**Development of a Species Diagnostic Molecular Tool for an Invasive Pest,** *Mythimna loreyi,* **Using LAMP**

**Hwa Yeun Nam 1 , Min Kwon 1 , Hyun Ju Kim 2 and Juil Kim 1,3,\***

Received: 20 October 2020; Accepted: 17 November 2020; Published: 19 November 2020 -

**Simple Summary:** *Mythimna loreyi* is a serious pest of grain crops and reduces yields in maize plantations. *M. loreyi* is a native species in East Asia for a long time. However, this species has recently emerged as a migration (or invasive from other Asian countries) pest of some cereal crops in Korea. Little is known about its basic biology, ecology, and it is difficult to identify the morphologically similar species, *Mythimna separate*, which occur at the cornfield in the larvae stage. Species diagnosis methods for invasive pests have been developed and utilized for this reason. Currently, the molecular biology method for diagnosing *M. loreyi* species is only using the mtCO1 universal primer (LCO1490, HCO2198) and process PCR and sequencing to compare the degree of homology. However, this method requires a lot of time and effort. In this study, we developed the LAMP (loop-mediated isothermal amplification) assay for rapid, simple, and effective species diagnosis. By analyzing the mitochondrial (mt) genome, the species-specific sequence was found at the coding region of the NADH dehydrogenase subunit 5 gene. A broad range of DNA concentration was workable in LAMP assay, in which the minimum detectable DNA concentration was 100 pg. DNA releasing method was applied, which took five minutes of incubation at 95 ◦C without the DNA extraction process, and only some pieces of tissue from larvae and adult samples were needed. The incidence of invasive pests is gradually diversifying. Therefore, this simple and accurate LAMP assay is possibly used in the intensive field monitoring for invasive pests and integrated management of *Mythimna loreyi*.

**Abstract:** The *Mythimna loreyi* (Duponchel) is one of the well-known invasive noctuid pests in Africa, Australia, and many Asian countries. However, it is difficult to identify the invasive and morphologically similar species, *Mythimna separate*, which occur at the cornfield in the larvae stage. Currently, the molecular biology method for diagnosing *M. loreyi* species is only using the mtCO1 universal primer (LCO1490, HCO2198), which requires a lot of time and effort, such as DNA extraction, PCR, electrophoresis, and sequencing. In this study, the LAMP assay was developed for rapid, simple, effective species identification. By analyzing the mitochondrial genome, the species-specific sequence was found at the coding region of the NADH dehydrogenase subunit 5 gene. Based on this unique sequence, four LAMP primers and two loop primers were designed. The F3 and B3 primers were able to diagnose species-specific, in general, and multiplex PCR and specifically reacted within the inner primers in LAMP assay. The optimal incubation condition of the LAMP assay was 61 ◦C for 60 min with four LAMP primers, though additional loop primers, BF and LF, did not significantly shorten the amplification time. The broad range of DNA concentration was workable in LAMP assay, in which the minimum detectable DNA concentration was 100 pg. DNA releasing method was applied, which took five minutes of incubation at 95 ◦C without the DNA extraction process. Only some pieces of tissue of larvae and adult samples were needed to extract DNA. The incidence of invasive pests is

gradually diversifying. Therefore, this simple and accurate LAMP assay is possibly applied in the intensive field monitoring for invasive pests and integrated management of *Mythimna loreyi*.

**Keywords:** *Mythimna loreyi*; rice armyworm; invasive pest; LAMP; diagnostic PCR

#### **1. Introduction**

The *Mythimna loreyi* (Duponchel) (often called the cosmopolitan) is a noctuid pest of grain crops found in Africa, Australia, the Near East, and the Middle East and undergoes multiple generations per year [1–4]. *M. loreyi* feeds on various host plants, including rice, wheat, maize, sugarcane, barley, sorghum, and others, which have a large effect on female fecundity. The fecundity of female moths is greatest when the larvae feed on maize in Egypt [5]. Since *M. loreyi* is facilitated to breed, some researchers focus on the identification of products secreted by the adult corpora allata [6]. Not only the physiological understanding of this species [7–10] but also ecology-based developmental characteristics [11] and flight performance [12] have been studied. For the biological control, *M. loreyi* densovirus (MIDNVs) was isolated in Egypt and characterized [13,14]. The outbreak of this pest and the damage to crops have been proliferated, particularly in some Asian countries. In Japan, *M. loreyi* typically occurs together with *M. separate* (Walker) that has significant negative effects on crop production [15]. Besides, *M. loreyi* has begun to occur and damage host plants together with *M. separate* [16].

During a couple of years in 2019 and 2020, there have been reports that cornfield has been damaged by the larvae of *M. loreyi* in Korea. This indicates that there is a possibility that *M. loreyi* can change into a sporadic pest, which can cause serious damage to crops. As per these cases, *M. loreyi* is occasionally damaged with its sister species, *M. separate*. However, it is difficult to distinguish two species at the cornfield in the larvae stage. Only one molecular diagnostic tool has been studied to distinguish *M. loreyi,* which is based on the sequencing of part of the mitochondrial COI gene [17]. However, only one mutation exists within the 658 bp amplicon. Moreover, high sequence similarity has been shown between *M. loreyi* and *M. separata*, which indicates the limitation of diagnosis of this pest.

In this study, we developed a simpler technique, termed loop-mediated isothermal amplification assay (LAMP). It is also widely used for the rapid and accurate identification of pest species [18–20]. The LAMP is a rapid, simple, effective, and specific amplification of DNA compared to real-time PCR based on the mitochondrial gene. It is performed under isothermal conditions that require a set of four primers, a strand-displacing DNA polymerase, and a water bath or heat block to maintain the temperature at about 65 ◦C following a one-time denaturation at 95 ◦C [21] or one-step incubation at about 65 ◦C [22].

Following the first infestation of *M. loreyi* in Korea, there is great demand from agricultural research, extension services, and farmers for diagnostic methods for these species. Therefore, we present a LAMP-based method for specimens collected in Korea and other sequences from GenBank. This method should be useful in assisting the effective pest management of *M. loreyi*.

#### **2. Materials and Methods**

#### *2.1. Sample Collection and Mitochondrial Genome Sequencing*

The larval stage of *Mythimna loreyi* Korean populations was collected from Hadong (35◦02′17" N, 127◦47′12" E) in a cornfield, 2019. Some larvae reared in the lab for morphological conformation in the adult stage, and the genomic DNA of several of the individual larva was directly extracted with DNAzol (Molecular Research Center, Cincinnati, OH, USA) and quantified by Nanodrop (NanoDrop Technologies, Wilmington, DE, USA). Besides, the genomic DNA of over 20 larvae or adults was extracted (population pooled genomic DNA). Universal primers (LCO1490 and HCO2198) were used

with *M. loreyi's* individual and pooled genomic DNA as templates in 20 µL PCR reaction containing 1 U TOYOBO KOD—FX TaqTM (Toyobo Life Science, Osaka, Japan), 2X buffer (with 15 mM MgCl2), 0.2 mM each dNTP, 0.5 µM each primer, and 100 ng genomic DNA [23]. The PCR products were directly sequenced (chromatogram) to verify the nucleotide polymorphism, and no mutation was found in intraspecies. For mitochondrial genome sequencing, the Miseq platform was used, and more than 1 Gb was sequenced. To assemble these data, the CLC Assembly Cell package (version 4.2.1) was used. After trimming raw data using CLC quality trim (ver. 4.21), the assembly was accomplished using the CLC de novo assembler with dnaLCW. Assembled sequences were confirmed by BLASTZ [24]. The GeSeq program was used for annotation [25], and the result was manually checked based on the alignment of other Noctuidae species mitochondrial genomes using MEGA 7 [26].

#### *2.2. Phylogenetic Analysis and Primer Design*

Molecular phylogenetic analysis of mitochondrion genomes was inferred by using the maximum likelihood method implemented by MEGA 7 with bootstrapping [26,27]. Mitochondrial genome sequences of other Noctuidae species were downloaded from GenBank, NCBI. For comparative analysis, mitochondrial genomes were aligned using mVISTA [28,29]. Based on the global alignment result, partial sequences were re-aligned for LAMP primer design using PrimerExplorer V5.

#### *2.3. LAMP and PCR*

WarmStart® LAMP Kit (New England Biolabs, Ipswich, UK) was used for the LAMP assay. The general protocol of LAMP was performed following the manufacture's guidelines in a 25 µL reaction mixture. For the general PCR, TOYOBO KOD—FX TaqTM (Toyobo Life Science, Osaka, Japan) was used in this study. Appropriate primers were used with the following PCR amplification protocol: a 2 min denaturing step at 94 ◦C and a PCR amplification cycle consisting of denaturing at 94 ◦C for 20 s, annealing at 60 ◦C for 20 s, and extension at 68 ◦C for 30 s, which was repeated 35 times. The amplified DNA fragments were separated using 1.5% agarose gel electrophoresis and visualized with SYBR green (Life Technologies, Grand Island, NY, USA). The larval stage of *M. loreyi* samples, which were collected from Hadong (Korea) in a cornfield (2019), was used. Pheromone traps were used for adults' sample collection, such as *M. separate*, *Agrotis segetum*, *Spodoptera frugiperda*, *Spodoptera exigua*, *Spodoptera litura*, and *Helicoverpa armigera*. Traps were set in Pyeongchang (37◦40′53" N, 128◦43′49" E), Hongchen (37◦43′35" N, 128◦24′33" E), and Gangneung (37◦36′56" N, 128◦45′59" E) [30]. DNA samples were prepared using DNAzol (Molecular Research Center, Cincinnati, OH, USA) from trapped adults. We used universal primers (LCO1490 and HCO2198) with each DNA sample and pooled genomic DNA as templates in a 20 µL PCR reaction containing 1 U TOYOBO KOD—FX TaqTM (Toyobo Life Science, Osaka, Japan), 2X buffer (with 15 mM MgCl2), 0.2 mM each dNTP, 0.5 µM each primer, and 100 ng genomic DNA [23]. The PCR products were directly sequenced to verify the mutation of each population using RAW\_F3S and RAW\_B3 primer set (MACROGEN). Three biological DNA samples that were collected from the fields were used in each LAMP and PCR to validate the reliability of the LAMP condition.

#### **3. Results**

#### *3.1. Mitochondrial Genome Sequencing and Primer Design*

A 15,320 bp of mitochondrial genome verified after trimming from about 2.2 Gb (7,312,504 reads) nucleotide sequences information was obtained through *Miseq.* The mitochondrial genome *of Mythimna loreyi* was assembled (GenBank MT506351). The mitochondrial genome included 13 protein-coding genes: NADH dehydrogenase components (complex *I*, ND), cytochrome oxidase subunits (complex VI, COX), cytochrome oxidase b (CYPB) and two ATP synthases, and two ribosomal RNA genes and 22 transfer RNAs (Supplementary Figure S1).

As a result of MegaBLAST, the most homologous species was an allied species, *Mythimna separate*, which showed 93.7% similarity. The genus of Spodoptera [31], such as *Spodoptera exigua*, *S. litura, S. frugiperda*, which are possibly found together with *M. loreyi* in the cornfields, showed about 89 to 90% homology based on the mitochondrial genome sequence. *Agrotis segetum* [32], which, in particular, occurs and damage at the corn seedling stage, showed 90.6% similarity to *M. loreyi* based on mitochondrial genome sequence (data not shown).

The phylogenetic relationship between 15 mitochondrial genomes of 14 species was examined (Figure 1A) to verify a specific nucleotide sequence that only *M. loreyi* possessed among the related species with high gene similarity or a morphologically similar pest. The result of the phylogenetic relationship was mostly similar to the megaBLAST result. Based on the mVISTA alignment results, conserved regions among Noctuidae species and variable regions were observed (Figure 1B). By combining the two results, the partial sequence in eight species of nine ND5 mitochondrial genome was re-aligned to design the specific primer of the *M. loreyi*. Finally, four essential primers and two loop primers were designed (Figure 2 and Table 1).

**Figure 1.** Comparison of entire mitochondrial genomes of some Noctuidae pests, including newly sequenced *Mythimna loreyi*. (**A**) Phylogenetic relationship inferred using maximum likelihood under MEGA7. (**B**) Schematic diagram of the genes and their flanking regions, showing the sequence diversity in mVISTA. UTR, D-loop denotes untranslated region and displacement-loop, respectively. Eight green colored mitochondrial genome sequences were re-aligned for primer design with that of the target species, *M. loreyi* (Figure 2).

**Figure 2.** Location of primers and primer binding regions on partial sequence of some Noctuidae pests' mtDNA for species identification of *M. loreyi*. Inner primer, FIP, consists of F1c (complementary sequences of F1) and F2. Another inner primer, BIP, is also composed of B1 and B2c (complementary sequences of B2). Essential four LAMP primers (F3, FIP, BIP, and B3) generate the dumbbell structure, and two loop primers, LF and LB, possibly accelerate the LAMP (loop-mediated isothermal amplification) reaction. Used primer information is documented in Table 1.

**Table 1.** Primer list for LAMP (loop-mediated isothermal amplification) and PCR in this study.

1 'G's at the 5′ end depicted in red were added to adjust the primer melting temperature.

Among the six primers, F3 is the specific primer that enables the diagnosis of *M. loreyi*, and only *M. loreyi* had the CCCC sequences in four priming regions, which are marked in the red box. A total of four populations that were collected from different regions had the same nucleotide sequence. Therefore, it could be sufficiently used for species diagnosis. The basic species diagnosis primer production strategy is the same as the previously reported species diagnosis development method of *S. frugiperda* [33]. We targeted the open reading frame (ORF) region because the ORF region is more intraspecies conserved than the non-coding region.

#### *3.2. Diagnostic LAMP and PCR*

As previously reported, the sensitivity of LAMP may vary depending on the temperature and reaction time [33]. Therefore, the reaction was performed at 65, 63, and 61 ◦C to find the optimal reaction conditions (Figure 3).

**Figure 3.** The sensitivity of the LAMP assay results in three temperature conditions, such as 65, 63, and 61 ◦C, for *M. loreyi* species detected under (**A1**, **B1**, and **C1**) visible light, (**A2**, **B2**, and **C2**) ultraviolet light with SYBR Green, and (**A3**, **B3**, and **C3**) gel electrophoresis. The original pink color of the reaction mixture turned yellow in a positive reaction when the product was formed but remained pink in negative reactions. (**D**) Conventional and multiplex PCR to distinguish *M. loreyi*. The 794 bp amplicon was amplified only in *M. loreyi,* and the conserved partial sequence of ace1 type acetylcholinesterase gene was targeted as an internal reference. Abbreviations are NTC (non-template control), RAW (rice armyworm *Mythimna loreyi*), OAW (oriental armyworm *Mythimna separata*), TM (Turnip moth *Agrotis segetum*), CBW (cotton bollworm *Helicoverpa armigera*), BAW (beet armyworm *Spodoptera exigua*), FAW (fall armyworm *Spodoptera frugiperda*), and TCW (tobacco cutworm *Spodoptera litura*).

As the amount of template DNA was quantified as 50 ng and reacted at each temperature with 25 µL reaction volume, the diagnosis result was confirmed in 120 min at 65 ◦C and 90 min at 63 ◦C. The diagnostic level of reaction did not occur when the reaction performed less than the corresponding time in each temperature. Despite the relatively low temperature, at 61 ◦C, the diagnostic level of reaction was confirmed in only 60 min, and the false-positive reaction did not appear only once in the results of more than three repetition tests. Despite the relatively low temperature, at 61 ◦C, the diagnostic level of reaction was confirmed in only 60 min, and the false-positive reaction did not appear only once in three repetition tests (Figure 3C). Under 100 bp sized bands were not the LAMP products and were aggregated primers and generated even in the negative control.

In the LAMP assay, to confirm the diagnosis primer F3, it was reacted with reverse primer B3 through PCR, and the 794 bp of PCR product was identified (Figure 3(D1)). Besides, a species-specific reaction was confirmed, as we performed the multiplex PCR with a universal positive control primer set that targets ace1-type acetylcholinesterase, which can produce a positive reaction in all samples (Figure 3(D2)). The two-loop primers were tested for possible enhancement of the LAMP reaction, as suggested by Nagamine et al. [22].

As a result of reacting two-loop primers with each or two together at 61 ◦C for 60 min, there was no difference between the reaction result of using only four primers and shortening the reaction time (Figure 4). Even when 10 µL of the reaction solution was verified by electrophoresis after the LAMP reaction, there was no difference in band intensity. It was possible to reliably diagnose up to 100 pg when reacting using four LAMP primers without adding a loop primer (Figure 5A). Therefore, *M. loreyi* species diagnosis LAMP method that we developed could diagnose under various DNA concentration conditions from 100 ng to 100 pg. To increase the usability in the field, we cut a part of the tissue of the adult antenna or leg and put it in 30 µL distilled water, which reacts at 95 ◦C for 5 min, securing the template DNA without a separate DNA extraction process (Figure 5(B4)). As we measured each sample of DNA concentration obtained through this DNA releasing method, each sample showed various measurements. The species-specific diagnosis was possible in the same method as using the template DNA, which was obtained through separate DNA extraction as a positive control since it was in the range of LAMP reaction (Figure 5B).

**Figure 4.** The LAMP assay results with (**A**) 4 primers and additional loop primers, (**B**) loop forward, LF, or (**C**) loop backward, LB, or (**D**) two-loop primers, LF and LB under (**A1**, **B1**, **C1**, and **D1**) visible light, (**A2**, **B2**, **C2**, and **D2**) ultraviolet light with SYBR Green, and (**A3**, **B3**, **C3**, and **D3**) gel electrophoresis. Abbreviations are NTC (non-template control), RAW (rice armyworm *Mythimna loreyi*), OAW (oriental armyworm *Mythimna separata*), TM (Turnip moth *Agrotis segetum*), CBW (cotton bollworm *Helicoverpa armigera*), BAW (beet armyworm *Spodoptera exigua*), FAW (fall armyworm *Spodoptera frugiperda*), and TCW (tobacco cutworm *Spodoptera litura*).

#### **4. Discussion**

Invasive pests, such as *S. frugiperda*, are increasing worldwide due to global warming and climate change [34,35]. Currently, *M. loreyi* is a serious grain pest in Africa, Australia, the Near East, and the Middle East and undergoes multiple generations per year [1–4]. Besides, *M. loreyi* is a species that originates from China and results in damage upon spreading gradually. In Korea, *M. loreyi* has been reported for a long time and maybe a native pest to Korea. Recently, there are several reports that the larvae of *M. loreyi* have damaged the cornfield in Korea. There is a possibility that *M. loreyi* can change into a sporadic pest, which can cause serious damage to crops in Korea.

As with other invasive pests found in the field, they are often similar in morphology to allied species, making the investigation of initial density and pest management difficult. Species diagnosis methods for invasive pests, such as *S. frugiperda*, have been developed and utilized for this reason [33]. Currently, the molecular biology method for diagnosing *M. loreyi* species is only using the mtCO1 universal primer (LCO1490, HCO2198) and process PCR and sequencing to compare the degree of homology. However, this method requires a lot of time and effort, such as DNA extraction, PCR, electrophoresis, and sequencing. Therefore, we developed a molecular diagnosis method that could diagnose species within a short time without a separate DNA extraction process, and only a heat block is needed that can control temperature (Figure 5). The basis diagnosis strategy is very similar to the *S. frugiperda* species diagnosis method [33]. This method can complete all experimental procedures and verify the results within 1 h and 30 min right after obtaining a sample. In this study, the results only specified adult samples, but it is possible to use larvae.

**Figure 5.** (**A**) Identification of the detection limit of genomic DNA in the LAMP assay from 100 ng to 100 fg under (**A1** and **B1**) visible light, (**A2** and **B2**) ultraviolet light with SYBR Green, and (**A3** and **B3**) gel electrophoresis. (**B**) The sensitivity of the LAMP assay results with the DNA releasing technique from insect tissue. Around 10 mg of the adult leg (or antenna) was incubated at 95 ◦C for 5 min (**B4**). Abbreviations are NTC (non-template control), RAW (rice armyworm *Mythimna loreyi*), OAW (oriental armyworm *Mythimna separata*), TM (Turnip moth *Agrotis segetum*), CBW (cotton bollworm *Helicoverpa armigera*), BAW (beet armyworm *Spodoptera exigua*), FAW (fall armyworm *Spodoptera frugiperda*), and PC (positive control, isolated DNA from *M. loreyi*).

The simplicity, accuracy, and adaptability for high throughput of the LAMP assay are distinct advantages [21,36,37]. Moreover, recently LAMP is utilized in various fields, such as many ecology studies, medical aspects, outside of the lab, and can be applied to diagnose plant viruses in insect body and insecticide-resistant gene mutation [38,39]. Besides, the diagnostic primer used for LAMP can be used for various diagnostic methods because it was possible to apply in general PCR and multiplex PCR (Figure 3). Therefore, it is feasible to diagnose a larger sample with positive control in the form of multiplex PCR, which is sufficiently modified and used in the laboratory. There are advantages and disadvantages to find a species diagnostic marker in the mitochondrial genome as well as in part of the genomic DNA. First of all, the disadvantage is that LAMP primer production is limited because there are many parts of AT-rich. It is complicated to design a primer that requires a minimum of four primers, and primers FIP and BIP are high-performance liquid chromatography (HPLC) purified primers. As an advantage, LAMP has higher amplification and efficiency and sensitivity compared to real-time PCR, and the results can be visually monitored by the naked eye either through turbidity or color change by fluorescent intercalating dye (Syber Green I) [40]. The number of copies is large, and it is possible to diagnose with a small amount of DNA or DNA releasing method, whose amplification can be accomplished with heating block. On the contrary, a species diagnosis marker in the genomic DNA should be found in exon rather than intron because it has a large variation. But it is difficult to develop marker in exon because it is often quite conserved within an allied species. Besides, the DNA releasing method can be used, but the efficiency is low when the copies of the gene are small [41].

#### **5. Conclusions**

In this study, a species diagnosis marker was designed within the mitochondrial genome to combine it with a DNA releasing method that is highly applicable to diagnose species in the field. Moreover, a significantly efficient method was developed, which targeted the ace1 gene as a positive control of the species to compare. This simple and accurate diagnosis using LAMP assay could be possibly applied in the intensive field to monitor and for the pest management of *M. loreyi*.

**Supplementary Materials:** The following are available online at http://www.mdpi.com/2075-4450/11/11/817/s1, Figure S1: Organization of the mitochondrial genome of *Mythimna loreyi* from Korea (GenBank MT506351). ND: NADH dehydrogenase components (Complex I) in yellow. COX: cytochrome oxidase subunits (Complex VI) in pink. ATP synthase in green. CYPB: cytochrome oxidase b in purple. Ribosomal RNA genes in red, tRNA genes in blue. Noncoding regions are not colored.

**Author Contributions:** Conceptualization, H.Y.N. and J.K.; methodology, H.Y.N. and J.K.; software, J.K.; validation, J.K.; formal analysis, J.K.; investigation, H.Y.N. and J.K.; resources, H.Y.N., M.K., H.J.K., and J.K.; data curation, H.Y.N. and J.K.; writing—Original draft preparation, H.Y.N. and J.K.; writing—Review and editing, H.Y.N. and J.K.; visualization, H.Y.N. and J.K.; supervision, J.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** The Cooperative Research Program supported this study for Agriculture Science and Technology Development (Project No. PJ01509301), the Rural Development Administration, Republic of Korea.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

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**1. Introduction**

The efficiencies of heat-engine operation employing various numbers (≥ 2) of heat reservoirs are investigated. In Section 2, we discuss heat-engine operation with the work output of the heat engines sequestered. In Section 3, we discuss heat-engine operation with the work output of the heat engines being totally frictionally dissipated. We consider mainly heat engines whose efficiencies depend on ratios of a higher and lower temperature or on simple functions of such ratios. Examples include heat engines operating not only via the Carnot cycle [1–9] but also via the Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [2–9], and endoreversible heat engines operating at maximum power output assuming Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9). But we also provide brief comments concerning more general cases. Endoreversible heat-engine operation assumes irreversible heat flows directly proportional to temperature differences but otherwise reversible operation [10–12]. Although we do not employ them in this chapter, we note that generalizations of the Curzon-Ahlborn efficiency, and also various related efficiencies, have also been investigated [13–21]. In particular, we note that alternative results [21] to the Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9) have been derived [21]. But for definiteness and for simplicity, in this chapter,

we employ the standard Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9) for cyclic heat engines operating at maximum power output.

We show that, if a hot reservoir supplies a heat engine whose waste heat is discharged *and* whose work output is totally frictionally dissipated into a cooler reservoir, which in turn supplies heat-engine operation that discharges waste heat into a still cooler reservoir, the total work output can exceed the heat input from the initial hot reservoir. This extra work output increases with increasing numbers (≥ 3) of reservoirs. We also show that this obtains within the restrictions of the First and Second Laws of Thermodynamics.

*W*1!2 þ *W*2!3 ¼ *Q*1ϵ1!2 þ *Q*1ð Þ 1 � ϵ1!2 ϵ2!3

*Improving Heat-Engine Performance by Employing Multiple Heat Reservoirs*

By contrast, if the heat engine operates in a single step at efficiency ϵ1!3, employing the reservoir at temperature *T*1 as a hot reservoir and the reservoir at

Anticipating that we will eventually deal with *n* heat reservoirs, let us consider

*Tj* � �*x*

2 � *T*2 *T*1 � �*x*

*T*2

� *T*3 *T*2 � �*x*

¼ *W*1!3*:*

� �*x* h i

ϵ*i*!*j* ¼ 1 � *Ti*

where *i* and *j* are positive integers in the respective ranges 1 ≤ *i* ≤ *n* � 1 and *i* < *j* ≤ *n* and where *x* is a positive real number in the range 0 < *x* ≤ 1. Applying

Eqs. (3) and (5), *W*1!3 = *W*1!2 + *W*2!3, as we will now show. We have

8 >><

>>:

8 >><

>>:

¼ *Q*1 1 � *T*2

¼ *Q*1 1 � *T*3

this additivity of *W* obtains for any number of steps, that is, we have

*W*1!*n* ¼ *W*1!2 þ *W*2!3 þ … þ *Wn*�1!*n* ¼ X*n*�1

¼ *Q*1

1 � *T*2 *T*1 � �*x* h i þ 1 � *T*3

� 1 � *T*2 *T*1 � �*x* h i 1 � *T*3

� 1 � *T*2 *T*1 � �*x*

> *T*1 � �*x T*3

*T*1 � � � �*x*

We note that *x* = 1 for the Carnot, Ericsson, Stirling, air-standard Otto, and air-standard Brayton cycles [1–9] and *x* = 1/2 for endoreversible heat engines operating at Curzon-Ahlborn efficiency [10–12] (see also Ref. [4], Section 4-9). For all of these cycles, the temperature in the numerator is that of the coldest available reservoir for a given cycle [1–12]. For the Carnot, Ericsson, and Stirling cycles, and for endoreversible heat engines operating at Curzon-Ahlborn efficiency, the temperature in the denominator is that of the hottest available reservoir for a given cycle [1–12]. For the air-standard Otto and air-standard Brayton cycles, the temperature in the denominator is that at the end of the adiabatic-compression process but before the addition of heat from the hottest available reservoir (substituting, in air-standard cycles, for combustion of fuel) [2–9] in a given cycle. The Second Law of Thermodynamics forbids *x* > 1 if the temperature in the numerator is that of the coldest available reservoir for a given cycle and the temperature in the denominator is that of the hottest available reservoir for a given cycle, because then the Carnot efficiency would be exceeded. Since for the aforementioned heat engines, and indeed for any heat engine for which Eq. (5) is applicable, *W*1!3 = *W*1!2 + *W*2!3,

� � � �*x*

temperature *T*3 as a cold reservoir, it can do work

*DOI: http://dx.doi.org/10.5772/intechopen.89047*

*W*1!2 þ *W*2!3 ¼ *Q*1

efficiencies of the form

**131**

¼ *Q*1ð Þ ϵ1!2 þ ϵ2!3 � ϵ1!2ϵ2!3 *:* (3)

*W*1!3 ¼ *Q*1ϵ1!3*:* (4)

*T*2 � �*x* h i 9 >>=

>>;

*T*2 � �*x* h i

> � *T*3 *T*2 � �*x*

> > þ *T*2 *T*1 � �*x T*3 *T*2

> > > *j*¼1

*Wj*!*j*þ1*:* (7)

*,* (5)

9 >>=

>>;

(6)

We fill in details and correct a few mistakes in an earlier, briefer, consideration of the efficiencies of heat-engine operation employing various numbers (≥ 3) of heat reservoirs [22]. We note that heat-engine operation employing various numbers (≥ 3) of heat reservoirs [22] should not be confused with recycling heat engines' frictionally dissipated work outputs into the hottest available reservoir [22–37], which is a *different* process that has been thoroughly investigated and discussed previously [22–37], and which we further investigate in another chapter [38] in this book.

We consider only cyclic heat engines. Noncyclic (necessarily one-time, singleuse) heat engines are not limited by the Carnot bound and can in principle operate at unit (100%) efficiency. A simple example is the one-time expansion of a gas pushing a piston. Other examples include rockets: the piston (payload) is launched into space by a one-time power stroke (but typically most of the work output accelerates the exhaust gases, not the payload) and firearms: the piston (bullet) is accelerated by a one-time power stroke and then discarded (but some, typically less than with rockets, of the work output accelerates the exhaust gases resulting from combustion of the propellant). Even if the work output of a noncyclic engine could be frictionally dissipated and the resulting heat returned to the system, there would be, at best, restoration of temperature to its initial value but not restoration of the piston to its initial position. Hence the method investigated in this chapter is useless with respect to noncyclic heat engines.

General remarks, especially concerning entropy, are provided in Section 4. Concluding remarks are provided in Section 5.


**6. Studies on Hematopoietic System**

Although interest in stem cell treatment increased over the last two decades, stem cell transplantation started more than half a century ago with bone marrow transplantation by Lorenz et al. [5] followed by Barnes et al. [53]. These authors demonstrated that transplantation of bone marrow cells could protect mice against ionising radiation. This was the pioneering process of bone marrow transplantation that developed as a routine clinical procedure, where whole marrow or marrow cells extracted from bone marrow are transplanted into myeloablated host in the treatment of both malignant and non-malignant diseases such as leukaemia, lymphoma, and certain types of anaemia [54].

The effect of transplantation of bone marrow-derived mononuclear cells in nonhuman primates were studied by Bertho et al. [55]. These authors demonstrated that cell transplantation 24 h after 8 Gy total body irradiation shortened the period and severity of pancytopenia. Acute radiation syndrome (ARS), besides multi-organ failure, causes pancytopenia too. The efficacy of transplantation of human UC-MSCs to combat the effects of ARS was also studied [56]. However, in this study, UC-MSCs were modified to to express human extracellular superoxide dismutase. The regenerative potential of MSCs combined with the antioxidant effect of human extracellular superoxide dismutase was intended to produce a rapid and effective strategy for the treatment of radiation accident victims.

The protective effects of allogenic stem cell transplantation against acute radiation syndrome was demonstrated by transplantation of human umbilical cord-derived MSCs in mice [57].


**9. Sexual health service and the type to use**

Several clients are considered to be having special needs for contraception, especially those that may have some conditions that may prevent them from accessing sexual and reproductive health services. Sometimes these groups of individuals may be at risk of unplanned pregnancy and STIs. Young people form 41% of 'new adult infections' globally [40]. They may include adolescents, young people, post-abortion clients as well as people with disability. Although they are biologically or emotionally tortured, they all have equal rights to information and services on contraception [41]. It is highly important to provide sexual and reproductive health service information to these groups of people as well as help them to make decisions that can lead to positive sexual health outcomes. This call for the provision of sexual health services (SHS), a system that provides increased attention to SRH information to young people. The intention is to target young individuals with increased risk and provide them with information on sexual and reproductive health care to prevent them from unplanned pregnancy, maternal deaths and STIs (Ibid). This is because they have the right to decide on which contraception to use, obtain and where to get them. On that account, it is essential to make service delivery sites more comfortable and friendlier. The intention of this inauguration is to decrease maternal and infant mortality and encourage the promotion of reproductive health so as to increase the prevalence of contraceptive uptake.

Sexual health service is described as systems that make provision of preventive health services to youth on-site or in their community or within the health facility [42]. These are healthcare services provided by healthcare providers and school health nurses. So, increasing access to these services will help improve adolescents' and young women's immediate and lifelong health significantly, where other health issues would be detected early and services provided on time.


*Edited by Edmir Daniel Carvalho, Gianmarco Silva David and Reinaldo J. Silva*

Aquaculture has been expanding in a fast rate, and further development should rely on the assimilation of scientific knowledge of diverse areas such as molecular and cellular biology, and ecology. Understanding the relation between farmed species and their pathogens and parasites, and this relation to environment is a great challenge. Scientific community is involved in building a model for aquaculture that does not harm ecosystems and provides a reliable source of healthy seafood. This book features contributions from renowned international authors, presenting high quality scientific chapters addressing key issues for effective health management of cultured aquatic animals. Available for open internet access, this book is an effort to reach the broadest diffusion of knowledge useful for both academic and productive sector.

ISBN 978-953-51-0497-1

Health and Environment in Aquaculture

Photo by Kondor83 / iStock


*Molecular Challenges and Advances in Clinical Islet Transplantation DOI: http://dx.doi.org/10.5772/intechopen.108571*

islet cell transplantation. In islet transplantation, the requirement for immunosuppressive drug treatment to protect alloislets from alloimmune rejection and recurrent autoimmunity introduces additional risks that may be specific to the individual immunosuppressive drug agents as well as related more generally to immunosuppression. Thus recognition of inter-individual differences is gaining importance and is becoming possible through integration of pharmacogenetics, pharmacoproteomics, epigenetics, and noncoding RNAs data into clinical practice, thus emphasizing that *'Precision medicine is vital in transplantation'.* A panel of genetic variants for transplant recipients and donors can function as an additional tool at disposition of transplant physicians to provide individualized care.


*2.4. Image Preprocessing*

All axial images were resized to 256 × 256 voxels for neural network training. The axial slices were set to have a distance of 3 mm between each other. The standard deviation and mean values of each image were calculated when retrieved from the database. The image signal intensity was then normalized and applied voxelwise to each image. From all the available mpMRI sequences, only the T2W images were used for training and validation.

### *2.5. Convolutional Neural Network*

The CNN used for this study was a custom modified U-Net. The algorithm's base architecture was derived from a standard U-Net, which is a fully convolutional contracting and expanding architecture [17]. The customized U-Net has a symmetric architecture and uses the same number of layers during subsampling and upsampling. U-Net also employs skip connections that allow the CNN to combine features for the image contraction and expansion pathways. These skip connections enabled the U-Net to use spatial information that could potentially be lost after the image is further downsampled in the contraction pathway. The entire image was trained during a single forward pass and the U-Net classified each image per pixel.

Our customized U-Net was extended to incorporate three dimensions during training and then produce outputs in two dimensions (Figure 1). Five layers were chosen empirically. In each layer, the image was processed by batch normalization, convolution, rectified linear unit (ReLU) activation, and downsampling with strided convolutions by a factor of 2. The 5 layers used 4, 8, 16, 32, and 32 filters per convolution. The image was downsampled until it became a 1 × 1 × 1 matrix before it underwent expansion. During the expansion pathway, the image was upsampled and a skip connection allowed the upsampled image to combine spatial information from the contraction pathway.

**Figure 1.** All neural network runs were completed on a U-Net with 5 layers. The number of channels used were 4, 8, 16, 32, and 32 for the 5 layers.


*Article* **Poroacoustic Traveling Waves under the Rubin–Rosenau–Gottlieb Theory of Generalized Continua**

#### **Pedro M. Jordan**

Acoustics Division, U.S. Naval Research Laboratory, Stennis Space Ctr., Hancock County, MS 39529, USA<EMAIL_ADDRESS>
Received: 16 November 2019; Accepted: 4 March 2020; Published: 14 March 2020

**Abstract:** We investigate linear and nonlinear poroacoustic waveforms under the Rubin–Rosenau– Gottlieb (RRG) theory of generalized continua. Working in the context of the Cauchy problem, on both the real line and the case with periodic boundary conditions, exact and asymptotic expressions are obtained. Numerical simulations are also presented, von Neumann–Richtmyer "artificial" viscosity is used to derive an exact kink-type solution to the poroacoustic piston problem, and possible experimental tests of our findings are noted. The presentation concludes with a discussion of possible follow-on investigations.

**Keywords:** poroacoustics; Rubin–Rosenau–Gottlieb theory; solitary waves and kinks

#### **1. Introduction**

What is known today as the "RRG theory" was put forth by Rubin et al. [1] in 1995. This phenomenological-based theory of generalized continua is thought capable of modeling dispersive effects caused by the introduction of a medium's characteristic length, which Rubin et al. denote as *α*. Under RRG theory, *α* is regarded as an inherent material property. From the modeling standpoint, this theory exhibits a number of appealing features, the two most important of which are the following: (i) it is only the pressure stress (i.e., isotropic) part of the Cauchy (i.e., total) stress tensor and the specific Helmholtz free energy that are modified, but these modifications are achieved by adding perburtative terms, which must satisfy certain constraint equations, to the constitutive relations of the former and latter; and (ii), no additional boundary nor initial conditions, beyond those required to solve classically formulated problems, are needed ([1], p. 4063).

To date, RRG theory has only been applied to single-phase media; see, e.g., Ref. [2] and those cited therein. Hence, there is an obvious need to investigate the nature of the solutions, e.g., those of the traveling wave type, predicted by this theory in the case of multi-phase media.

Accordingly, the aim of this communication is to carry out a *preliminary* investigation of RRG theory in the context of acoustic problems involving propagation in dual-phase (specifically, fluid + solid) media—dual-phase media being, of course, the simplest case of multi-phase media. Employing both analytical and numerical methodologies, we consider linear and finite-amplitude poroacoustic propagation under the RRG-based generalization of what some refer to as the *Brinkman poroacoustic model* (BPM) (Although he does not refer to it as such, the general, multi-D, version of the BPM follows on setting *C* = 0 in Burmeister [3].). Here, it should be noted that the original version of the drag law on which the BPM is based reads (see, e.g., Refs. [4,5])

$$
\nabla P = \tilde{\mu}\chi\nabla^2 \mathbf{u} - (\mu\chi/K)\mathbf{u}.\tag{1}
$$

In this relation, **u** is the intrinsic average velocity of the fluid, which it is related to **v**, the Darcy velocity, via the Dupuit–Forchheimer relationship **v** = *χ***u** ([4], p. 5); *P* is a pressure, an intrinsic

quantity, which is not the thermodynamic pressure ([4], §1.4.1); *μ* is the usual shear viscosity coefficient; *μ*˜ is an effective viscosity ([4], §1.5.3), which is often referred to as the *Brinkman viscosity*; and *K* > 0 and *χ* ∈ (0, 1), the permeability and porosity of the solid matrix, are assumed to be constants. We should also note that Equation (1) reduces to Darcy's law on setting *μ*˜ := 0.

Before beginning our analysis, we should point out that when traveling wave solutions (TWS)s in the form of kinks are encountered below, their shock thicknesses shall be expressed using Prandtl's definition (see, e.g., (Ref. [6], p. 680)), viz.:

$$\text{shock thickness} := \frac{\mathfrak{F}(-\infty) - \mathfrak{F}(+\infty)}{\max\_{\mathfrak{z} \in \mathbb{R}} |\mathrm{d}\mathfrak{F}(\mathfrak{z})/\mathrm{d}\mathfrak{z}|},\tag{2}$$

where z represents the wave (i.e., similarity) variable. Herein, all traveling wave profiles shall be taken to be propagating to the right along the axis corresponding to the wave variable under consideration.

#### **2. Mathematical Formulations**

#### *2.1. Poroacoustic Model Systems*

When entropy production in the fluid, which we hereafter take to be a *perfect gas* [7], is ignored (i.e., we take the flow to be *homentropic* ([7], p. 60)) and the porous matrix is regarded as being both stationary and composed of a thermally non-conducting rigid solid, the 1D versions of the RRG-based model we propose *and* the BPM become, in the case of propagation along the *x*-axis,

$$
\varrho\_t + \mathfrak{u}\varrho\_x + \mathfrak{e}\mathfrak{u}\_x = 0,\tag{3a}
$$

$$\varrho(\mu\_l + \mu u\_x) = \overline{\mu}\chi u\_{xx} - (\mu \chi / K)u - \begin{cases} \wp\_{X'} & \text{BPM,} \\ [\wp - 2a^2 \varrho(u\_{xt} + \mu u\_{xx})]\_{x'} & \text{RRG,} \end{cases} \tag{3b}$$

$$
\xi \wp \approx \wp\_0 (\not q / \wp\_0)^\gamma \qquad (\mathfrak{n} \not\simeq \mathfrak{n}\_0). \tag{3c}
$$

In System (3), ℘(> 0) is the thermodynamic pressure; (> 0) is the mass density of the gas; n is the specific entropy of the gas; the parameter *γ* denotes the ratio of specific heats, where *γ* ∈ (1, 5/3] in the case of perfect gases; we take *α*(> 0), which carries the unit of length, to be a constant (That is, we have assumed the *simplest* version of RRG theory; see (Ref. [1], Equation (20)).); the problem geometry dictates that, here and henceforth, **u** = (*u*(*x*, *t*), 0, 0), ℘ = ℘(*x*, *t*), and = (*x*, *t*); and a zero ("0") subscript attached to a dependent variable denotes the (constant) equilibrium state value of that variable, where we note that **u**0 = (0, 0, 0).

Here, we observe that since the flow has been assumed homentropic, our RRG-based poroacoustic model is obtained by perturbing *only* the pressure tensor term in the BPM. Also, we record for later reference that *c*0 = *γ*℘0/0 is the (constant) equilibrium state value of the sound speed, i.e., the speed of sound in the undisturbed gas; see, e.g., (Ref. [7], §4.3).

#### *2.2. Finite-Amplitude Equation of Motion: The Case μ* := *const.*

We begin this sub-section with the following observation: Because ∇ × **u** = (0, 0, 0) holds identically under the present (1D) geometry, it follows that **u** = ∇*φ*; therefore, *u*(*x*, *t*) = *φx*(*x*, *t*), where *φx* denotes the scalar velocity potential.

Hence, on invoking the finite-amplitude approximation, and introducing the following dimensionless variables:

$$\mathbf{u}^{\diamond} = \mathbf{u} / \mathcal{U}\_{\mathbb{P}^\diamond} \quad \mathbf{s} = (\varrho - \varrho\_0) / \varrho\_0, \quad \boldsymbol{\Phi}^{\diamond} = \boldsymbol{\Phi} / (\mathcal{U}\_{\mathbb{P}} \mathcal{L}), \quad \mathbf{x}^{\diamond} = \mathbf{x} / \mathcal{L}, \quad \mathbf{t}^{\diamond} = \mathbf{t} (\mathbf{c}\_0 / \mathcal{L}), \tag{4}$$

where the positive constants *L* and *U*p respectively denote a macro-length scale characteristic of the propagation domain and the magnitude of the peak particle velocity in the gas, it is not difficult to establish (See, e.g., the derivation performed in (Ref. [8], §2), and note that (Ref. [8], Equation (10)) is the *σ*, *δ* := 0 special case of Equation (5) herein.) that the *μ* := const. case of System (3) reduces to the weakly-nonlinear, *bi-directional*, equation of motion (EoM)

$$
\phi\_{tt} - \left[1 - 2\varepsilon(\beta - 1)\phi\_t\right]\phi\_{xx} + \varepsilon\partial\_t(\phi\_x)^2 = \sigma\phi\_{txx} + a\_0^2\phi\_{ttxx} - \delta\phi\_{t\prime} \tag{5}
$$

where here and henceforth all diamond () superscripts have been suppressed for convenience. In Equation (5), which we note reduces to the corresponding EoM for the (1D) BPM on setting *a*0 := 0,  = *U*p/*c*0 is the Mach number, where  1 is assumed; *δ* = *νχL*/(*c*0*K*) is the dimensionless Darcy coefficient, where *ν* = *μ*/0 is the kinematic viscosity of the gas; *a*0, the dimensionless version of *α*, is given by *a*0 = *α* √2/*L*; we have set *σ* := *χ*/*Re*B, where *Re*B = *c*0*L*/*ν*˜ is a Reynolds number, and where *ν*˜ = *μ*˜/0; and *β*(> 1) denotes the *coefficient of nonlinearity* [9], which in the case of a perfect gas is given by

$$
\beta = (\gamma + 1)/2. \tag{6}
$$

In deriving Equation (5) we have assumed that *δ*, *σ*, *a*0, |*s*|∼O() and, in accordance with the finite-amplitude approximation, only *nonlinear* terms O(2) have been neglected.

#### *2.3. Right-Running Equations of Motion for the Case μ* := *const.*

Although derived under the finite-amplitude approximation, Equation (5) is still too complicated for treatment by analytical means. Fortunately, however, the nature of the problems to be considered below is such that we may employ the uni-directional approximation to reduce the order of Equation (5) by one and confine its nonlinearity to a single (quadratic) term. Omitting the details, we find that under, say, the right-running case of this approximation (See, e.g., Crighton's ([10], p. 16) derivation of the acoustic version of Burgers' equation.), which in the present setting reads *φx* −*φt*, our EoM becomes, after making use of the relation *u*(*x*, *t*) = *φx*(*x*, *t*) and simplifying,

$$
\epsilon \mu\_t + \mu\_x + \epsilon \beta \mu u\_x - \frac{1}{2} a\_0^2 u\_{txx} + \frac{1}{2} \delta u = \frac{1}{2} \sigma u\_{xx\prime} \tag{7}
$$

which on switching to the variables *X* = *x* − *t* and *T* = *t* is further reduced to

$$
\epsilon \mu\_T + \epsilon \beta \mu \iota\_X - \frac{1}{2} a\_0^2 \mu\_{TXX} + \frac{1}{2} \delta \iota = \frac{1}{2} \sigma \iota\_{XX}.\tag{8}
$$

If we once again make use of the right-running approximation, which now takes the form *uT* −*uX*, to re-express *only* the third order term in Equation (8), which is justified since *a*0 ∼ O() (i.e., (*a*2 0/2)*uTXX* is a "small" term), then Equation (8) assumes its final form, specifically,

$$
\epsilon\_1 u\_T + \epsilon \beta u u \chi + \frac{1}{2} a\_0^2 u \chi \chi \chi + \frac{1}{2} \delta u = \frac{1}{2} \sigma u \chi \chi,\tag{9}
$$

a PDE which we term the *damped Burgers–KdV* (dBKdV) equation.

In closing this sub-section we stress that Equations (7)–(9) apply only to right-running waveforms; i.e., to problems wherein reflection (to the left) is not possible.

#### **3. Comparison of Linearized EoMs: The Cauchy Problem**

In this section we compare the BPM with its RRG-based counterpart under the linear approximation, which at the EoM level corresponds to setting  := 0. We do so in the context of what is perhaps the best known problem from classical PDE theory.

To this this end, we consider the linearized version of Equation (9) in the setting of the following initial value problem (IVP), i.e., in the setting of the classical *Cauchy problem*:

$$
\sigma \mu\_T + \frac{1}{2} a\_0^2 \mu\_{XXX} + \frac{1}{2} \delta \mu = \frac{1}{2} \sigma \mu\_{XX}, \quad X \in \mathbb{R}, T > 0,\tag{10a}
$$

*Water* **2020**, *12*, 807

$$
\mu(X,0) = f(X), \quad X \in \mathbb{R}.\tag{10b}
$$

Here, we take *f*(*X*), our initial condition (IC), to be defined on the real line and such that its Fourier transform exists.

On applying the Fourier transform to both Equation (10a) and the IC, and then solving the resulting (first order) ODE, it is readily shown that

$$\text{fit}(k, T) = \hat{f}(k) \exp\left[ -\frac{1}{2} \left( \delta + \sigma k^2 - \text{i}a\_0^2 k^3 \right) T \right],\tag{11}$$

where *k* is the Fourier transform parameter and a hat over a quantity denotes the Fourier transform of that quantity. In turn, applying F −1(·), the inverse Fourier transform, to Equation (11) gives

$$\mu(X,T) = \frac{1}{2\pi} \exp(-\delta T/2) \int\_{-\infty}^{\infty} \hat{f}(k) \exp\left[-\frac{1}{2}\left(\sigma k^2 - \mathrm{i}a\_0^2 k^3\right)T\right] \exp(\mathrm{i}kX) \,\mathrm{d}k \tag{12}$$

*3.1. The RRG Case: a*0 > 0

Using the convolution theorem, and letting Ai(·) denote the Airy function of the first kind, the RRG (i.e., *a*0 > 0) case of Equation (12) can be recast in the more explicit form

$$\begin{split} u(X,T) = \exp\left(-\delta T/2\right) \left(\frac{2}{3a\_0^3T}\right)^{1/3} \\ &\times \int\_{-\infty}^{\infty} \mathcal{F}^{-1} \quad \left[\not\!f(k) \exp\left(-\frac{1}{2}\sigma k^2\right)\right] \text{Ai}\left[\left(X-\mathcal{Y}\right)\left(\frac{2}{3a\_0^2T}\right)^{1/3}\right] \text{d}\mathcal{Y} \quad (T>0). \end{split} \tag{13}$$

For obvious reasons, the following two special cases of *f*(*X*) are of particular interest:

$$\begin{split} \mathcal{U}(X,T) &= \exp(-\delta T/2) \left(\frac{2}{3a\_0^2 T}\right)^{1/3} \\ &\times \begin{cases} \frac{1}{\sqrt{2\pi\sigma T}} \int\_{-\infty}^{\infty} \exp\left(-\frac{1}{2\sigma^2} \mathcal{Y}^2/T\right) \text{Ai}\left[ (X-\mathcal{Y}) \left(\frac{2}{3a\_0^2 T}\right)^{1/3} \right] \text{d}\mathcal{Y}, & f(X) = \mathfrak{d}(X) \\\\ \frac{1}{\sqrt{1+b\sigma T}} \int\_{-\infty}^{\infty} \exp\left[ -\frac{1}{2} b (1+b\sigma T)^{-1} \mathcal{Y}^2 \right] \text{Ai}\left[ (X-\mathcal{Y}) \left(\frac{2}{3a\_0^2 T}\right)^{1/3} \right] \text{d}\mathcal{Y}, & f(X) = e^{-bX^2/2} \end{cases} (T > 0). \end{split}$$

Here, d(·) denotes the Dirac delta function and *b*(> 0) is a (dimensionless) constant.

#### *3.2. The BPM Case: a*0 := 0

If we assume instead the BPM, then the solution of IVP (10) is readily obtained on setting *a*0 := 0 in Equation (12); for the two aforementioned cases of *f*(*X*), we find that

$$u(X,T) = \begin{cases} \left[\frac{\exp(-\delta T/2)}{\sqrt{2\pi\sigma T}}\right] \exp\left(-\frac{1}{2\sigma}X^2/T\right), & f(X) = \mathfrak{d}(X) \\\\ \left[\frac{\exp(-\delta T/2)}{\sqrt{1+b\sigma T}}\right] \exp\left[-\frac{1}{2}b(1+b\sigma T)^{-1}X^2\right], & f(X) = \mathfrak{e}^{-bX^2/2} \end{cases} (T > 0). \tag{15}$$

#### *3.3. Remarks: RRG vs. BPM*

With regard to the Gaussian IC, the primary difference between the linearized RRG and BPM cases is that the pulse profile corresponding to the former instantly becomes oscillatory about the *X*-axis, due to the Airy function in its integrand, while that of the latter maintains, for all *T* > 0, the shape and strict positivity of the initiating Gaussian. The clearly contrasting behaviors exhibited by these two models should, therefore, allow researchers to experimentally determine which of the two best describes propagation in a given poroacoustic system.

#### **4. Comparison of Right-Running, Weakly-Nonlinear, EoMs: Special Cases with** *μ* := **const.**

Before examining it in its most general form, and for the benefit of those readers who are not well acquainted with the intricacies of nonlinear evolution equations, it is instructive to first review selected special cases of Equation (9). The right-running models discussed in the next three sub-sections, all of which, it should be noted, have applications beyond poroacoustics, will each have a role to play in the analysis performed in Section 4.4.

#### *4.1. Case (I): Damped KdV (dKdV) Equation*

This case follows on setting *σ* := 0 (i.e., setting *μ*˜ := 0), under which Equation (9) reduces to

$$
\epsilon\_{\!\!\!\!\/} \mu\_T + \epsilon \beta \mu u\_X + \frac{1}{2} a\_0^2 \mu\_{XXX} + \frac{1}{2} \delta u = 0. \tag{16}
$$

This PDE, we observe, is the RRG-modified version of the right-running Darcy–Jordan model; see Section 4.3 below.

Since we have assumed *δ* 1, applying the *Kryloff–Bogoliubov* asymptotic expansion method to the dKdV equation yields, as Ott and Sudan [11] have shown, the large-*T* expression

$$\mu(X,T) \sim \exp(-2\delta T/3) \operatorname{sech}^2\left[\left(\zeta/a\_0\right)\sqrt{\frac{c\delta}{\delta}\exp\left(-2\delta T/3\right)}\right], \qquad T \sim \mathcal{O}(1/\delta), \tag{17}$$

where we have taken *N*(0) = 1 (see Ref. [11]), and we let

$$\mathcal{Z} = X - \frac{c\beta}{2\delta} \left[ 1 - \exp\left(-\frac{2}{3}\delta T\right) \right];\tag{18}$$

see also Ref. [12], as well as Ref. [13] (Ref. [13] contains a number of recently identified typographical errors; see Appendix A below.) and those cited therein. Equation (17) represents a damped, and *decelerating*, solitary waveform, and as such it cannot be a soliton in the classical sense [14]. Note, however, that the acoustic version of the classic soliton solution of the KdV equation (see Ref. [8]) is recovered as the limiting case

$$\mu(X,T) = \mathrm{sech}^2\left[a\_0^{-1}(X - \epsilon\beta T/3)\sqrt{\frac{\epsilon\beta}{6}}\right] \qquad (\delta \to 0). \tag{19}$$

#### *4.2. Case (II): Damped Burgers' Equation*

This case, which corresponds to setting *a*0 := 0, reads

$$
\varepsilon\mu\_T + \varepsilon\beta\mu\iota\_X + \frac{1}{2}\delta\iota = \frac{1}{2}\sigma\iota\_{XX}.\tag{20}
$$

Equation (20) is the right-running EoM stemming from the BPM, and in this context it has recently been investigated by Rossmanith and Puri [15].

As shown by Nimmo and Crighton [16], this generalization of Burgers' equation does not admit a linearizing (i.e., Cole–Hopf type) transform. As shown by Malfliet [17], however, its TWS, which assumes the form of a damped kink, is readily approximated. To the order expressed explicitly in Ref. [17], the TWS of Equation (20) is given by

$$u(X,T) \approx \frac{1}{2} \exp(-\delta T/2) [1 - Y(X,T)] \{1 + a\_5(T)[1 + Y(X,T)]Y^3(X,T) \\ \begin{aligned} & \quad \begin{aligned} &u(X,T) \\ &\quad + a\_5(T) \end{aligned} \} \end{aligned} \tag{21}$$

Here,

$$Y(X,T) := \tanh\left[\frac{2}{\lambda\_B}\left(X - \frac{\epsilon\beta[1 - \exp(-\delta T/2)]}{\delta}\right)\right],\tag{22}$$

where *λ*B = 4*σ*/( *β*) is the shock thickness exhibited by the TWS given below in Equation (25);

$$a\_3(T) = -(1 - \exp(-\delta T/2)) / 3;\tag{23}$$

and

$$a\_{\mathbb{S}}(T) = -[128 - 160 \exp(-\delta T/2) + \delta \lambda\_{\mathbb{B}}^2 \exp(-\delta T/2)/\sigma + 32 \exp(-\delta T)]/240. \tag{24}$$

In Ref. [17], the parameter *c*, which herein has the value *c* = 2/*λ*B, is defined so that Equation (21) yields the limiting case

$$\mu(X,T) = \frac{1}{2}\left\{1 - \tanh\left[\frac{2}{\lambda\_B}\left(X - \frac{1}{2}\epsilon\beta T\right)\right]\right\} \qquad (\delta \to 0). \tag{25}$$

Equation (25) and *λ*B are the TWS, which we note takes the form of a *Taylor shock*, and corresponding shock thickness, which was determined using Equation (2), respectively, admitted by the classic Burgers equation.

#### *4.3. Case (III): Damped Riemann Equation*

In the poroacoustic context, this case corresponds to the right-running version of the weakly-nonlinear *Darcy–Jordan model* (Also known as the *Jordan–Darcy model*; see Ciarletta and Straughan [18], as well as Straughan [5].) (DJM) [19]; specifically, the first order PDE [20]

$$
\epsilon \mu\_T + \epsilon \beta u \mu\_X + \frac{1}{2} \delta u = 0,\tag{26}
$$

which follows on setting *a*0 := 0 *and σ* := 0 in Equation (9).

In the setting of the Cauchy problem, the exact solution of the damped Riemann equation is readily determined; see, e.g., Crighton ([21], p. 196). In the particular case of Equation (26), this solution can be expressed as [20]

$$
\mu(X, T) = \mu\_0(\xi) \exp(-\delta T/2),
\tag{27a}
$$

$$a^\star(X - \xi) = u\_0(\xi)[1 - \exp(-\delta T/2)].\tag{27b}$$

Here, *ξ* = *ξ*(*X*, *T*) is the wave variable; *u*0(*ξ*) is the IC; and *α*, the critical amplitude value for acceleration waves under the DJM, is given by [19]

$$
\alpha^\star = \frac{\delta}{2\varepsilon\beta}.\tag{28}
$$

For the particular case *u*0(*X*) = cos(2*πX*), it can be shown (see, e.g., (Ref. [20], p. 3)) that under System (27)

$$T\_B^\* = -2\delta^{-1} \ln \left( 1 - \frac{a^\*}{2\pi} \right). \tag{29}$$

If *T*∗ *B* ∈ R+, then *T*∗ *B* is the time at which the solution of the Cauchy problem involving Equation (26) suffers (finite-time) *gradient catastrophe* ([22], p. 36), where it is expected that *α*∗ < 2*π* ( =⇒ *T*∗ *B* ∈ R+) in all cases of practical interest.

#### *4.4. Numerical Results*

Inspired by, and closely following, Zabusky and Kruskal's [14] analysis of the classic KdV equation, in this subsection we perform numerical experiments on Equation (9), and its special cases listed above as Cases (I) and (II), in the setting of the following initial-boundary value problem (IBVP) with *periodic* boundary conditions:

$$
\sigma u\_T + \epsilon \beta \mu u\_X + \frac{1}{2} a\_0^2 \mu\_{XXX} + \frac{1}{2} \delta u = \frac{1}{2} \sigma u\_{XX}, \quad |X| < 1, T > 0,\tag{30a}
$$

$$u(-1, T) = u(1, T), \quad T > 0,\tag{30b}$$

$$
\mu(X,0) = \cos(2\pi X), \quad |X| < 1. \tag{30c}
$$

In (Ref. [14], Figure 1), snapshots of the evolution of the KdV's solution profile were displayed in units of (dimensionless) time *TB*, where Zabusky and Kruskal used *TB* to denote the "breakdown time" (i.e., the time at which finite-time gradient catastrophe occurs) of the solution to the Cauchy problem involving the classic (i.e., undamped) Riemann equation. In our analysis of IBVP (30), *T*∗ *B* shall play the role of *TB*.

The graphs presented in Figures 1–3 were computed and plotted using MATHEMATICA (ver. 11.2). Except for the value of *β*(= 1.2), which corresponds to diatomic gases (e.g., air) [9], all other parameter values were selected based on our desire to produce clear, illustrative, graphs and the need to satisfy the assumptions under which Equation (9) was derived.

From Figure 1 it is easy to see that, except for attenuation of the profile (caused by the Darcy term) and a slight phase shift, the dKdV profiles are qualitatively similar to those of the classic KdV equation in the setting of IBVP (30). And, as is also true in the case of the latter, reducing (resp. increasing) *a*0 increases (resp. decreases) the number of pulses seen in Figure 1b.

In contrast, the plots shown in Figures 2 and 3 highlight the fact that, like that of the damped Burgers equation, the dBKdV profile suffers attenuation, again due to the Darcy term, and it also develops a "dull sawtooth" appearance as it shocks-up (to the right), but never breaks since *σ* > 0. More interesting, however, is the fact that for large-*T*, both the damped Burgers equation and dBKdV profiles are seen to *re-assume* the periodic form of the IC. As Figures 2c and 3c illustrate, both profiles evolve to become damped, and in the dBKdV case slightly phase-shifted (to the left), versions of the IC given in Equation (30c). This suggests that for sufficiently large values of *T*, one may employ the approximations *u*(*X*, *T*) ≈ *u*1,2(*X*, *T*), where

$$u\_{1,2}(X,T) := \begin{cases} \exp[-\triangle\_1(T)]\cos[2\pi X + \psi\_1(T)], & \text{dBKdV equation} \\ \exp[-\triangle\_2(T)]\cos[2\pi X + \psi\_2(T)], & \text{damped Burers' equation} \end{cases} \tag{31}$$

and where we require -1,2(*T*) > 0. Comparing the blue-broken curve in Figure 2c with its counterpart in Figure 3c we see that -2(20*T*∗ *B*) > -1(20*T*∗ *B*) > 0 while *ψ*1(20*T*∗ *B*) > *ψ*2(20*T*∗ *B*) := 0; here, for simplicity, we have *assumed* -1,2(*T*) and *ψ*1,2(*T*) to be linear functions of *T*. In the setting of IBVP (30), then, the presence of the third order (i.e., RRG) term in the dBKdV equation gives rise to both a phase shift and slightly less attenuation vis-à-vis the damped Burgers equation.

While their usefulness may be limited to certain "windows" of *T*-values, the functions -1,2(*T*) and *ψ*1,2(*T*) should be constructible based on Equation (31) and numerically generated, large-*T*, data sets using one of the many data-fitting methodologies found in the literature.

**Figure 1.** The dKdV case of IBVP (30). (**a**–**c**) correspond to *T* = *T*∗ *B*, 3.6*T*∗ *B*, and 20*T*∗ *B*, respectively, where *T*∗ *B* ≈ 1.382. Red curves: *u* vs. *X* for *a*0 = (0.005) √2, *σ* = 0, *β* = 0.12, *δ* = 0.12, and *α* = 0.5. Blue curves: IC given in Equation (30c).

**Figure 2.** The dBKdV case of IBVP (30). (**a**–**c**) correspond to *T* = *T*∗ *B*, 3.6*T*∗ *B*, and 20*T*∗ *B*, respectively, where *T*∗ *B* ≈ 1.382. Purple curves: *u* vs. *X* for *a*0 = (0.005) √2, *σ* = 0.005, *β* = 0.12, *δ* = 0.12, and *α* = 0.5. Blue curves (solid): IC given in Equation (30c). Blue-broken curve: *u*1(*X*, 20*T*∗ *B*) vs. *X* (see Equation (31)), where we have set -1(20*T*∗ *B*) := (29.9)*δT*∗ *B* and *ψ*1(20*T*∗ *B*) := (0.1013)*T*∗ *B* based on a series of trial-and-error "visual fits".

**Figure 3.** The damped Burgers equation case of IBVP (30). (**a**–**c**) correspond to *T* = *T*∗ *B*, 3.6*T*∗ *B*, and 20*T*∗ *B*, respectively, where *T*∗ *B* ≈ 1.382. Green curves: *u* vs. *X* for *a*0 := 0, *σ* = 0.005,  *β* = 0.12, *δ* = 0.12, and *α* = 0.5. Blue curves (solid): IC given in Equation (30c). Blue-broken curve: *u*2(*X*, 20*T*∗ *B*) vs. *X* (see Equation (31)), where we have set -2(20*T*∗ *B*) := (29.946)*δT*∗ *B* and *ψ*2(20*T*∗ *B*) := 0 based on a series of trial-and-error "visual fits".

#### **5. The RRG Case with "Artificial"** *μ*

In 1950, von Neumann and Richtmyer (vNR) [23] introduced their artificial viscosity coefficient. In this section, we make use of this celebrated artifice not to regularize numerical schemes used to calculate shock profiles, as was vNR's aim, but rather to obtain an *analytical* solution to the poroacoustic version of the piston problem (Unlike Ref. [23], wherein Lagrangian coordinates were used, in this communication we employ the Eulerian description; see, e.g., (Ref. [24], §*V-D-1*) wherein vNR's system is recast under the latter.).

To this end, we return to the RRG case of System (3) and assume that

$$
\mu \approx \mathfrak{a}^2 \varrho |u\_x|\,, \tag{32}
$$

but continue to regard *μ*˜ as a positive constant. Here, we have expressed the length-scale factor in vNR's artificial viscosity coefficient as *α*, instead of some grid spacing Δ*x*.

For simplicity, we now assume that the porous solid in question is comprised of packed beds of rigid solid spheres, all of radius *r*(> 0), which are fixed in place. For such a configuration, the permeability is given by the well known *Kozeny–Carmen* relation [4]:

$$K = \frac{r^2 \chi^3}{45(1 - \chi)^2}.\tag{33}$$

As these spheres are scatters of acoustic waves, we take *α* to be proportional to the characteristic length now associated with our dual-phase medium; i.e., we take *α* = *b*1*r*, where *b*1(> 0) is an "adjustable" (dimensionless) constant ([24], p. 233).

If, moreover, we limit our focus to kink-type waveforms, as physical intuition suggests, and have the piston located at *x* = −∞ and moving to the right along the *x*-axis, then *ux* < 0 and Equation (32) becomes

$$
\mu = -b\_1^2 r^2 \varrho \phi\_{xx} \tag{34}
$$

where we have used the relation *u* = *φx*. Returning to our dimensionless variables, and once again applying the finite-amplitude approximation, it is readily established that, under the aforementioned assumptions, the following (simpler) weakly-nonlinear PDE replaces Equation (5) as our bi-directional EoM:

$$
\sigma \phi\_{tt} - \left[1 - 2\epsilon(\beta - 1)\phi\_t\right] \phi\_{xx} + \epsilon(1 - \delta\_1)\partial\_t(\phi\_x)^2 = \sigma \phi\_{txx} + a\_1^2 \phi\_{ttxx} \qquad \text{(arrival } \mu\text{)}.\tag{35}
$$

In Equation (35), which we observe applies *only* to the RRG case, we have set *a*1 := *b*1*r* √2/*L*, where we require that *a*1 ∼ O(), and

$$\delta\_1 := \frac{\epsilon b\_1^2 45 (1 - \chi)^2}{\chi^2} \qquad (0 < \delta\_1 < 1), \tag{36}$$

where the requirement *δ*1 ∈ (0, 1) implies that *b*1 must satisfy the inequality

$$0 < b\_1 < \frac{1}{3\sqrt{5\epsilon}} \left(\frac{\chi}{1-\chi}\right). \tag{37}$$

Assuming the gas at *x* = +∞ is in its equilibrium state, and thus motionless, and observing that in the present context *U*p is the *dimensional* speed of the piston, we let *φ*(*x*, *t*) = *G*(*η*), where *η* = *x* − *v*1*t* and the (dimensionless) shock speed *v*1 is taken to be a positive constant, and then substitute into Equation (35). On integrating once with respect to *η* and then imposing/enforcing the asymptotic conditions *g* → 1, 0 as *η* → ∓∞, respectively, Equation (35) is reduced to the ODE

$$
\sigma\_1^2 \upsilon\_1 \mathbf{g}^{\prime\prime} - \sigma \mathbf{g}^{\prime} - \epsilon \beta\_1 \mathbf{g}(1 - \mathbf{g}) = 0,\tag{38}
$$

where we note that the resulting constant of integration is zero. In Equation (38), *g*(*η*) = *G* (*η*), where a prime denotes d/d*η*; we have defined

$$
\beta\_1 := \beta - \delta\_1. \tag{39}
$$

recalling that *β* is the coefficient of nonlinearity (see Equation (6)); and

$$
v\_1 = \frac{1}{2}\epsilon\beta\_1 + \sqrt{1 + \frac{1}{4}\epsilon^2\beta\_1^2} \qquad (v\_1 > 1), \tag{40}$$

which we observe is the positive root of

$$
\sigma\_1^2 - \epsilon \beta\_1 \upsilon\_1 - 1 = 0. \tag{41}
$$

To apply the solution methodology employed in (Ref. [2], §2) to Equation (38), the following condition must be satisfied:

$$25a\_1^2(v\_1^2 - 1) = 6v^2.\tag{42}$$

In (Ref. [2], §2), satisfying Equation (42) required that the value of the Mach number be fixed, a constraint which clearly limits the usefulness of the TWSs presented in that article. Here, however, we shall use this restriction to our advantage; specifically, in the following sense: Since the value of *μ*˜ for a given poroacoustic flow is, in general, not known, and we are seeking a kink-type TWS, then the only possible solution of Equation (42) in the present context is

$$
\sigma = 5a\_1 \sqrt{\frac{v\_1^2 - 1}{6}} \qquad \implies \qquad \overline{\mu} = \frac{5q\_0 b\_1 r}{\chi} \sqrt{\frac{v\_1^2 - c\_0^2}{3}}, \tag{43}
$$

where we observe that v1 = *v*1*c*0 is the *dimensional* shock speed and, moreover, that *v*1 > 1 implies v1 > *c*0.

On imposing the usual wave front condition *g*(0) = 1/2, but otherwise referring the reader to (Ref. [2], §2) for details regarding its derivation, the TWS we seek is

$$g(\eta) = \frac{1}{4}\text{sech}^2\left(\frac{27\eta}{16\lambda\_1} + \mathcal{K}\right) + \frac{1}{2}\left[1 - \tanh\left(\frac{27\eta}{16\lambda\_1} + \mathcal{K}\right)\right],\tag{44}$$

where K = tanh−1 −1 + √2 . Letting *λ*1 = 1/*L* denote the *dimensionless* shock thickness (Recall Equation (2)) admitted by Equation (44), it is easily established that

$$\lambda\_1 = \frac{135a\_1^2 v\_1}{8\sigma} = \frac{81a\_1 v\_1}{4\sqrt{6(v\_1^2 - 1)}} \qquad \implies \qquad \ell\_1 = \frac{81\mathfrak{v}\_1 b\_1 r}{4\sqrt{3(\mathfrak{v}\_1^2 - \mathfrak{c}\_0^2)}},\tag{45}$$

where 1 is the corresponding *dimensional* shock thickness. Also, with regard to computing *λ*1, it should be noted that *g*(*η*∗) = 0, where *η*∗(< 0) is given by

$$\eta^\* = \frac{10a\_1^2 \upsilon\_1 \left[\tanh^{-1}(1/3) - \mathcal{K}\right]}{\sigma},\tag{46}$$

and where it should also be noted that *g*(*η*∗) = 5/9.

The usefulness of Equation (45) might be ascertained as follows. Assume that v1 and 1 can both be determined, either directly or indirectly, from experimental measurements and, moreover, that both are (at most) slowly varying functions of time. With v1 known, *b*1 can, of course, be computed using Equations (36), (39) and (40). If this (inferred) value of *b*1 satisfies the inequality in Equation (37), *a*1 ∼ O() is also satisfied, and the measured value of 1 is in agreement with that computed from

Equation (45) over, say, some span of time *t* ∈ T , then we can expect Equation (45) to prove useful as an approximation within the transition region of our kink-type traveling wave profile for *t* ∈ T .

#### **6. Discussion: Possible Follow-On Studies**

In addition to gaining a better understanding of how the solution of IBVP (30) behaves for large-*T*, in particular, determining to what extent (if any) the recurrence behavior seen in Figures 2c and 3c is related to the functional form of a given IC, future work on poroacoustic RRG theory could included the use of homogenization methods in problems wherein *K* and/or *χ* vary with position. Other possible extensions include the poroacoustic generalization of the study carried out in Ref. [25], wherein *α* was taken to be a function of (*ux*)2, and also the case in which *μ*˜ is a power-law function of the shear rate tensor. Follow-on work might also include the study of poroacoustic signaling problems involving sinusoidal and/or shock input signals, as well as problems in which changes in entropy and temperature are taken into account.

**Funding:** This work was supported by U.S. Office of Naval Research (ONR) funding.

**Acknowledgments:** The author thanks Ivan Christov, for kindly providing him with the specialized MATHEMATICA code used to generate the data sets from which Figures 1–3 were plotted, and Markus Scholle, for the kind invitation to contribute to this special issue. The author also thanks the two anonymous reviewers for the critiques they provided; their constructive comments have led to a marked improvement in this paper.

**Conflicts of Interest:** The author declares no conflict(s) of interest.

#### **Appendix A. Corrections to Ref. [13]**

As kindly confirmed by Prof. Leibovich [26], Equations (4b), (4d), and (13), and the caption of FIG. 2, in Ref. [13] contain typographical errors; the required corrections, also provided by Prof. Leibovich [26], read as follow:

#### **References**

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