The Impact of Heat Transfer and a Magnetic Field on Peristaltic Transport with Slipping through an Asymmetrically Inclined Channel

Abstract

This theoretical investigation explores the intricate interplay of slip, heat transfer, and magneto-hydrodynamics (MHD) on peristaltic flow within an asymmetrically inclined channel. The channel walls exhibit sinusoidal undulations to simulate flexibility. The governing equations for continuity, momentum, and energy are utilized to mathematically represent the flow dynamics. Employing the perturbation method, these nonlinear equations are systematically solved, yielding analytical expressions for key parameters such as stream function, temperature distribution, and pressure gradient. This study meticulously examines the influence of various physical parameters on flow characteristics, presenting comprehensive visualizations of flow streamlines, fluid axial velocity profiles, and pressure gradient distributions. Noteworthy findings include the observation that the axial velocity of the fluid increases by 55% when the slip parameter is increased from 0 to 0.1, indicative of enhanced fluid transport. Furthermore, the analysis reveals that the pressure gradient amplifies by 80% with increased magnetic field strength from 0.5 to 4, underscoring the significant role of MHD effects on overall flow behavior. In essence, this investigation elucidates the complex dynamics of peristaltic flow in an asymmetrically inclined channel under the combined influence of slip, heat transfer, and magnetohydrodynamics. It sheds light on fundamental mechanisms that govern fluid dynamics in complex geometries and under diverse physical conditions.

1. Introduction

In recent years, significant attention has been directed towards the investigation of peristaltic motion, characterized by oscillations induced by transverse progressive waves propagating through compliant walls. This phenomenon is represented by a sinusoidal wave pattern. Peristaltic mechanisms are ubiquitous in various biological systems, including bile movement in the duodenum, food transit in the esophagus, urine flow from the kidney to the bladder, sperm transport in the male reproductive tract, ovum propulsion in the fallopian tubes, and blood circulation through arteries, venules, and capillaries. Additionally, peristaltic motion plays a role in fluid mixing and transport within biological tracts. In biomedical applications, the peristaltic pump is an effective tool, particularly in open-heart surgery, where conventional pumps may damage blood cells. The peristaltic pump mimics the natural motion of blood within arteries, circulating blood throughout the body and lungs without such risks. Industrially, peristaltic pumps are used in chemical laboratories for handling sterile or corrosive fluids, pumping intravenous fluids (IVs), or slurries with high fluid content. They also find applications in crude oil extraction from porous rock, fluid mixing, hygienic fluid transport, and handling erosive fluids.

This study delves into the dynamics of peristaltic motion considering various significant physical parameters. Latham [1] conducted both theoretical and experimental investigations on peristaltic fluid pumps, demonstrating how variations in velocity profiles highlight the role of viscosity in peristaltic pump operation. Notably, even in the absence of pressure differentials across the pump, both forward and backward flow occur, although Latham’s analysis neglected inertia forces. Fung and Yih [2] expanded on this work by incorporating nonlinear convective acceleration into their theoretical model. Subsequently, Yin and Fung [3] extended the analysis to axisymmetric flow within cylindrical tubes. Jaffrin and Shapiro [4] explored the influence of low and high Reynolds numbers (Re) on peristaltic motion, finding a mere 17% reduction in maximum time-mean flow under varied Reynolds numbers. Raju and Devanathan [5] investigated the peristaltic flows of non-Newtonian fluids, focusing on power-law fluids and comparing velocity profiles for Newtonian, pseudoplastic, and dilatant fluids. Finally, Siddiqui and Schwarz [6] examined the mechanics of non-Newtonian fluid peristaltic pumping through axisymmetric conduits, contrasting the behavior of non-Newtonian fluids with that of Newtonian fluids.

Recent studies have focused on the influence of internal parameters on motion dynamics, including fluid type, channel geometry, and Reynolds numbers. While these parameters are crucial in modulating motion dynamics, realistic scenarios require a consideration of external influences, such as magnetic fields (Lorentz force), which can significantly impact peristaltic flow behavior. The effects of magnetic fields span various scientific and engineering domains, including polymer extrusion, liquid crystal solidification, petroleum industries, magnetotherapy, MHD generators, cancer therapy, MHD pumps, biological fluid dynamics, artificial dialysis, hyperthermia, and arterial flow. In modern medical practices, particularly in magnetic resonance imaging (MRI), understanding the interplay between magnetic fields and peristaltic motion is of paramount importance. MRI techniques enable non-invasive, three-dimensional anatomical and physiological imaging by subjecting patients to transverse radio frequency (RF) and magnetic fields to capture internal organ images [7]. Given the prevalence of magnetic field exposure from various medical devices, comprehending their impact on peristaltic motion is essential. This understanding is critical for mitigating potential adverse effects on human health caused by device-induced perturbations in peristaltic motion dynamics.

Vishnyakov and Pavlov [8] pioneered research into the impact of magnetic fields on peristaltic flow through viscous fluids, conducting both theoretical analyses and experimental investigations. Mekheimer [9] explored the peristaltic flow of coupled stress fluids under the influence of an induced magnetic field within an asymmetric channel, determining that an increase in the magnetic parameter correlates with a rise in the pressure gradient. Nadeem and Akbar [10] investigated the effects of induced magnetic fields on peristaltic flow within vertical symmetric channels, focusing on the Johnson Segalman fluid model. Their findings indicated a reduction in the size of trapped boluses as the magnetic field intensity increased. Additionally, for the Williamson fluid model, Nadeem and Akbar [11] examined the influence of an inclined magnetic field on peristaltic flow within an inclined asymmetric channel. Rashid et al. [12] delved into the impact of induced magnetic fields on the peristaltic flow of incompressible Williamson fluid, highlighting a significantly greater pressure increment compared to viscous fluids. Hayat et al. [13] investigated the peristaltic flow of a second-order fluid in the presence of an induced magnetic field. Furthermore, Bhatti and Abbas [14] analyzed the effects of magnetohydrodynamics (MHD) on peristaltic flow, specifically focusing on the Jeffrey fluid model across a porous medium. Their results demonstrated a decrease in fluid velocity with an increase in magnetic field strength.

The influence of magnetic fields on peristaltic blood flow has garnered significant attention within biofluid dynamics. Pioneering works in this area include the study by Sud et al. [15], which explores the impact of magnetic flux on blood temperature regulation. Their findings suggest a direct proportionality between the rise in blood temperature and the wavelength of the applied magnetic field, albeit with a marginal temperature differential. Tzirtzilakis [16] developed a mathematical framework to analyze blood flow within magnetic fields, aiming to elucidate the effects on circulation. His analysis indicates a pronounced impact on deoxygenated blood within spatially varying magnetic fields, potentially reducing flow rates by up to 40% under strong magnetic influence. Further investigations by Mekheimer and Al-Arabi [17] examined the nonlinear transport dynamics of magnetohydrodynamic (MHD) blood flow. They concluded that intensified magnetic fields alter flow characteristics, with the disappearance of centerline trapped eddies and the emergence of fluid rigidity. Building upon this research, Mekheimer [18] revisited the study of peristaltic blood flow in magnetic fields, focusing on non-uniform channels and considering the fluid as a couple stress fluid. These investigations enhance our understanding of how applied magnetic fields affect various aspects of blood flow dynamics. Numerous other studies have explored the ramifications of magnetic fields on blood flow dynamics, as documented in subsequent references [19,20,21,22,23,24]. Collectively, these inquiries contribute to the evolving comprehension of the interplay between magnetic fields and blood flow behavior.

Heat transfer analysis has garnered significant attention recently due to its pivotal role in various cooling processes across different applications. In the human body, biological processes operate within specific temperature ranges, tightly regulated by blood circulation. Deviations from this range can result in tissue damage. Cooling mechanisms in the human body primarily involve free convection and radiation from the skin, aimed at maintaining optimal blood temperature. When blood temperature rises, the body initiates mechanisms to cool down, such as increasing skin cooling through perspiration or initiating fat metabolism to generate heat. Conversely, if blood temperature decreases, mechanisms like shivering are activated to generate heat. Moreover, heat transfer plays a crucial role in medical procedures such as oxygenation and hemodialysis. Recent advancements in utilizing heat (hyperthermia), radiation (laser therapy), and cold (cryosurgery) for tissue manipulation, particularly in treating conditions like cancer, have spurred interest in the thermal modeling of tissues. Hyperthermia, for instance, targets tissue destruction through controlled heating, typically in the range of 42–45 °C [25]. Earlier research investigated heat transfer within blood circulation. Charm et al. [26] examined heat transfer effects in small-diameter tubes (0.6 mm), followed by Victor and Shah [27], who studied heat transfer in fully developed blood flow within a tube with thermal convection. Ogulu and Abbey [28] simulated heat transfer during the peristaltic motion of blood flow through a stenosed porous artery, finding that magnetic fields could influence temperature. Misra et al. [29] investigated magnetic fields’ impact on heat transfer in electrically conducting viscous fluids, highlighting their role in enhancing blood temperature. Zaman et al. [30] studied heat and mass transfer effects on blood flow in tapered tubes. Heat transfer analysis extends beyond blood circulation to other fluid dynamics scenarios. Vajravelu et al. [31] analyzed peristaltic motion with heat conduction in vertical porous annuli, while Srinivas and Kothandapani [32] investigated heat transfer’s influence on the peristaltic transport of Newtonian fluids in vertical annuli under magnetic fields. El-Masry [33] explored free convection heat transfer effects on Eyring–Powell fluids with peristaltic motion, considering varying magnetic fields.

In numerous recent studies, the consideration of slip effects in fluid models has become imperative, particularly when investigating fluids that demonstrate macroscopic wall slip phenomena. This is due to the inadequacies of assuming a no-slip condition in such scenarios. Slip effects manifest in various practical applications, spanning from internal cavities to the refinement of artificial heart valves. Navier [34] introduced a boundary condition delineating fluid slip effects at solid boundaries, positing that the fluid velocity at the boundary is linearly proportional to the shear stress at that interface. In the context of a porous channel featuring elastic wall properties, Srinivas et al. [35] scrutinized the repercussions of slipping conditions on the peristaltic flow of magnetohydrodynamic (MHD) Newtonian fluids. Their findings indicated that an increasing slipping factor correlates with heightened axial velocity at both the channel’s boundaries and central region. Similarly, Ramesh and Devakar [36] explored slip effects within Casson fluid model flows, deriving analytical solutions. Gudekote and Choudhari [37] delved into the impact of slip on Casson fluid peristaltic transport within an inclined elastic tube. Examining the influence of incompressible flow, Hayat et al. [38] assessed the effects of slipping on the peristaltic motion of an incompressible viscous fluid within a non-symmetric channel. They demonstrated that increasing the slip parameter reduces the size of trapped boluses, thereby potentially improving fluid flow dynamics. In the realm of biomedical applications, Bhatti et al. [39] elucidated the influence of slip-on blood flow driven by peristaltic waves in the presence of endoscopy. Their study revealed a notable reduction in pressure difference as the slip parameter increased. Further extending this line of inquiry, Bhatti et al. [40] investigated the combined effects of magnetohydrodynamics and slip on blood flow dynamics. Their work contributes to a comprehensive understanding of how slip phenomena interplay with various fluid dynamics scenarios, particularly within biological and biomedical contexts. Alsabery et al. [41] introduce a study that develops a fluid–structure interaction model to investigate the effects of thermal treatment on blood flow within abdominal aortic aneurysms. Utilizing numerical simulations, the research meticulously examines the thermal influence on the dynamic interaction between the blood flow and the arterial wall, aiming to enhance the understanding of mechanical behaviors and improve therapeutic interventions for management. The study by Aghakhani and Hajatzadeh [42] investigates the flow of water/alumina nanofluids in a channel obstructed by hot obstacles. Employing numerical methods, they examine the impact of varying obstacle height and spacing on temperature distribution and heat transfer. By applying a downward magnetic field beneath the obstacles and the hot wall, they optimize parameters to enhance heat transfer efficiency and increase the outlet nanofluid temperature. The results demonstrate that smaller obstacles and closer spacing correlate with higher rates of heat transfer, resulting in a significant elevation of the Nusselt number compared to scenarios with larger obstacles and wider spacing.

The introduction highlights the significant attention drawn to certain effects due to their vital importance in practical applications. However, previous research predominantly relies on mathematical models with varied assumptions. Therefore, this study aims to conduct a mathematical comparison between two models: one incorporating slip conditions and the other excluding them. The objective is to discern the disparities between these models. Both models are tackled using semi-analytical methods (perturbation techniques) to minimize reliance on assumptions and better approximate real solutions.

Perturbation techniques are widely recognized as powerful tools for solving nonlinear equations. Our team has effectively employed these techniques in diverse applications, including biofluid mechanics [42,43,44] and nanofluid [45,46].

The present study delves into analyzing the effects of slip, heat transfer, and magnetic fields on peristaltic flow within an inclined asymmetric channel. This research holds practical significance, especially in biomedical applications like radiosurgery and targeted magnetic resonance radiation in medical procedures. The mathematical model is built upon conservation equations governing mass, momentum, and energy, with viscosity modeled as a function of fluid temperature. To address this model, we employ a perturbation technique using a small parameter, commonly referred to as the wave number. Our investigation explores the influence of different variables on streamlines, velocity, and pressure gradient profiles.

2. Mathematical Modeling of the Problem

In the present paper, we study the motion of a fluid through a non-symmetrical inclined channel with the width of ($d_1+d_2$) as shown in Figure 1 [47]. On the length of the channel walls, sinusoidal wave sequences propagating at constant speed (c) and wavelength (λ) produce this motion.

The equations describing the walls’ geometries are as follows:

$$\mathrm{upper}\mathrm{wall}:{\overline{h}}_1\left(\overline{X},\overline{t}\right)=d_1+a_0\cos \left[\frac{2\pi }{\lambda }\left(\overline{X}-c\overline{t}\right)\right]$$

(1)

$$\mathrm{lower}\mathrm{wall}:{\overline{h}}_2\left(\overline{X},\overline{t}\right)=-d_2-a_0\cos \left[\frac{2\pi }{\lambda }\left(\overline{X}-c\overline{t}\right)\right]$$

(2)

where ($a_0$) denotes the wave amplitude of the walls, (λ) represents the wavelength, (c) represents the peristaltic wave’s velocity, and ($\overline{t}$) stands for time. ($\overline{X}$,$\overline{Y}$) are the Cartesian coordinates, where ($\overline{X}$) stands for the direction of wave propagation and ($\overline{Y}$) is normal to it.

2.1. Fundamental Computation of Lorentz Force

To determine the Lorentz force, a uniform magnetic field $\overrightarrow{B}$ with strength (B₀) is applied in the normal direction of the axial flow (i.e., with a direction of ($\overline{Y}$)), and the influence of the magnetic field on the flow is investigated. Assume that the fluid is electrically conducting in the presence of a uniform magnetic field as $\overrightarrow{B}=\left(0,B_0,0\right)$.

$$\overrightarrow{J}=\sigma \left(\overrightarrow{V}\times \overrightarrow{B}\right)=\sigma \left[\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ 0& 0& B_{0}u\\ 0& B_0& 0\end{array}\right|\right]$$

(3)

Then, by Ohm’s law, we obtain the following:

$$\overrightarrow{J}\times \overrightarrow{B}=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ 0& 0& \sigma B_{0}u\\ 0& B_0& 0\end{array}\right|=-\sigma B_0^{2}u\overrightarrow{i}$$

(4)

where $\left(\overrightarrow{J}\right)$ denotes the current density vector, $\left(\sigma \right)$ is the electrical conductivity of the fluid, and (B₀) stands for the magnetic field strength. The effect of the magnetic field on the fluid flux is significant in the ($\overline{X})$ direction.

2.2. The Governing Equations

To describe the model mathematically, the governing equations of motion for an incompressible fluid model, through an inclined non-symmetric channel, are as follows:

The continuity equation.

$$\frac{\partial \overline{U}}{\partial \overline{X}}+\frac{\partial \overline{V}}{\partial \overline{Y}}=0$$

(5)

The momentum equations.

$$\rho \left[\frac{\partial \overline{U}}{\partial \overline{t}}+\overline{U}\frac{\partial \overline{U}}{\partial \overline{X}}+\overline{V}\frac{\partial \overline{U}}{\partial \overline{Y}}\right]=-\frac{\partial \overline{P}}{\partial \overline{X}}+2\frac{\partial }{\partial \overline{X}}\left[\overline{\mu }\left(\overline{T}\right)\frac{\partial \overline{U}}{\partial \overline{X}}\right]+\frac{\partial }{\partial \overline{Y}}\left[\overline{\mu }\left(\overline{T}\right)\left(\frac{\partial \overline{V}}{\partial \overline{X}}+\frac{\partial \overline{U}}{\partial \overline{Y}}\right)\right]-\sigma B_0^2\overline{U}+\rho g\alpha \left(\overline{T}-T_0\right)\sin \left(\phi \right)$$

(6)

$$\rho \left[\frac{\partial \overline{V}}{\partial \overline{t}}+\overline{U}\frac{\partial \overline{V}}{\partial \overline{X}}+\overline{V}\frac{\partial \overline{V}}{\partial \overline{Y}}\right]=-\frac{\partial \overline{P}}{\partial \overline{Y}}+2\frac{\partial }{\partial \overline{Y}}\left[\overline{\mu }\left(\overline{T}\right)\frac{\partial \overline{V}}{\partial \overline{Y}}\right]+\frac{\partial }{\partial \overline{X}}\left[\overline{\mu }\left(\overline{T}\right)\left(\frac{\partial \overline{V}}{\partial \overline{X}}+\frac{\partial \overline{U}}{\partial \overline{Y}}\right)\right]$$

(7)

The energy equation.

$$\rho C_p\left[\frac{\partial \overline{T}}{\partial \overline{t}}+\overline{U}\frac{\partial \overline{T}}{\partial \overline{X}}+\overline{V}\frac{\partial \overline{T}}{\partial \overline{Y}}\right]=K\left[\frac{{\partial }^2\overline{T}}{\partial {\overline{X}}^2}+\frac{{\partial }^2\overline{T}}{\partial {\overline{Y}}^2}\right]+\frac{\partial {\overline{q}}_r}{\partial \overline{Y}}+Q_0$$

(8)

where ($\overline{U},\overline{V}$) denote velocity components in the ($\overline{X})$ and ($\overline{Y})$ directions, respectively.

$\left[\overline{T},\rho ,\overline{P},Q_0,K,\alpha ,{\overline{q}}_r\right]$ stand for the temperature, the density, the pressure, constant heat addition/absorption, thermal conductivity, the coefficient of linear thermal expansion, and heat flux, respectively.

2.3. Overcome Heat

The heat flux can be expressed using Roseland approximation as follows:

$${\overline{q}}_r=-\frac{4\mathrm{ἐ}}{3K^{\prime }}\frac{\partial T^4}{\partial \overline{Y}}$$

(9)

where $\left(\mathrm{ἐ}\right)$ represents the mean absorption coefficient and ($K\prime$) stands for the Stefan–Boltzmann constant. By using Taylor expansion and eliminating higher-order variables, we may write, given that the temperature variation within the fluid mass that is flowing is sufficiently modest, the following:

$$T^4\approx 4T_0^3\overline{T}-3T_0^4$$

(10)

Substituting Equation (10) in Equation (9) results in the following:

$${\overline{q}}_r=-\frac{16\mathrm{ἐ}{T}_0^3}{3K^{\prime }}\frac{\partial T}{\partial \overline{Y}}$$

(11)

The boundary conditions, including wall slipping and convection, are as follows:

$$\mathrm{upper}\mathrm{wall}\overline{U}-A\frac{\partial \overline{U}}{\partial \overline{Y}}=0, \overline{T}=T_1 \mathrm{at}\overline{Y}={\overline{h}}_1\left(\overline{X},\overline{t}\right)$$

(12)

$$\mathrm{lower}\mathrm{wall}\overline{U}+A\frac{\partial \overline{U}}{\partial \overline{Y}}=0, \overline{T}=T_0 \mathrm{at}\overline{Y}={\overline{h}}_2\left(\overline{X},\overline{t}\right)$$

(13)

The flow phenomenon is essentially unstable in the laboratory coordinate system $\left(\overline{X},\overline{Y},\overline{t}\right)$, but it may be represented as a steady flow in a coordinate system $\left(\overline{x},\overline{y}\right)$, rotating the two frames as a laboratory coordinate, which travels with the speed of the wave with equal angular velocity. The following equations describe the connection between the two frames:

$$\overline{y}=\overline{Y},\overline{x}=\overline{X}-c\overline{t},\overline{u}=\overline{U}-c,\overline{v}=\overline{V},\overline{p}\left(\overline{x},\overline{y}\right)=\overline{P}\left(\overline{X},\overline{Y},\overline{t}\right),T=\overline{T}$$

(14)

in which $\overline{u},\overline{v},\overline{p},$ and T designate velocity components, pressure, and temperature, respectively.

By substituting Equation (14) into Equations (1) and (2), the equations of the walls’ geometry become the following:

$$\mathrm{upper}\mathrm{wall}{\overline{h}}_1\left(\overline{x},\overline{t}\right)=d_1+a_0\cos \left[\frac{2\pi }{\lambda }\overline{x}\right]$$

(15)

$$\mathrm{lower}\mathrm{wall}{\overline{h}}_2\left(\overline{x},\overline{t}\right)=-d_2-a_0\cos \left[\frac{2\pi }{\lambda }\overline{x}\right]$$

(16)

Applying Equation (14) in Equations (6)–(9), (12), and (13), the governing equations become the following:

$$\frac{\partial \left(\overline{u}+c\right)}{\partial \left(\overline{x}+c\overline{t}\right)}+\frac{\partial \overline{v}}{\partial \overline{y}}=$$

(17)

$$\begin{array}{l}\rho \left[\left(\overline{u}+c\right)\frac{\partial \left(\overline{u}+c\right)}{\partial \left(\overline{x}+c\overline{t}\right)}+\overline{v}\frac{\partial \left(\overline{u}+c\right)}{\partial \overline{y}}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-\frac{\partial \overline{p}}{\partial \left(\overline{x}+c\overline{t}\right)}+2\frac{\partial }{\partial \left(\overline{x}+c\overline{t}\right)}\left[\overline{\mu }\left(T\right)\frac{\partial \left(\overline{u}+c\right)}{\partial \left(\overline{x}+c\overline{t}\right)}\right]+\frac{\partial }{\partial \overline{y}}\left[\overline{\mu }\left(T\right)\left(\frac{\partial \overline{V}}{\partial \left(\overline{x}+c\overline{t}\right)}+\frac{\partial \left(\overline{u}+c\right)}{\partial \overline{y}}\right)\right]-\sigma B_0^2\left(\overline{u}+c\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\rho g\alpha \left(T-T_0\right)\sin \left(\varphi \right)\end{array}$$

(18)

$$\rho \left[\left(\overline{u}+c\right)\frac{\partial \overline{v}}{\partial \left(\overline{x}+c\overline{t}\right)}+\overline{v}\frac{\partial \overline{v}}{\partial \overline{y}}\right]=-\frac{\partial \overline{p}}{\partial \overline{y}}+2\frac{\partial }{\partial \overline{y}}\left[\overline{\mu }\left(T\right)\frac{\partial \overline{v}}{\partial \overline{y}}\right]+\frac{\partial }{\partial \left(\overline{x}+c\overline{t}\right)}[\overline{\mu }\left(T\right)(\frac{\partial \overline{v}}{\partial \left(\overline{x}+c\overline{t}\right)}+\frac{\partial \left(\overline{u}+c\right)}{\partial \overline{y}})]$$

(19)

$$\rho c_p\left[\left(\overline{u}+c\right)\frac{\partial T}{\partial \left(\overline{x}+c\overline{t}\right)}+\overline{v}\frac{\partial T}{\partial \overline{y}}\right]=\mathrm{K}\left[\frac{{\partial }^{2}T}{\partial {\left(\overline{x}+c\overline{t}\right)}^2}+\frac{{\partial }^2\overline{\mathrm{T}}}{\partial {\overline{\mathrm{y}}}^2}\right]+\frac{\partial {\overline{\mathrm{q}}}_{\mathrm{r}}}{\partial \overline{\mathrm{y}}}+{\mathrm{Q}}_0$$

(20)

and the boundary conditions become the following:

$$\mathrm{upper}\mathrm{wall}\left(\overline{u}+c\right)-A\frac{\partial \left(\overline{u}+c\right)}{\partial \overline{\mathrm{y}}}=0, T=T_1\mathrm{at}\overline{\mathrm{y}}={\overline{h}}_1\left(\overline{x}\right)$$

(21)

$$\mathrm{lower}\mathrm{wall}\left(\overline{u}+c\right)+A\frac{\partial \left(\overline{u}+c\right)}{\partial \overline{\mathrm{y}}}=0,T=T_2\mathrm{at}\overline{\mathrm{y}}={\overline{h}}_2\left(\overline{x}\right)$$

(22)

To simplify the equations, the following dimensionless variables are presented:

$$\begin{array}{l}y=\frac{\overline{y}}{d_1},x=\frac{\overline{x}}{\lambda },t=\frac{c\overline{t}}{\lambda },u=\frac{\overline{u}}{c},v=\frac{\lambda \overline{v}}{d_{1}c},p=\frac{d_1^2\overline{p}}{c\lambda {\mu }_0},\\ h_1=\frac{{\overline{h}}_1\left(x\right)}{d_1},h_2=\frac{{\overline{h}}_2\left(x\right)}{d_1},a=\frac{a_0}{d_1},d=\frac{d_2}{d_1},\mu \left(\theta \right)=\frac{\overline{\mu }\left(T\right)}{{\mu }_0},\\ \theta =\frac{T-T_0}{T_1-T_0},\delta =\frac{d_1}{\lambda },Kn=\frac{A}{d_1},Pr=\frac{{\mu }_0{c}_p}{K},Gr=\frac{\rho g{d}_1^2\alpha \left(T_1-T_0\right)}{c{\mu }_0},\\ M^2=\frac{\sigma d_1^2{B}_0^2}{{\mu }_0},S=\frac{d_1^2{Q}_0}{K\left(T_1-T_0\right)},Re=\frac{2\rho d_{1}c}{{\mu }_0},Nr=\frac{16\mathrm{ἐ}{T}_0^3}{3K^{\prime }k}\end{array}$$

(23)

(t) is the dimensionless time, (u, v) denote the dimensionless axial and normal component of velocity, (p) is the dimensionless pressure, (a) is the amplitudes of the lower wall and the upper wall, ($\theta$) denotes the dimensionless temperature, and (δ) is the wave number. (Kn) represents the Knudsen number, (A) is the molecules’ mean free path, (Gr) is called the Grashof number, (M) is the Hartmann number, (S) is the heat source/sink parameter, (Re) is the Reynolds number, (Nr) is the thermal radiation parameter, and (Pr) is the Prandtl number.

Using the quantities from Equation (23) in Equations (15)–(22), the equations of walls geometry become the following:

$$\mathrm{upper}\mathrm{wall}{h}_1=1+a\cos \left(2\pi x\right)$$

(24)

$$\mathrm{lower}\mathrm{wall}{h}_2=-d-a\cos \left(2\pi x\right)$$

(25)

The governing equations become the following:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(26)

$$Re\delta \left[\left(u+1\right)\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}\right]=-\frac{\partial p}{\partial x}+2{\mathsf{\delta }}^2\frac{\partial }{\partial x}\left[\mu \left(\theta \right)\frac{\partial u}{\partial x}\right]+\frac{\partial }{\partial y}\left[\mu \left(\theta \right)\left({\delta }^2\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)\right]-{\mathrm{M}}^2\left(\mathrm{u}+1\right)+Gr\theta \sin \left(\varphi \right)$$

(27)

$$Re{\delta }^3\left[\left(u+1\right)\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right]=-\frac{\partial p}{\partial y}+2{\mathsf{\delta }}^2\frac{\partial }{\partial y}\left[\mu \left(\theta \right)\frac{\partial u}{\partial y}\right]+\frac{\partial }{\partial x}{\delta }^2\left[\mu \left(\theta \right)\left({\delta }^2\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)\right]$$

(28)

$$Re\delta Pr\left[\left(u+1\right)\frac{\partial \theta }{\partial x}+v\frac{\partial \theta }{\partial y}\right]=\left[{\delta }^2\frac{{\partial }^2\mathsf{\theta }}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathsf{\theta }}{\partial {\mathrm{y}}^2}\right]+\mathrm{Nr}\frac{{\partial }^2\theta }{\partial y^2}+\mathrm{S}$$

(29)

and the boundary conditions become the following:

$$\mathrm{upper}\mathrm{wall}\mathrm{u}-Kn\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=-1, \theta =1\mathrm{at}y=h_1$$

(30)

$$\mathrm{lower}\mathrm{wall}\mathrm{u}+Kn\frac{\partial \mathrm{u}}{\partial \mathrm{y}}=-1, \theta =0\mathrm{at}y=h_2$$

(31)

For two-dimensional flow, the dimensionless stream function (ψ) expressed as

$$u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x}$$

(32)

Using the quantities in Equation (32) in Equations (26)–(29), we obtain the following:

$$\frac{{\partial }^2\psi }{\partial x\partial y}+\frac{{\partial }^2\psi }{\partial x\partial y}=0$$

(33)

$$\begin{array}{l}Re\delta \left[\left(\frac{\partial \psi }{\partial y}+1\right)\frac{{\partial }^2\psi }{\partial x\partial y}-\frac{\partial \psi }{\partial x}\frac{{\partial }^2\psi }{\partial y^2}\right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-\frac{\partial p}{\partial x}+2{\mathsf{\delta }}^2\frac{\partial }{\partial x}\left[\mu \left(\theta \right)\frac{{\partial }^2\psi }{\partial x\partial y}\right]+\frac{\partial }{\partial y}[\mu \left(\theta \right)(-{\delta }^2\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2})]-{\mathrm{M}}^2\left(\frac{\partial \mathsf{\psi }}{\partial \mathrm{y}}+1\right)+Gr\theta \sin \left(\varphi \right)\end{array}$$

(34)

$$Re{\delta }^3\left[-\left(\frac{\partial \psi }{\partial y}+1\right)\frac{{\partial }^2\psi }{\partial x^2}+\frac{\partial \psi }{\partial x}\frac{{\partial }^2\psi }{\partial \mathrm{y}\partial \mathrm{x}}\right]=-\frac{\partial p}{\partial y}-2{\mathsf{\delta }}^2\frac{\partial }{\partial y}\left[\mu \left(\theta \right)\frac{{\partial }^2\psi }{\partial x\partial y}\right]+\frac{\partial }{\partial x}{\delta }^2\left[\mu \left(\theta \right)\left(-{\delta }^2\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}\right)\right]$$

(35)

$$Re\delta Pr\left[\left(\frac{\partial \psi }{\partial y}+1\right)\frac{\partial \theta }{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial \theta }{\partial y}\right]=\left[{\delta }^2\frac{{\partial }^2\mathsf{\theta }}{\partial {\mathrm{x}}^2}+\frac{{\partial }^2\mathsf{\theta }}{\partial {\mathrm{y}}^2}\right]+\mathrm{Nr}\frac{{\partial }^2\theta }{\partial y^2}+\mathrm{S}$$

(36)

By subtracting the two equations after differentiating Equation (34) with respect to (y) and Equation (35) with respect to (x), we arrive at the following:

$$\begin{gathered} Re\delta \left[\left(\frac{\partial \psi }{\partial y}+1\right)\frac{{\partial }^3\psi }{\partial x\partial y^2}-\frac{\partial \psi }{\partial x}\frac{{\partial }^3\psi }{\partial y^3}\right]-Re{\delta }^3\left[-\left(\frac{\partial \psi }{\partial y}+1\right)\frac{{\partial }^3\psi }{\partial x^3}+\frac{\partial \psi }{\partial x}\frac{{\partial }^3\psi }{\partial y\partial x^2}\right]=2{\mathsf{\delta }}^2\frac{{\partial }^2}{\partial x\partial y}\left[\mu \left(\theta \right)\frac{{\partial }^2\psi }{\partial x\partial y}\right]+\frac{{\partial }^2}{\partial y^2}\left[\mu \left(\theta \right)\left(-{\delta }^2\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}\right)\right] \\ -2{\mathsf{\delta }}^2\frac{{\partial }^2}{\partial y\partial x}\left[\mu \left(\theta \right)\frac{\partial \psi }{\partial y\partial x}\right]+{\delta }^2\frac{{\partial }^2}{\partial x^2}\left[\mu \left(\theta \right)\left(-{\delta }^2\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}\right)\right]-{\mathrm{M}}^2\frac{{\partial }^2\psi }{\partial y^2}+Gr\frac{\partial \theta }{\partial y}\sin \left(\varphi \right) \end{gathered}$$

(37)

$\mu \left(\theta \right)=1-\alpha \theta$ where α << 1.

Volume Flow Rate

In a laboratory frame, the fluid instantaneous volume flow rate can be calculated by

$$\overline{Q}={{\displaystyle \int }}_{{\overline{h}}_2\left(\overline{X},\overline{t}\right)}^{{\overline{h}}_1\left(\overline{X},\overline{t}\right)}\overline{U}\left(\overline{X},\overline{Y},\overline{t}\right)d\overline{Y}$$

(38)

Similarly, the volume flow rate in a wave frame can be calculated by the following:

$$\overline{q}={{\displaystyle \int }}_{{\overline{h}}_2\left(\overline{x},\overline{t}\right)}^{{\overline{h}}_1\left(\overline{x},\overline{t}\right)}\overline{u}\left(\overline{x},\overline{y}\right)d\overline{y}$$

(39)

Using the conversions from Equation (14) in Equation (38), and with Equation (39), the relation between volumetric flow rates becomes

$$\overline{Q}=\overline{q}+c\left({\overline{h}}_1\left(\overline{x},\overline{t}\right)-{\overline{h}}_2\left(\overline{x},\overline{t}\right)\right)$$

(40)

The mean flow over a period T = $\left(\frac{\lambda }{c}\right)$ at a fixed position is given by

$$\tilde{Q}=\frac{1}{T}{{\displaystyle \int }}_0^T\overline{Q}dt$$

(41)

By substituting (41) into (40), we obtain

$$\tilde{Q}=q+c\left(d_1+d_2\right)$$

(42)

Let (Q) be the dimensionless time mean flow, where

$$F=\frac{q}{c{d}_1},Q=\frac{\overline{Q}}{c{d}_1}$$

(43)

We derive the next relations by using Equation (42) as such:

$$Q=F+1+d$$

(44)

and

$$F={{\displaystyle \int }}_{h_2\left(x\right)}^{h_1\left(x\right)}\frac{\partial \psi }{\partial y}dy=\psi \left(h_1\right)-\psi \left(h_2\right)$$

(45)

3. Method of Solution

To tackle the complexities of the equations, the perturbation method comes into play. This involves leveraging a small parameter to navigate the intricacies of strictly nonlinear differential equations, given the inherent difficulty in obtaining an exact solution. This method proves beneficial as it allows for the derivation of an approximate solution by initially solving a simpler and analogous problem. This strategic approach stands out for its effectiveness, yielding a converging series of solutions that progressively refine our understanding of the problem at hand. In order to unveil this series solution, a crucial step involves expanding certain parameters (ψ, θ, p, and F) in terms of a small parameter denoted as δ. This enables us to articulate the flow quantities using the perturbation technique, providing a nuanced and iterative approach to approaching and solving complex problems in mathematical modeling. This expansion essentially breaks down the problem into manageable components, allowing for a systematic analysis of the system’s behavior under different conditions. By incrementally refining our approximation through successive terms in the perturbation series, we are able to gain insight into the intricacies of the system dynamics, ultimately leading to a more comprehensive understanding and potentially more accurate predictions of real-world phenomena. Therefore, the flow quantities can be written as follows using the perturbation technique:

$$\begin{array}{l}\psi ={\psi }_0+\delta {\psi }_0+{\delta }^2{\psi }_1+\cdots \\ \theta ={\theta }_0+\delta {\theta }_1+{\delta }^2{\theta }_2+\cdots \\ p=p_0+\delta p_1+{\delta }^2{p}_2+\cdots \\ F=F_0+\delta F_1+{\delta }^2{F}_2+\cdots \end{array}$$

(46)

Zero Order System

$$\frac{{\partial }^2{\theta }_0}{\partial y^2}=-\frac{S}{1+Nr}$$

(47)

$$\frac{{\partial }^4{\psi }_0}{\partial y^4}-M^2\frac{{\partial }^2{\psi }_0}{\partial y^2}=-Gr\frac{\partial {\theta }_0}{\partial y}\sin \left(\varphi \right)$$

(48)

$$\frac{\partial p_0}{\partial x}=\frac{{\partial }^3{\psi }_0}{\partial y^3}-M^2\left(\frac{\partial {\psi }_0}{\partial y}+1\right)+Gr\theta \sin \left(\varphi \right)$$

(49)

These are the boundary conditions:

$$\mathrm{upper}\mathrm{wall}{\psi }_0=\frac{F_0}{2},\frac{\partial {\mathsf{\psi }}_0}{\partial \mathrm{y}}-Kn\frac{{\partial }^2{\mathsf{\psi }}_0}{\partial {\mathrm{y}}^2}=-1,{\theta }_0=1\mathrm{at}y=h_1$$

(50)

$$\mathrm{lower}\mathrm{wall}{\psi }_0=-\frac{F_0}{2},\frac{\partial {\mathsf{\psi }}_0}{\partial \mathrm{y}}+Kn\frac{{\partial }^2{\mathsf{\psi }}_0}{\partial {\mathrm{y}}^2}=-1,{\theta }_0=0\mathrm{at}y=h_2$$

(51)

The solution of the zero-order system, obtained by solving Equations (47)–(49) using the boundary conditions in Equations (50) and (51), is as follows:

$${\theta }_0=-\frac{S}{2\left(1+\mathrm{Nr}\right)}{y}^2++{\mathrm{N}}_1\mathrm{y}+{\mathrm{N}}_2$$

(52)

$${\psi }_0={\mathrm{e}}^{My}\left[C_1+{\mathrm{e}}^{-2My}{\mathrm{C}}_2\right]+{\mathrm{C}}_{3}y+{\mathrm{C}}_4-\frac{{\mathrm{y}}^2\left(-3{\mathrm{N}}_4+{\mathrm{N}}_3\mathrm{y}\right)}{6{\mathrm{M}}^2}$$

(53)

$$\frac{\partial p_0}{\partial x}-\frac{{\mathrm{N}}_3+{\mathrm{M}}^4\left(1+{\mathrm{C}}_3\right)}{{\mathrm{M}}^2}+{\mathrm{N}}_1\mathrm{Gr}\sin \left(\varphi \right)$$

(54)

First order system:

$$\frac{{\partial }^2{\mathsf{\theta }}_1}{\partial {\mathrm{y}}^2}=\frac{1}{1+Nr}\left(PrRe\left[\frac{\partial {\psi }_0}{\partial y}\frac{\partial {\theta }_0}{\partial x}+\frac{\partial {\theta }_0}{\partial x}-\frac{\partial {\psi }_0}{\partial x}\frac{\partial {\theta }_0}{\partial y}\right]\right)$$

(55)

$$\frac{{\partial }^4{\psi }_1}{\partial y^4}-M^2\frac{{\partial }^2{\psi }_1}{\partial y^2}=-Gr\frac{\partial {\theta }_1}{\partial y}\sin \left(\varphi \right)+Re\left[\frac{\partial {\psi }_0}{\partial y}\frac{{\partial }^3{\psi }_0}{\partial x\partial y^2}+\frac{{\partial }^3{\psi }_0}{\partial x\partial y^2}-\frac{\partial {\psi }_0}{\partial x}\frac{{\partial }^3{\psi }_0}{\partial y^3}\right]$$

(56)

$$\frac{\partial p_1}{\partial x}=\frac{{\partial }^3{\psi }_1}{\partial y^3}-M^2\frac{\partial {\psi }_1}{\partial y}+Gr{\theta }_1\sin \left(\varphi \right)-Re\left[\frac{\partial {\psi }_0}{\partial y}\frac{{\partial }^2{\psi }_0}{\partial x\partial y}+\frac{{\partial }^2{\psi }_0}{\partial x\partial y}-\frac{\partial {\psi }_0}{\partial x}\frac{{\partial }^2{\psi }_0}{\partial y^2}\right]$$

(57)

These are the boundary conditions:

$$\mathrm{upper}\mathrm{wall}{\psi }_1=\frac{F_1}{2},\frac{\partial {\mathsf{\psi }}_1}{\partial \mathrm{y}}-Kn\frac{{\partial }^2{\mathsf{\psi }}_1}{\partial {\mathrm{y}}^2}=0,{\mathsf{\theta }}_1=1\mathrm{at}y=h_1$$

(58)

$$\mathrm{lower}\mathrm{wall}{\psi }_1=-\frac{F_1}{2},\frac{\partial {\mathsf{\psi }}_1}{\partial \mathrm{y}}+Kn\frac{{\partial }^2{\mathsf{\psi }}_1}{\partial {\mathrm{y}}^2}=0,{\mathsf{\theta }}_1=0\mathrm{at}y=h_2$$

(59)

The solution of the first-order system is as follows:

$${\theta }_1=\frac{\begin{array}{c}{\mathrm{e}}^{-\mathrm{My}}\mathrm{PrRe}(60{\mathrm{e}}^{2\mathrm{My}}\left(\left(-2+My\right){\mathrm{N}}_7+M{\mathrm{N}}_9\right)+60\left(\left(2+My\right){\mathrm{N}}_8+M{\mathrm{N}}_{10}\right)\\ +{\mathrm{e}}^{\mathrm{My}}{\mathrm{M}}^3\left(60y{\mathrm{C}}_5+60{\mathrm{C}}_6+3{\mathrm{y}}^5{\mathrm{N}}_{11}+5{\mathrm{y}}^2\left({\mathrm{y}}^2{\mathrm{N}}_{12}+2y{\mathrm{N}}_{13}+6{\mathrm{N}}_{14}\right)\right))\end{array}}{60{\mathrm{M}}^3\left(1+\mathrm{Nr}\right)}$$

(60)

$$\begin{array}{l}{\psi }_1=\frac{1}{60{\mathrm{M}}^6}{\mathrm{e}}^{-\mathrm{My}}\left(5{\mathrm{e}}^{2\mathrm{My}}\mathrm{M}\right(12{\mathrm{M}}^5{\mathrm{C}}_7+\mathrm{y}(51+\mathrm{My}(-15+2\mathrm{My}\left)\right){\mathrm{V}}_1+3\mathrm{My}(-5+\mathrm{My}){\mathrm{V}}_3+6\mathrm{M}(-2+\mathrm{My}){\mathrm{V}}_5)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-5\mathrm{M}(-12{\mathrm{M}}^5{\mathrm{C}}_8+\mathrm{y}(51+\mathrm{My}(15+2\mathrm{My})){\mathrm{V}}_2+3\mathrm{M}(\mathrm{My}(5{\mathrm{M}}^7+\mathrm{y}){\mathrm{V}}_4+2(2+\mathrm{My}){\mathrm{V}}_6\left)\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathrm{e}}^{\mathrm{My}}(60{\mathrm{M}}^6{\mathrm{yC}}_9+60{\mathrm{M}}^6{\mathrm{C}}_{10}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\mathrm{y}}^2\left(2\right(360+30{\mathrm{M}}^2{\mathrm{y}}^2+{\mathrm{M}}^4{\mathrm{y}}^4){\mathrm{V}}_7+{\mathrm{M}}^2(3\mathrm{y}(20+{\mathrm{M}}^2{\mathrm{y}}^2){\mathrm{V}}_8+60{\mathrm{V}}_9+5{\mathrm{M}}^2({\mathrm{y}}^2{\mathrm{V}}_9+2{\mathrm{yV}}_{10}+6{\mathrm{V}}_{11})\left)\right)\left)\right)\end{array}$$

(61)

$$\begin{array}{l}\frac{\partial p_1}{\partial x}=\frac{1}{60{\mathrm{M}}^4\left(1+\mathrm{N}\mathrm{r}\right)}{\mathrm{e}}^{-\mathrm{M}\mathrm{y}}\left(\right(1\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{N}\mathrm{r}\left)\right(60{\mathrm{e}}^{2\mathrm{M}\mathrm{y}}\mathrm{M}(2{\mathrm{V}}_1\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{M}(-{\mathrm{V}}_3-2{\mathrm{V}}_1\mathrm{y}+\mathrm{M}({\mathrm{e}}^{-\mathrm{M}\mathrm{y}}{\mathrm{M}}^3\mathrm{R}\mathrm{e}{\mathrm{C}}_1({\mathrm{C}}_2+{\mathrm{e}}^{\mathrm{M}\mathrm{y}}({\mathrm{e}}^{\mathrm{M}\mathrm{y}}{\mathrm{C}}_1+\mathrm{y}{\mathrm{C}}_3+{\mathrm{C}}_4\left)\right)+{\mathrm{V}}_5+\mathrm{y}({\mathrm{V}}_3+{\mathrm{V}}_1\mathrm{y})\left)\right))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-30\mathrm{M}(-2{\mathrm{e}}^{\mathrm{M}\mathrm{y}}{\mathrm{M}}^5\mathrm{R}\mathrm{e}{\mathrm{C}}_1{\mathrm{C}}_2-2{\mathrm{e}}^{-\mathrm{M}\mathrm{y}}{\mathrm{M}}^5\mathrm{R}\mathrm{e}{\mathrm{C}}_2+4{\mathrm{V}}_2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{M}(-2{\mathrm{M}}^4\mathrm{R}\mathrm{e}{\mathrm{C}}_2(\mathrm{y}{\mathrm{C}}_3+{\mathrm{C}}_4)-3{\mathrm{V}}_4+5{\mathrm{M}}^8{\mathrm{V}}_4+4\mathrm{y}{\mathrm{V}}_2+2\mathrm{M}({\mathrm{V}}_6+\mathrm{y}({\mathrm{V}}_4+\mathrm{y}{\mathrm{V}}_2)\left)\right))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathrm{e}}^{\mathrm{M}\mathrm{y}}(-360{\mathrm{V}}_8+60{\mathrm{M}}^2\mathrm{R}\mathrm{e}({\mathrm{e}}^{\mathrm{M}\mathrm{y}}{\mathrm{C}}_1+{\mathrm{e}}^{-\mathrm{M}\mathrm{y}}{\mathrm{C}}_2\left)\right({\mathrm{H}}_1-\mathrm{y}{\mathrm{H}}_2)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathrm{M}}^2\left(60{\mathrm{H}}_1\mathrm{R}\mathrm{e}\right(\mathrm{y}{\mathrm{C}}_3+{\mathrm{C}}_4)-60{\mathrm{M}}^4{\mathrm{C}}_9-60({\mathrm{V}}_{10}+{\mathrm{H}}_0\mathrm{R}\mathrm{e}\mathrm{y}(\mathrm{y}{\mathrm{C}}_3+{\mathrm{C}}_4))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathrm{M}}^2\mathrm{y}(60{\mathrm{V}}_{11}+\mathrm{y}(30{\mathrm{V}}_{10}+\mathrm{y}(20{\mathrm{V}}_9+15{\mathrm{V}}_8\mathrm{y})\left)\right)\left)\right))\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\mathrm{G}\mathrm{r}\mathrm{R}\mathrm{e}\left(60\mathrm{M}\mathrm{P}\mathrm{r}\right(2{\mathrm{N}}_2+{\mathrm{N}}_4\mathrm{M}+\mathrm{y}{\mathrm{N}}_2\mathrm{M})+30{\mathrm{M}}^3(1+\mathrm{N}\mathrm{r}\left)\mathrm{y}{\mathrm{C}}_2\right(2+\mathrm{M}\mathrm{y}){\mathrm{N}}_1\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathrm{e}}^{\mathrm{M}\mathrm{y}}\left({\mathrm{M}}^4\mathrm{P}\mathrm{r}\right(60{\mathrm{C}}_6+60\mathrm{y}{\mathrm{C}}_5+30{\mathrm{N}}_8{\mathrm{y}}^2+10{\mathrm{N}}_7{\mathrm{y}}^3+5{\mathrm{N}}_6{\mathrm{y}}^4)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-30\mathrm{y}\left(1+\mathrm{N}\mathrm{r}\right)\left(2{\mathrm{M}}^2\right(1+{\mathrm{C}}_3)+{\mathrm{H}}_1\mathrm{y}){\mathrm{N}}_1)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+30{\mathrm{e}}^{2\mathrm{M}\mathrm{y}}\mathrm{M}\left(2\mathrm{P}\mathrm{r}\right(-2{\mathrm{N}}_1+{\mathrm{N}}_3\mathrm{M}+{\mathrm{N}}_1\mathrm{M}\mathrm{y})+{\mathrm{M}}^2(1+\mathrm{N}\mathrm{r}\left){\mathrm{C}}_1\mathrm{y}\right(-2+\mathrm{M}\mathrm{y}\left){\mathrm{N}}_1\right)\left)\sin \left(\mathsf{\varphi }\right)\right)\end{array}$$

(62)

$$\mathsf{\Delta }p={{\displaystyle \int }}_0^1\frac{\partial p}{\partial x}dx$$

(63)

where the symbols ${\mathrm{N}}_1$ through ${\mathrm{N}}_{14}$, ${\mathrm{C}}_1$ through ${\mathrm{C}}_{10}$, and ${\mathrm{V}}_1$ through ${\mathrm{V}}_{11}$ represent constants with their respective values detailed in the Supplementary Materials.

4. Discussion

4.1. Model Validation

The results are compared to those presented by Kothandapani and Srinivas [48]. Both models were brought to the same conditions for comparison by considering equal parametric values in the absence of a magnetic field and neglecting slip effects in the current study. Heat transfer effects were neglected by setting Gr = 0 for comparison purposes. However, in the analytical solution of the velocity profiles presented by Kothandapani and Srinivas [48], the porous media effect was neglected by considering a large value of the permeability parameter. Figure 2 demonstrates that the results of the current research are in good agreement with those of the prior study.

4.2. Graphical Results and Discussions

The main purpose of this study is concerned with the combined interaction of the most important physical parameters with the streamline, axial velocity, temperature profile, and pressure gradient. The graphical results of the solutions are displayed and disused in this section. The present study included different cases for $\mathrm{Kn}=0$ (absences of slipping) and $\mathrm{Kn}\ne 0$ (presence of slipping).

Figure 3 illustrates axial velocity profiles observed at various positions along the channel. Within the region where 0 ≤ x ≤ 0.5, there is an elevation in velocity attributed to a constricted flow area. Conversely, in the range of (0.5 ≤ x ≤ 1), a deceleration in velocity is discerned owing to an expanded cross-sectional area. This phenomenon can be elucidated by considering the principle of conservation of mass. As the channel narrows (0 ≤ x ≤ 0.5), the same mass of fluid must pass through a reduced cross-sectional area, resulting in an increase in velocity to maintain a constant mass flow rate. Conversely, in the enlarged cross-section (0.5 ≤ x ≤ 1), the fluid experiences a larger area through which to pass, necessitating a reduction in velocity to uphold mass conservation principles. The symmetrical velocity profiles at x = 0.25 and x = 0.75 can be attributed to the identical cross-sectional areas at these positions, indicating a balanced distribution of fluid velocity. This symmetry exemplifies the influence of channel geometry on fluid flow characteristics, providing valuable insights for engineering and fluid dynamics applications.

Figure 4 stands for the effect of the magnetic parameter (M) on fluid flow, where Figure 4a,b represent the velocity profiles of the fluid at various values of the magnetic field parameters. It is observed that the axial velocity in the no-slip condition, Figure 4a, increases with increasing M in $\left(-0.9\le y\le -0.4\right)$. Then, the velocity decreases in the interval $\left(-0.4\le y\le 0.4\right)$. Finally, the velocity increases in $\left(0.4\le y\le 0.9\right)$. In the slipping condition, Figure 4b, the same effect appears but with an increase in the magnitude of the axial velocity. The change in the velocity in the first and third intervals is small; the meaningful change appears in the second interval. Figure 4c,d illustrate the impact of a magnetic field on total velocity; a magnetic field parallel to the lower wall increases transverse velocity in the field’s direction, which raises the total velocity in the channel’s lower half, as shown in Figure 4c.

Figure 5 stands for the effect of the magnetic parameter (M) on the flow pressure gradient, and Figure 6 stands for the effect of the magnetic parameter (M) on the flow pressure difference. It is noted that with increasing (M), $\left(\frac{\partial p}{\partial x}\right)$ the increases, especially with the increase at (Kn = 0.05), as the slipping increases the flow velocity, as shown in Figure 4, so the pressure gradient between two points increases, as shown in Figure 5. The effect of magnetic field (M) on pressure difference (Δp) is shown in Figure 6a,b. It is noted that the (Δp) increases with the increase in (M).

In Figure 7, we can see how the magnetic field (M) affects the flow patterns. When the strength of the magnetic field increases, a swirling motion, known as a vortex, starts to form near the lower wall due to the presence of the magnetic field near that surface. This vortex causes a slowdown in the forward velocity, resulting in an overall decrease in the speed, as shown in Figure 4. At the same time, the growing magnetic field makes the vortex wider, creating more resistance in the flow direction. This heightened resistance contributes to an increase in pressure, as depicted in Figure 6.

Figure 8 represents the influence of the walls’ amplitude (a) on the flow velocity, where Figure 8a,b depicts the behavior of fluid axial velocity profile plotted versus different amplitude values (a). It is noticed that the axial velocity rises in the range $\left(-0.5\le y\le 0.5\right)$ with increasing amplitude (a), but close to the walls, the velocity reduces with the increase in amplitude (a). Meanwhile, Figure 8c,d demonstrate the effect of (a) on fluid total velocity. It is noted that with increasing (a), the total velocity increases in the center of the channel and near the walls, and this effect increases with an increasing slipping parameter, as shown in Figure 8b,d.

Figure 9 represents the influence of the walls’ amplitude (a) on the flow pressure gradient. In Figure 9a, the pressure gradient is shown to grow with the increment in amplitude (a), and it takes on the same form as a peristaltic wave. When sliding occurs, the pressure gradient increases as the velocity increases because of slipping. With slipping, the value of the pressure gradient increases as the velocity increases due to slipping, as shown in Figure 9b.

Figure 10a,b illustrate the impact of amplitude (a) on the pressure difference, demonstrating how the pressure difference (Δp) increases as the amplitude of the wall increases (a), this effect increases by increasing slipping as shown in Figure 10b.

Figure 11a,b elucidate the streamlines within the context of varied amplitudes of the wall (a), thereby delineating a discernible alteration in the geometric characteristics of the fluid conduit. This alteration manifests as a reduction in the cross-sectional area at the contraction segment, juxtaposed with a concomitant augmentation in the cross-sectional area at the expansion segment. This pronounced modification in geometry instigates a consequential shift in the profile of the pressure gradient, as exemplified in Figure 10. The diminution of the flow cross-section, as highlighted in the aforementioned figures, precipitates an escalation in fluid velocity, as substantiated by the observations in Figure 8. This augmentation in fluid velocity is concurrently associated with a corresponding reduction in flow pressure, as explicated in Figure 10. The interplay of these parameters underscores the intricate dynamics governing the fluid flow within the defined system, providing valuable insights into the impact of varying wall amplitudes on flow characteristics.

In Figure 12, the discernible impact of the heat source/sink parameter (S) on fluid flow velocity is depicted. Figure 12a specifically illustrates the variation in axial velocity because of employing the heat source/sink parameter (S), considering both no-slip and slipping conditions. Under the no-slip condition, an observable augmentation in axial velocity is observed with the introduction of the heat source/sink (S). Conversely, in Figure 12b, the effect of slipping is investigated, revealing a proportional increase in the velocity profile as the degree of slipping intensifies. This experimental investigation underscores the intricate interplay between heat source/sink parameters and fluid dynamics. The observed changes in axial velocity provide valuable insights into the nuanced relationship between thermal influences and the resulting fluid flow characteristics. Such findings contribute to a deeper understanding of heat transfer mechanisms and their repercussions on fluid behavior in the context of various slip conditions, thereby enriching the scientific comprehension of these complex phenomena.

The effect of ($\beta$) on pressure difference (Δp) is shown in Figure 13a,b. The pressure difference (Δp) linearly reduces with the increase in the heat source ($\beta$), because increasing the flow velocity increases the heat source ($\beta$).

The significant influence of the heat source/sink (S) appears on temperature distribution, as shown in Figure 14, where increasing the heat source enhances the heat transfer curve and the flow of heat transfer, leading to reduced fluid viscosity and leading to increased fluid velocity, which reduces the flow pressure as in Figure 13.

The effect of geometric parameters (δ) on the flow behavior is demonstrated in Figure 15, where Figure 15a,b elucidate that the pressure difference increases with increasing (δ) in the slipping and no-slipping conditions.

The effect of gravity (Gr) on the flow behavior is demonstrated in Figure 16, where increasing gravity (Gr) reduces the pressure difference, as demonstrated in Figure 16a,b.

Figure 17 demonstrates the influence of thermal radiation (Nr) on the flow pressure, whereas Figure 17a illustrates the pressure gradient reduction with the increment of the thermal radiation parameter. At the same value of thermal radiation, the value of the pressure gradient increases with slipping (Kn = 0.05), as shown in Figure 17b.

As shown in Figure 18, the temperature decreases with thermal radiation (Nr) for the slipping and non-slipping conditions. Without thermal radiation (Nr = 0), the temperature reaches its maximum value at $\left(y=0.5\right)$; this maximum value decreases with the increase in (Nr).

Figure 19 represent the effect of the inclination of the pipe (ϕ) on the fluid’s pressure difference. It is noted that increasing incidence of the pipe, the pressure decreases, as the gravity force on the flow increases in the flow direction.

Figure 20 depicts the effect of flow rate (Q) on fluid velocity, where the axial velocity increases with flow rate (Q) at the same cross-sectional area, as shown in Figure 20a. With slipping, the velocity is enhanced more, as shown in Figure 20b.

Figure 21 shows that, despite the effect of flow rate (Q) on the flow pressure, as shown in Figure 21a,b, as the flow rate (Q) increases, the pressure gradient is observed to rise. For that, the pressure difference increase is as shown as well in Figure 22a,b.

Figure 23a,b demonstrate the influence of increasing the flow rate on streamlines. It is observed that increasing fluid flow rate (Q) smooths fluid flow in the slip and non-slip conditions.

The effect of slipping (Kn) on the flow velocities is shown in Figure 24. The increment of slipping leads to increasing axial velocity in the interval (−0.5 ≤ y ≤ 0.5) in the channel medial and increasing velocity in the opposite direction, as shown in Figure 24a, because increasing slipping leads to the separation of the fluid of the wall. Figure 24b illustrates the impact of slipping on the total velocity of the fluid. Specifically, the axial velocity within the core increases in the positive direction, while in proximity to the wall, it rises in the negative direction. Consequently, the total velocity exhibits an increase in both directions, leading to the emergence of a distinct point where the influence of slipping diminishes. In other words, at this point, the velocity remains unaffected by the progressive increase in the slipping parameter. This observation underscores the existence of a critical juncture where the inherent dynamics of slipping cease to significantly alter the fluid velocity, thereby providing a nuanced understanding of the intricate interplay between slipping conditions and total velocity within the fluid system under consideration.

Figure 25 shows the pressure gradient at various values of slipping. The pressure gradient is observed to increase due to the increase in fluid velocity. The presence of slipping parameters leads to enhancing the flow characteristics as it reduces the friction near the channel walls, leading to reduced flow resistance at the walls.

Figure 26 shows the pressure difference at various values of slipping. The pressure difference is observed to increase due to the increasing pressure gradient, as shown in Figure 25.

5. Conclusions

This paper presents a comprehensive theoretical investigation of the transport phenomena exhibited by Newtonian biofluids, with a specific focus on the intricate interplay between heating conditions and the presence of a magnetic field. The study is centered on peristaltic flow occurring in an asymmetrically inclined channel, designed to emulate the fluid motion observed in living organisms. The peristaltic motion along the channel walls is mathematically represented as a sinusoidal wave, providing a basis for understanding fluid flow dynamics. Governing equations including continuity, momentum, and energy equations are employed to construct a mathematical framework for the model. The analysis rigorously examines fundamental fluid flow characteristics such as fluid velocity, temperature distribution, pressure gradient, and streamlines. External factors such as the magnetic field, heat source, and Reynolds number are carefully considered, with special attention given to slipping conditions within the channel.

Due to the nonlinear nature of the governing equations, specialized methods for solving are necessary. Perturbation methods are utilized in this context to analytically solve the model equation. This systematic approach aims to elucidate the complexities of peristaltic flow in biofluids, offering insights into the influence of slipping conditions on transport processes within living organisms. The primary objective of this study is to advance understanding regarding the implications of slipping conditions on biofluid flow, with potential applications spanning various industries. The scientific analysis presented contributes significantly to a broader comprehension of fluid dynamics in biological systems, holding promise for practical applications across diverse fields.

Empirical findings derived from our comprehensive investigation are summarized as follows:

• As the heat source (S) increases, the temperature distribution increases.
• An increase in thermal radiation (Nr) reduces the temperature.
• Increasing thermal radiation (Nr) lowers the temperature of the fluid.
• The axial velocity in the middle of the pipe decreases as the magnetic field is increased, while it increases close to the walls.
• As the flow rate increases, the axial velocity increases.
• The presence of slipping leads to smoothing the flow by reducing the existing vortex.
• An increase in slipping (Kn) leads to an increase in axial velocity.
• The Prandtl number (Pr) and Reynolds number (Re) have a small influence on the peristaltic flow.
• The pressure difference is reduced by increasing gravity (Gr).
• An increase in magnetic field (M), heat source (S), thermal radiation (Nr), flow rate (Q), and flow rate leads to an increasing pressure difference (Δp).

Future research:

Comparative analysis of mathematical models for non-Newtonian fluids, focusing on accurately representing biological fluids like blood.

Investigation of various forces (damping, wall tension, expansion) acting on fluid boundaries in conduits for both Newtonian and non-Newtonian fluids.

• Study of the effects of nanoparticles on flow characteristics and the impact of peristaltic motion and fluid rotation within circular channels.