friction—at least relative to that of the heat transfer irreversibility. The variation of the Brinkmann number Br is sketched in **Figure 7F**. This figure shows that the Bejan number is reduced, as we have to enhance the Br. **Figures 7G,H** shows the influence of thermophoresis (Nt) and Brownian motion (Nb) parameters on the Bejan number. From these plots, we note that Ntand Nb have inverse effects on the Bejan number profile.

### Impact on Physical Entities

**Figures 8**–**11** illustrate the effects of different physical parameters on the local Nusselt number and local Sherwood number. The influence of fw and γ on Nux is shown in **Figure 8**: heat transfer decays for higher values of fw and γ . The same phenomena can be observed for larger values Hg and γ on the Nusselt number profile, as presented in **Figure 9**. The combined effects of Nb and M as well as Nt and M are shown in **Figures 10**, **11**, respectively. From these figures, we conclude that thermophoresis (Nt) and Brownian motion (Nb) parameters have produce the converse trend in the mass transfer rate.

### NUMERICAL CODE VALIDATION

In this segment, we examine the code validation of early published works of literature. **Table 2** validates the results of

TABLE 2 | Validation of −*f* ′′ (0) and −θ ′ (0) for the limiting case *M* = 0, *K* = 0, *Nb* = 0, *Nt* = 0, *Rd* = 0, *S* = 0, and *Bi* → ∞.

−f ′′(0) and −θ ′ (0) for the limiting case M = 0, K = 0, Nb = 0, Nt = 0, Rd = 0, S = 0, and Bi → ∞ with Imtiaz et al. [37]. Moreover, the skin friction rate is also validated by the same literature [37] when M = K = 0 (see **Table 3**). **Table 4** exhibits the matching results of reduced Nusselt number with the references [20, 49–51]. From the above validation, results show that the current simulation is considered an efficient one.

### KEY RESULTS

The present research work examines the entropy generation influence on third-grade nanoliquid flow caused by a stretching sheet in the appearance of Magnetic field, radiation, and convective heating effects. Christov-Cattaneo heat flux replaces

TABLE 3 | Comparison of *Re*0.5*Cf* for different values when *M* = *K* = 0.

TABLE 4 | Matching results of reduced Nusselt number with the restricting case *Rd* = *Ec* = *M* = *Nt* = *Nb* = γ = *S* = *K* = Ŵ = *fw* = 0, and *Bi* → ∞.

ordinary heat flux. HAM is employed to validate the nonlinear governing equations. Results of velocity, temperature, nanoparticle volume concentration, the system of entropy, the Bejan number, mass, and heat transfer rates are presented graphically. We obtained the following main upshots:

### REFERENCES

### DATA AVAILABILITY STATEMENT

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

### AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

### FUNDING

This research was supported by the Research Group Project (RGP), University of Tabuk, Tabuk 71491, Saudi Arabia, under Grant No. RGP-0207-14440.

convective boundary condition. Int J Heat Mass Transfer. (2013) 64:384–92. doi: 10.1016/j.ijheatmasstransfer.2013.04.032

embedded in a porous medium. Int Commun Heat Mass. (2006) 33:1096– 102. doi: 10.1016/j.icheatmasstransfer.2006.04.009

**Conflict of Interest:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2020 Loganathan, Muhiuddin, Alanazi, Alshammari, Alqurashi and Rajan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

### LIST OF SYMBOLS

*(Continued)*

ζ Dimensionless concentration difference

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