**9. Conclusion**

Therapeutic keratoplasty is generally an emergency, high-risk procedure that challenges the surgical and medical skills of the corneal surgeon. It requires meticulous attention to detail and careful postoperative monitoring. Therapeutic keratoplasty play a definitive role in the treatment of microbial keratitis refractory to medical therapy. Advances in microsurgical technique, antimicrobial therapy new and more powerful antibiotics (fourth generation quinolones), and control of inflammation have resulted in an improved prognosis for therapeutic keratoplasty in cure and improved visual outcomes.


**4. Use of biotelemetry in other fields**

Other fields have preceded poultry in the use of biotelemetry and studies of the use of biotelemetry in other species are available for wildlife, livestock, fish, laboratory animals and humans. A quick survey of some of these studies may be a useful source of information for poultry research as they contain interesting equipment and methodologies.

A broad survey of the literature seems to indicate that the most advanced use of biotelemetry is in human medicine. There has been increased interest in the medical field in remote patient monitoring driven by the need for real-time patient data and the ability to monitor multiple patients simultaneously (Tan et al., 2009). Several studies in the literature have surveyed advances in biotelemetry in the medical field and they give insight into the

Advances in Management of Poultry Production Using Biotelemetry 179

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Brown–Brandl, T. M., Yanagi, Jr., T., Xin, H., Gates, R. S., Bucklin, R. A., & Ross, G. S. (2003).

Budinger, T. F. (2003). Biomonitoring with wireless communications. *Annual Review of Biomedical Engineering*, Vol. 5, No. 1, (August 2003), pp. 383-412, ISSN 1523-9829 Byrne, C., & Lim, C. L. (2007). The ingestible telemetric body core temperature sensor: a

Coenen, A. M. L., Lankhaar, J., Lowe, J. C., & McKeegan, D. E. F. (2009). Remote monitoring

Cross, P. S., Künnemeyer, R., Bunt, C. R., Carnegie, D. A., & Rathbone, M. J. (2004). Control,

Crowther, C., Davies, R., & Glass, W. (2003). The effect of night transportation on the heart

Eigenberg, R. A., Brown-Brandl, T. M., & Nienaber, J. A. (2008). Sensors for dynamic

Green, A. R. & Xin, H. (2009). Effects of stocking density and group size on

Güler, I. & Kara, S. (1996). A low-cost biotelemetry system for long time monitoring of

Güler, N. F., & Übeyli, E. D. (2002). Theory and applications of biotelemetry. *Journal Of Medical Systems*, Vol. 26, No. 2, (April 2002), pp. 159-178, ISSN 0148-5598 Hamrita, T. K., Hamrita, S. K., Van Wicklen, G., Czarick, M., and Lacy, M. (1997). Use of

Hamrita, T. K., Van Wicklen, G., Czarick, M., & Lacy, M. (1998). Monitoring poultry deep

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bioelectric monitoring of electrocardiograms and temperature in male broilers at risk for sudden death syndrome. *Poultry Science*, Vol. 81, No. 1, (June 2002) pp. 887-

A new telemetry system for measuring core body temperature in livestock and poultry. *Applied Engineering in Agriculture*, Vol. 19, No. 5, (September 2003), pp. 583-

review of validity and exercise applications. *British Journal of Sports Medicine*, Vol.

of electroencephalogram, electrocardiogram, and behavior during controlled atmosphere stunning in broilers: Implications for welfare. *Poultry Science*, Vol. 88,

communication and monitoring of intravaginal drug delivery in dairy cows. *International Journal of Pharmaceutics*, Vol. 282, No. 1-2, (September 2004), pp. 35-44,

rate and skin temperature of ostriches during real transportation. *Meat Science*, Vol.

physiological measurements. *Computers and Electronics in Agriculture*, Vol. 62, No. 1,

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advanced state of medical biotelemetry equipment and its applications (Akyildiz et al., 2002; N. F. Güler & Übeyli, 2002; Budinger, 2003; Lewis & Goldfarb, 2003; Strydis, 2005; Byrne & Lim, 2007; Luong et al., 2008; Ruiz-Garcia et al., 2009; Lin et al., 2010; Yilmaz et al., 2010).


**5.2 Seed certification process**

Quality control objectives are two: (a) Ensure that farmers receive quality seed in terms of input to maximize their production at harvest, and (b) Ensure farmers are not at risk of receiving low-quality seed (inputs) from fraudulent traders [24].

The Seed Certification process involves the following steps: Field inspections, Seed Treatment; The seed test, Label and seal, Post controls, and Post Certification Survey

*Production of Potato Quality Seeds in Mountainous Region of Central Africa DOI: http://dx.doi.org/10.5772/intechopen.107126*

#### *5.2.1 Field inspections*

This is the first step in the certification process. It consists of:

The inspection is conducted to ensure that harvested crop is the true type, it has no contamination, and culture is healthy (disease free/pest free).


**7. Conclusion**

In this Chapter we considered different approaches to development of multi-channel adaptive measurement systems based on multiplexing of dynamic holograms in a photorefractive crystal. We show that such systems can provide (i) high sensitivity to detection of ultra-small physical quantities (close to the classical homodyne detection limit), (ii) cross-talk free performance, and (iii) adaptive properties which cancel uncontrollable environmental influence on the system. For development of such systems different approaches to dynamic holograms multiplexing, including spatial, angular, spectral and their combinations, can be effectively used. Moreover, dynamic gratings recorded in a photorefractive crystal can also be used for development of such elements of multi-channel systems as spectral tunable filters which can provide effective demultiplexing of signals with different wavelengths.


**1. Introduction**

Schiff base ligands (SBLs) are known as "privileged ligands" [1] due to their facile synthesis by the condensation of aldehydes with imines. In 1864, Hugo Schiff first introduced the condensation of an aldehyde and an amine, which leads to the formation of a Schiff base (SB) [2]. SBLs are capable to form coordination bonds with metals *via* imine nitrogen and group, which is generally connected to the aldehyde. In recent days, Schiff bases (SBs) are prepared where efficient and well-synthesized SBLs are labeled as "privileged ligands." SBs can stabilize various metals with different oxidation states and control the behavior of metals in diverse catalytic transformations,

which are beneficial to most industries [3]. SBLs are easily synthesized and can combine with practically any metal ion to form complexes [4]. At high temperatures (>100°C) and with moisture, several Schiff base complexes (SBCs) exhibit strong catalytic activity in various organic reactions. There have been numerous studies in recent years explaining application in catalysis; therefore, this chapter highlights the catalytic activity of SBCs. Almost all SBCs exhibit strong catalytic activity [4]. SBs are often bi- or tridentate ligands that can bind to transition metals to produce incredibly stable complexes. In some cases, liquid crystals are also employed for this purpose as most liquid crystals have an azomethine group. The formation of carbon-nitrogen bonds can be accomplished through Schiff base reactions in organic synthesis [5]. Coordination compounds were once thought to be an infrequent and unique group, and later they were acknowledged as the most important compounds, which are capable to transform straightforward inorganic molecules to form organic matter. To give a good example of a coordination component, chlorophyll (the magnesium porphyrin complex found in plants) converts CO2 and H2O into carbohydrates.

The basic structural formula for SBs is RRC=N-R, where R, R', here R is labeled as hydroxyalkyl, alkyl, cyclohexyl, hydroxyaryl, etc. SBs consist of azomethine moiety (C=N) an azomethine group. In some cases, -SH or –OH functional groups also participate in complexation, when they are enough close to azomethine moiety. This is due to the lone pair of electron's being donated to the electron deficient metal ion. Overall, this facilitates the development of a metal complex with good stability. With salicylideneimine, Ettling created the copper (II) complex in 1840. Since then, there has been a significant quantity of material published on SBs and related metal complexes (MCs) [6].

### **2. Synthetic routes to Schiff base complexes**

From the past two decades, there is significant progress in the formation of SB transition MCs, with an increased focus on synthesis and structure. The characteristic of the SB and metal ion determines the synthesis, characteristics, and structure of the SBCs. Mostly, the following techniques have been employed to create MCs of SBs [7].

Aldehydes and amines can condense with one another under various reaction circumstances in the presence of various solvents. Methanol and ethanol are the typical solvents used to prepare the SBs, either at room temperature or under refluxing conditions. SBs are typically more likely to form when dehydrating chemicals such as magnesium sulfate are present. If the syntheses are conducted in organic solvents, the water generated during the reaction can be extracted easily. During the purification process, the SBs may deteriorate. It is preferable in these circumstances to crystallize the SBs and purify them. Here, in a suitable solvent, SB is permitted to interact with metal ions. The use of a binary azeotropic mixture of water and an organic solvent has been documented in some situations, and using an organic solvent is recommended to prevent the hydrolysis of the azomethine group.

#### **2.1 Direct synthesis**

Here, a suitable solvent is used to facilitate the reaction between SB and metal ions. Despite the fact that there have been instances of mixing organic solvent and water, the use of organic solvent is recommended to prevent the hydrolysis of the azomethine group [8].


2.2.4. HPLC Analysis

The HPLC analysis was conducted using a VWR Hitachi Chromaster 600 chromatograph with a spectrophotometric detector (DAD) and EZChrom Elite software (Merck) on a Discovery C18 reversed-phase column (25 cm × 4.0 mm i.d., 5 μm particle size) (Supelco, Sigma-Aldrich, St. Louis, MO, USA). Mobile phase consisted of acetonitrile-water—1% phosphoric acid (90:10:0.5 *v*/*v*/*v*). Flow rate of eluent was 1 mL/min and column temperature was 10 ◦C [18]. Chromatograms were recorded from 200 to 400 nm. The triterpenic acids were quantified at 205 nm.

### **3. Results and Discussion**

### *3.1. Characteristics of Si-PANI*

The morphology of Si-PANI particles was assessed using a confocal microscope and automated procedure of computer image analysis (Figure 1). Blue color of Si-PANI particles showed that polyaniline was successfully deposited on silica (Figure 1a). Moreover, no changes of particle shape were observed and this proved that the synthesis conditions did not cause the destruction of silica. The analysis of diameters (Figure 1b) showed the slight increase of Si-PANI particle diameter (average diameter was 10.7 μm) comparing to bare silica (average diameter was 10 μm) as a result of covering the surface with polyaniline film. The intensity of the PANI signal recorded during Raman analysis proved that polyaniline was deposited more intensively inside the adsorbent grain (Figure 1c). This may be explained by the fact that silica has a porous structure and the surface of grain inner pores is larger than the outside pores.

**Figure 1.** The morphology of Si-PANI particles: (**a**) microscope picture of silica and Si-PANI; (**b**) particle diameter distribution of silica and Si-PANI; (**c**) the examples of spatial distribution of polyaniline on the surface within the particle.

Polyaniline may occur in various forms [1]; therefore, in order to establish its form after protonation and deprotonation of Si-PANI bed, the Raman spectra were recorded (Figure 2). Both spectra matched the spectral pattern of the emeraldine [32]. Although minor shifts in some peak positions were observed, the differences between spectra were irrelevant.

**Figure 2.** Smoothed Raman spectra of Si PANI sorbent: protonated (red line) and deprotonated (blue line).

### *3.2. Optimization of d-SPE Parameters*

In order to establish the optimal conditions for d-SPE of triterpenic acids, the main parameters affecting the extraction efficiency and recovery of analytes were investigated using standard solution of UA, OA and BA.

### 3.2.1. Form of PANI and Impregnation Solution

The interaction of sorbent with analyte strongly affect the ability of trapping the analyte from solution. Four variants of experiment, using protonated (Si-PANI (+)) and deprotonated (Si-PANI) sorbent impregnated with methanol or water, were conducted to establish the optimal conditions to bond the highest amount of investigated compounds. The results are presented in Figure 3.

**Figure 3.** Percentage of retained analytes depending on Si-PANI form and impregnation solution.

As can be seen, the form of polyaniline and impregnation have an impact on efficiency of trapping the triterpenic acids from solution. Surprisingly, a high percentage of extraction efficacy was obtained both for protonated and deprotonated sorbent; however, the different impregnation solutions were required for particular form (water and methanol, respectively).

The structure of Si-PANI and Si-PANI (+) was modeled (Figure 4) to compare the charge density of the surface what could be helpful to explain the observed effect.

**Figure 4.** Modeled chain of polyaniline: (**a**) Deprotonated and (**b**) Protonated form.

Based on modeled structures, significant differences in distribution of charge on polyaniline were noted. As a result, the protonation of PANI with HCl, surface of sorbent was positively charged, the anions (Cl−) were accumulate to compensate and the electrical double layer was formed (on Figure 4b the density of charge at Cl− is visible). Presumably, Si-PANI (+) gained the ability to anion exchange. Impregnation with water occurred optimal for Si-PANI (+) because in water the ionization of analytes increased and then, the ability to exchange Cl- on anionic analytes was possible (ion exchange mode of retention).

In turn, on deprotonated Si-PANI, the charges were focused on nitrogen (Figure 4a) and the retention was probably mostly caused by π-π interactions between aromatic rings of analytes and aromatic rings of sorbent which were enhanced by methanol [33].

### 3.2.2. Time of Extraction

The bonding of analyte in d-SPE strongly depends on time. The partition of analyte between solution and sorbent is dynamic process and appropriate time is necessary to obtain equilibrium [26,27]. Our investigation showed that the amount of bonded triterpenic acids increased up to 3 min and then remained constant (the plateau effect was observed) (Figure 5). No statistically significant differences between the investigated compounds or between both forms of sorbent were noted.

**Figure 5.** Effect of extraction time on percentage of retained analytes.

### 3.2.3. Ratio of Sorbent to Analyte

Since the number of active sites on adsorbent surface should be sufficient to trap the total amount of target compound, the sorbent/analyte ratio is a significant factor affecting the extraction efficiency. As can be seen on Figure 6 the ratio 160:1 (mg of sorbent/mg analytes) was found to be optimal for all investigated triterpenic acids. Moreover, we noticed that the curves of relationship between sorbent/analyte ratio and percentage of retained compound were similar for protonated and deprotonated Si-PANI.

**Figure 6.** Effect of sorbent/analytes ratio on percentage of retained analytes.


**Michael T. Davies-Coleman 1,\* and Clinton G. L. Veale 2**

Academic Editor: Sylvia Urban Received: 25 July 2015; Accepted: 29 September 2015; Published: 14 October 2015

**Abstract:** Recent developments in marine drug discovery from three South African marine invertebrates, the tube worm *Cephalodiscus gilchristi*, the ascidian *Lissoclinum* sp. and the sponge *Topsentia pachastrelloides*, are presented. Recent reports of the bioactivity and synthesis of the anti-cancer secondary metabolites cephalostatin and mandelalides (from *C. gilchristi* and *Lissoclinum* sp., respectively) and various analogues are presented. The threat of drug-resistant pathogens, e.g., methicillin-resistant *Staphylococcus aureus* (MRSA), is assuming greater global significance, and medicinal chemistry strategies to exploit the potent MRSA PK inhibition, first revealed by two marine secondary metabolites, *cis*-3,4-dihydrohamacanthin B and bromodeoxytopsentin from *T. pachastrelloides*, are compared.

**Keywords:** cephalostatin; mandelalide; methicillin resistant *Staphylococcus aureus*; MRSA PK; bisindole alkaloids


**1. Introduction**

It is a well-established fact that a moving body reduces drag if it is elevated by a layer of air. This phenomenon is used in air-cushioned vehicles and in air hockey, in which the frictional resistance of moving objects is reduced. Skalak and Wang [1] were the pioneers of studying the three-dimensional flow that arises between a moving porous flat plate and the ground, and they later on wrote an erratum on their own paper [2]. Wang also studied elliptical porous sliders [3]. In the case of Newtonian fluids, past studies have included porous circular, long, inclined, and elliptical sliders. R. C. Bhattacharjee studied a porous slider bearing lubricated with a coupled stress (a magneto-hydrodynamic (MHD) fluid) [4]. Jimit made a comparison of the different porous structures on the performance of a magnetic fluid [5]. Prawal Sinha analyzed the thermal effects of a long porous rough slider bearing [6]. Mohmmadrayian analyzed a rough porous inclined slider bearing lubricated with a ferrofluid in consideration of slip velocity [7]. Ji Lang both theoretically and experimentally investigated the transient squeezing flow in a highly porous film [8]. Similarly, a large amount of literature is available in relation to long porous sliders (LPSs) [1,6,9–13] and circular porous sliders (CPSs) [2,14–18]. Awati investigated the lubrication of a long porous slider by using the homotopy analysis method (HAM) [10]. In a separate study, Khan studied the e ffects of Reynolds numbers by using di fferent analytical methods [11,12]. Naeem studied the influence of Reynolds numbers (R) on long [13] and circular porous sliders [18]. Ghoreishi studied the circular slider [14]. Madani investigated the circular porous slider by using HPM, and also analyzed its lift and drag [19].

All the above mentioned studies were done without a slip condition on either the immobile ground or slider. However, a slip condition is essential for super-hydrophobic planes, as it is di fficult to have a zero mean tangential velocity from where the fluid is injected when there is a slip. Furthermore, in order to minimize adhesion, the fluid could be a rarefied gas, where the compact exterior could be coated with a material, or the ground could be uneven so that an equivalent slip exists or there is a slip flow regime. Wang [16] discussed slip e ffects, but didn't consider the e ffects of a transverse magnetic field. Therefore, the goal of the current work is to examine the impact of slip and Reynolds numbers when a transverse magnetic field is a ffecting the performance of a porous slider. Through the literature survey, it is assumed that a three-dimensional flow with slip and a uniform magnetic field does not exist. Hence, the goal of the current research is to analyze the performance of porous sliders in the presence of slip and a Reynolds number with a constant magnetic field, and to assess their e ffects on the components of velocity lift and drag.

The structure of the article is as follows: In the introduction, a brief history of the problem of the porous slider and its application is presented. In the second section, the formulation of the problems are given, while in the third section the formulation of a homotopic solution is presented [20]. The fourth section deals with the convergence criteria of the HAM. Results and discussions are given in the fifth section. Finally, the conclusion is given in the sixth section, with a list of nomenclature.

As discussed above, the velocity slip condition is considered in this study. Navier introduced the slip condition for the first time as follows:

$$\mathbf{x}\_1 = H\boldsymbol{\zeta} \tag{1}$$

In Equation (1), tangential velocity *u* is proportional to the shear stress and *H* is the constant of proportionality, which is actually a slip coe fficient. In order to ignore the end e ffects, it is assumed that the gap between slider and ground is quite small as compared to the slider's lateral dimension. Both circular and long porous sliders are considered in this study.

### **2. Problem Formulation of Long and Circular Sliders**

In this study, the incompressible and steady flow of a viscous fluid between porous (long and circular) sliders and the ground is considered in the presence of a uniform magnetic field, as shown in Figure 1.

**Figure 1.** (**a**) Schematic diagram of the movement of a long porous slider (LPS). (**b**) Schematic diagram of the movement of a circular porous slider (CPS).

Length and width are quite big compared to height *d*. The slider moves with the velocity components and is elevated because of the injection of fluid from below with a magnetic field applied externally. In order to avoid the induced magnetic field formed by the movement of the fluid, it is assumed that the magnetic Reynolds number is not very big. Furthermore, the induced and imposed electric field are supposed to be negligible, and therefore the electromagnetic body force per unit volume simplifies F*em* = σ0(v × B) × B, where B =(0, 0, *B*0) is the magnetic field.

Under the above-stated assumptions and conditions, Navier–Stokes equations take the following form:

$$
\phi\_1 \frac{\partial \phi\_1}{\partial \mathbf{x}\_1} + \phi\_2 \frac{\partial \phi\_1}{\partial \mathbf{x}\_2} + \phi\_3 \frac{\partial \phi\_1}{\partial \mathbf{x}\_3} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_1} + \nu \left( \frac{\partial^2 \phi\_1}{\partial \mathbf{x}\_1^2} + \frac{\partial^2 \phi\_1}{\partial \mathbf{x}\_2^2} + \frac{\partial^2 \phi\_1}{\partial \mathbf{x}\_3^2} \right) - \frac{\sigma\_0}{\rho} B\_0^2 \phi\_1 \tag{2}
$$

$$
\phi\_1 \frac{\partial \phi\_2}{\partial \mathbf{x}\_1} + \phi\_2 \frac{\partial \phi\_2}{\partial \mathbf{x}\_2} + \phi\_3 \frac{\partial \phi\_2}{\partial \mathbf{x}\_3} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_2} + \nu \left( \frac{\partial^2 \phi\_2}{\partial \mathbf{x}\_1^2} + \frac{\partial^2 \phi\_2}{\partial \mathbf{x}\_2^2} + \frac{\partial^2 \phi\_2}{\partial \mathbf{x}\_3^2} \right) - \frac{\sigma\_0}{\rho} B\_0^2 \phi\_2 \tag{3}
$$

$$
\phi\_1 \frac{\partial \phi\_3}{\partial \mathbf{x}\_1} + \phi\_2 \frac{\partial \phi\_3}{\partial \mathbf{x}\_2} + \phi\_3 \frac{\partial \phi\_3}{\partial \mathbf{x}\_3} = -\frac{1}{\rho} \frac{\partial p}{\partial \mathbf{x}\_3} + \nu \left( \frac{\partial^2 \phi\_3}{\partial \mathbf{x}\_1^2} + \frac{\partial^2 \phi\_3}{\partial \mathbf{x}\_2^2} + \frac{\partial^2 \phi\_3}{\partial \mathbf{x}\_3^2} \right) \tag{4}
$$

Velocity components are expressed as (φ1, φ2, φ3), where ρ, *p*, and υ are density, pressure and kinematic viscosity, respectively. Law of conservation of mass is as follows:

$$\frac{\partial \phi\_1}{\partial \mathbf{x}\_1} + \frac{\partial \phi\_2}{\partial \mathbf{x}\_2} + \frac{\partial \phi\_3}{\partial \mathbf{x}\_3} = 0 \tag{5}$$

According to Naeem [13], the following transform has been used:

$$
\phi\_1 = lI\psi\_1(\boldsymbol{\varsigma}) + \frac{\mathcal{W}}{d} \mathbf{x}\_1 \psi\_3 \;/\; (\boldsymbol{\varsigma}),\\\phi\_2 = V\psi\_2(\boldsymbol{\varsigma}),\\\phi\_3 = -\mathcal{W}\psi\_3(\boldsymbol{\varsigma}).\tag{6}
$$

where ς = *x*3*d* . By adding Equation (6) into Equations (2)–(4), the following ordinary differential equations are obtained:

$$
\psi\_3^{\rm ir} = R \big( \psi\_3^{\prime} \psi\_3^{\prime \prime \prime} - \psi\_3 \psi\_3^{\prime \prime \prime \prime} \big) + M^2 \psi\_3^{\prime} \tag{7}
$$

$$
\Psi\_1^{\prime \prime} = \mathcal{R} \{ \psi\_1 \psi\_3^{\prime} - \psi\_3 \psi\_1^{\prime} \} + \mathcal{M}^2 \psi\_1 \tag{8}
$$

$$
\psi\_2^{\prime\prime\prime} = -\mathcal{R}(\psi\_1 \psi\_2^{\prime}) + M^2 \psi\_2 \tag{9}
$$

where *R* is the Reynolds number (*R* = *Wd*/υ). Boundary conditions at *x*3 = 0 and *x*3 = *d* are given in Equations (10) and (11), respectively.

$$
\phi\_1 = \mathcal{U} + H\_1 \mu \frac{\partial \phi\_1}{\partial \mathbf{x}\_3}, \quad \phi\_2 = V + H\_1 \mu \frac{\partial \phi\_2}{\partial \mathbf{x}\_3}, \quad \phi\_3 = 0 \tag{10}
$$

$$
\phi\_1 = -H\_2 \mu \frac{\partial \phi\_1}{\partial \mathbf{x}\_3}, \quad \phi\_2 = -H\_2 \mu \frac{\partial \phi\_2}{\partial \mathbf{x}\_3} = 0, \quad \phi\_3 = -W \tag{11}
$$

where *H*1, *H*2, and μ = ρυ are slip coefficients and viscosity, respectively. Equations (10) and (11) take the following form:

$$\begin{aligned} \psi\_3'(0) &= \beta\_1 \psi\_3'(0) = , \psi\_3(0) = 0, \\ \psi\_3(1) &= 1, \psi\_3'(1) = -\beta\_2 \psi\_3'(1), \\ \psi\_1(1) &= -\beta\_2 \psi\_1'(1), \psi\_1(0) - 1 = \beta\_1 \psi\_1'(0), \\ \psi\_2(1) &= -\beta\_2 \psi\_2'(1), \psi\_2(0) - 1 = \beta\_1 \psi\_2'(0). \end{aligned} \tag{12}$$

where β1 = *H*1μ/*d*, β2 = *H*2μ/*d* are slip factors. Equations (7)–(9) and (12) will be solved by the HAM. The expression for pressure can be deduced from Equations (2)–(4) as follows:

$$-\frac{p}{\rho} = \frac{\mathcal{W}^2 \Lambda \mathbf{x}\_1^2}{2d} + \frac{1}{2} \phi\_3^2 - \gamma \phi\_{3, x\_3} + A \tag{13}$$

where Λ, *A* are constants and

$$
\Lambda = \left(\psi\_3^{\prime}\right)^2 - \psi \circ \psi\_3^{\prime \prime \prime} - \frac{1}{R} \psi\_3^{\prime \prime \prime \prime} = \left(\psi\_3^{\prime}(0)\right)^2 - \frac{1}{R} \psi\_3^{\prime \prime \prime \prime}(0). \tag{14}
$$

If 2*l* is the width of the slider with ambient pressure ρ0, then Equation (13) gives

$$p - p\_0 = -\rho \frac{\Lambda \mathcal{W}^2 \{\mathbf{x}\_1^2 - l^2\}}{2d^2}. \tag{15}$$

The relationship between depth and lift can be expressed as follows:

$$L = \int\_{-1}^{1} (p - p\_0) dx = \frac{2\rho W^2 l^3}{3d^2} \Lambda. \tag{16}$$

where 2ρ*W*2*l*3/-3*d*2is normalized factor. The relationship between depth and drag in the *x*1− direction is

$$D\_{\mathbf{x}\_1} = -\int\_{-1}^{1} \mu \frac{\partial \phi\_1}{\partial \mathbf{x}\_3} \vert\_{z=d} d\mathbf{x}\_1 = -\frac{2\mu Ul}{d} \psi\_1'(1). \tag{17}$$

Similarly, 2μ*Ul*/*d* is the normalized factor of drag in the *x*1− direction, which is −ψ/1 (1), while −ψ/2(1) is normalized drag for the *x*2− direction:

$$D\_{x2} = -\int\_{-1}^{1} \mu \frac{\partial \phi\_2}{\partial x\_3} \vert\_{z=d} d\mathbf{x}\_1 = -\frac{2\mu V l}{d} \psi\_2'(1). \tag{18}$$

Similarly, from Figure 1b, a circular slider can be seen, where *L* is the radius of the slider (which can be assumed to be comparatively bigger than the width). Since the slider is levitated, the axes on the slider can be fixed so that the ground is moving with a velocity component in the *x*1− direction. For the circular slider, a similar transform [18] helps to reduce the partial differential equations into ordinary differential equations:

$$\mathbf{x}\_1 = \mathcal{U}\boldsymbol{\psi}\_5(\boldsymbol{\varsigma}) + \frac{\mathcal{W}}{d} \mathbf{x}\_1 \boldsymbol{\psi}\_4^{\prime}(\boldsymbol{\varsigma}),\\\mathbf{x}\_2 = \frac{\mathcal{W}}{d} \mathbf{x}\_2 \boldsymbol{\psi}\_4^{\prime}(\boldsymbol{\varsigma}),\\\mathbf{x}\_3 = -2\mathcal{W}\boldsymbol{\psi}\_4(\boldsymbol{\varsigma}).\tag{19}$$

With the help of Equation (19), Equations (2)–(4) take the following form

$$
\psi\_4^{\dot{v}\upsilon} - 2R\psi\_4 \psi\_4^{\upsilon'/'} - M^2 \psi\_4^{\upsilon'} = 0 \tag{20}
$$

$$
\psi\_5^{\prime \prime} - R \Big( \psi\_5 \psi\_4^{\prime} - 2 \psi\_4 \psi\_5^{\prime} \Big) - M^2 \psi\_5 = 0 \tag{21}
$$

$$-\frac{p}{\rho} = \frac{\mathcal{W}^2 \Lambda(\mathbf{x}\_1^2 + \mathbf{x}\_2^2)}{2d} + \frac{1}{2}\mathbf{x}\_3^2 - \chi \mathbf{x}\_{3,x\_3} + \mathcal{C} \tag{22}$$

in which Λ, *C* are constants and

$$
\Lambda\_1 = \left(\psi\_4^{'}(0)\right)^2 - \frac{1}{R} \psi\_4^{'//} (0). \tag{23}
$$

The boundary conditions on *x*3 = 0&*d* :

$$\begin{aligned} \psi\_4'(0) &= \beta\_1 \psi\_4'^{'/}(0) = , \psi\_4(0) = 0, \\ \psi\_4(1) &= 1/2, \psi\_4'(1) = -\beta\_2 \psi\_4^{'/}(1), \\ \psi\_5(1) &= -\beta\_2 \psi\_5'(1), \psi\_5(0) - 1 = \beta\_1 \psi\_5'(0). \end{aligned} \tag{24}$$

To normalize the lift, integrating the bottom of the slider as a result of the normalized factor can be expressed as πρ*W*2*l*4/4*d*.

$$L = \frac{4d}{\pi \rho W^2 l^4} \iint\limits\_{\text{s}} (p - p\_0) ds = \frac{1}{R^3} \Lambda. \tag{25}$$

The relationship between depth and drag in the *x*1− direction is

$$D\_{\mathbf{x}\_1} = \frac{d}{\pi \mu l L l^2} \iint\_s H\_{\mathbf{x}\_3 \mathbf{x}\_1} ds = -\frac{1}{R^3} \psi\_5^{'}(1). \tag{26}$$

*Mathematics* **2019**, *7*, 748