Peristaltic flow of magnetized hyperbolic tangent fluid through an asymmetric channel in the presence of homogeneous-heterogeneous reactions

Abstract

The hyperbolic tangent fluid model accurately captures the shear-thinning phenomenon and is widely used in both laboratory experiments and industrial applications. The primary aim of this research is to investigate the key characteristics of the peristaltic flow of electrically conducting hyperbolic tangent fluid through an asymmetric channel, considering the effects of homogeneous and heterogeneous reactions. Slip conditions at the sinusoidal channel walls are also incorporated. The resulting nonlinear equations are solved using the lubrication theory approximation, which is particularly relevant for predicting physiological characteristics, such as blood flow dynamics. Analytical solutions for velocity, pressure gradient, streamlines, and concentration distributions are obtained using the regular perturbation technique. The influence of various parameters on these distributions is presented through graphical analysis. Additionally, the pressure rise per wavelength along the channel walls is computed numerically. The results indicate that homogeneous and heterogeneous reaction parameters exhibit opposing effects on the concentration field. Furthermore, increasing the magnetic parameter leads to a reduction in fluid velocity. This study has practical applications, including the use of peristaltic pumps in the chemical and pharmaceutical industries, particularly for drug delivery systems.

Keywords

Peristaltic flow hyperbolic tangent fluid MHD slip conditions homogeneous-heterogeneous reactions

Introduction

Peristalsis is a process of moving fluid along a channel by deforming the surface. It involves squeezing and relaxing a flexible vessel to push the fluid through it. This process occurs in the human body, for example, when blood and urine are transported from the kidney to the bladder by the contraction and expansion of muscles. Peristalsis is also used to transfer fluid that should not come into contact with the mechanical parts of a pump, such as radioactive, toxic, or corrosive fluids in nuclear reactors. The earliest study of peristalsis flow was probably done by Latham¹ for his MS thesis. Later, Shapiro et al.² improved the understanding of peristalsis by examining the effects of reverse flow and pumping waves. Following their contributions, a substantial study has been undertaken on various flow patterns. Hayat et al.³ investigated peristaltic motion within an angled tube using a magnetized Williamson liquid while accounting for heat and mass transmission. Using nanotechnology, Abbasi et al.⁴ looked at how hall current and entropy optimization affected peristaltic activity. Riaz et al.⁵ scrutinized the peristaltic motion of nano-sized elements in a curved conduit within a porous media, accounting for second-order slip phenomena. The heat transfer phenomena on peristaltic transportation of the hydromagnetic flow of non-Newtonian fourth-grade fluid in a tapered asymmetric channel filled with porous media was inspected by Rafiq et al.⁶ The significance of double-diffusivity convection and an inclined magnetic field on peristaltic propulsion of fourth-grade nanofluids through an inclined asymmetric channel was reported by Akram et al.⁷ Khan et al.⁸ investigated the impact of double-diffusive convection and inclined magnetic field in nanofluids on the peristaltic pumping of fourth-grade fluid in a non-uniform channel. Rafiq et al.⁹ examined the mixed convective movement of magnetized couple stress fluid across a curved structure, utilizing the lubrication concept as a foundation. Kada et al.¹⁰ researched the shear-dependent peristalsis movement of non-Newtonian liquid in a microchannel with ciliated edges. Shaheen et al.¹¹ examined metachronal waves within a ciliated porous channel, with an emphasis on Williamson liquid with viscous dissipation. The effect of homogeneous and heterogeneous reactions on the peristaltic blood transportation of electrically conducting Williamson fluid in an asymmetric microchannel under velocity slip conditions was reported by Abbas et al.¹² Recently, Reddy et al.¹³ discussed the electro-osmotic flow of entropy generation analysis for peristaltic movement in a nanofluid with temperature-dependent viscosity under lubrication theory.

The study of non-Newtonian fluids has gained significant attention due to their relevance in various engineering and industrial applications.^14–16 Examples of such fluids include coal-oil slurries, paints, clay suspensions, coatings, cosmetic products, grease, custard, and physiological liquids. The tangent hyperbolic fluid model is one such non-Newtonian model, widely used to describe shear-thinning behavior and the effect of normal stresses. This model employs the hyperbolic tangent function, often denoted as “tanh,” to capture the relationship between shear stress and shear rate. It is particularly useful for fluids that demonstrate either shear-thinning or shear-thickening behavior, where the viscosity changes with the shear rate. Unlike Newtonian fluids, whose viscosity remains constant regardless of the shear rate, non-Newtonian fluids such as hyperbolic tangent fluids exhibit variable viscosity, which can be accurately modeled using the tanh function. This allows researchers and engineers to better predict and simulate fluid behavior in various applications, including in the food industry, cosmetics, and polymer processing. Several studies have focused on the peristaltic flow of hyperbolic tangent fluids. Nadeem and Akram¹⁷ discussed the peristaltic motion of tangent hyperbolic tangent fluid in an asymmetric channel under the lubrication approximation theory. The heat and mass transfer phenomena in peristaltic transportation of hyperbolic tangent fluid through a tapered channel were discussed by Abbas and Rafiq.¹⁸ Naduvinamani et al.¹⁹ investigated the peristaltic transport of hyperbolic tangent fluid in a tapered Asymmetric channel. The impact of electro-osmotic flow, radiation, and Joule heating through a porous medium on the peristaltic motion of tangent hyperbolic fluid in an endoscope with the assumptions of low Reynolds number and long wavelength was reported by Kotnurkar and Talawar.²⁰ Elogail²¹ analyzed the peristaltic flow of a hyperbolic tangent fluid with variable viscosity and thermal conductivity through a vertical asymmetric channel under the lubrication approximation hypothesis. Naduvinamani and Guttedar²² studied the two-dimensional peristaltic flow of a hyperbolic tangent fluid in an asymmetric channel through a porous medium, utilizing long wavelength and low Reynolds number assumptions. Bartwal et al.²³ investigated the impact of melting heat transfer on the stagnation point flow of an electrically conducting hyperbolic tangent fluid over a porous rotating disk. Additionally, Bartwal et al.²⁴ examined the effects of irreversibility on the stagnation point flow of electrically conducting tangent hyperbolic fluid over a permeable, stretchable rotating disk, considering viscous dissipation effects.

The phenomenon in which an electrically conducting fluid is propelled by an externally applied magnetic force is referred to as Magnetohydrodynamic (MHD). MHD finds significant applications in different fields, including industrial engineering, geophysics, and physiology. It plays a crucial role in inflammation, aerodynamic heating process, electrostatic precipitation, glass production, generators, tumor detection, paper production, cancer treatment, MRI, blood reduction, and ulcer therapy. Furthermore, the investigation of peristalsis in relation to heat transfer is important, particularly in light of its applications in medical procedures such as oxygenation and hemodialysis. In 1942, the discipline of MHD used in fluid dynamics was initially established by the esteemed physicist Alfvén.²⁵ He received the Nobel Prize in 1970 for his significant contributions to this field. In the same way, Abbas et al.^26,27 investigated the influence of variable viscosity on the peristaltic movement of viscous fluid under lubrication approximation theory. The peristaltic transport of blood-based nanofluid comprising cylindrical-shaped Gold (Au) nanoparticles in a non-uniform tube under the influence of a magnetic field was analyzed by Ashraf et al.²⁸ Abbasi et al.²⁹ inspected the impact of entropy generation on the peristaltic motion of a hybrid nanofluid within a 2D asymmetric channel. Hussain et al.³⁰ investigated the impact of a magnetic field on the peristaltic motion of sisko fluid through a channel having sinusoidal walls.

The slip effect is a crucial factor that needs to be determined in peristalsis flow. It refers to the change in angle when fluids leave the pumping material. Due to their wide applications in physiology, slip-flow studies have become significant today. The fluid slip behaviors in peristalsis have important uses for the process of polishing internal cavities, polymer technology, and heart valves. Fluid slips against the walls due to surface wetness. Slippage on the flow of different configurations under various effects has been studied by various authors in the literature. For instance, Akram et al.³¹ analyzed the impacts of slip constraints on the peristaltic motion of magneto-Prandtl nanofluid through an asymmetric channel under the influence of double-diffusive convection, which includes both thermal and concentration gradients. The peristaltic flow of Prandtl fluid in the presence of mixed convection across a curved channel under slip conditions was discussed by Zahir.³² Gangavathi et al.³³ investigated the consequences of Hall as well as slip taking place on the peristaltic stream of a Jeffrey solution through a permeable middling within an inclined 2D strait below the extensive wavelength estimation. The entropy generation and thermal characteristics of a couple stress blood nanofluid on the peristaltic slip flow of gold nanoparticles being conveyed through an asymmetric channel through a porous medium were inspected by Thabet et al.³⁴ Kumar and Yadav³⁵ discussed the peristaltic flow of non-miscible couple stress and Newtonian fluids flowing through a horizontal channel filled with a porous material with slip boundary conditions under the impact of an inclined magnetic field. Kanwal and Khan³⁶ investigated the influence of slip constraints on the peristaltic flow of Walter’s B fluid through an asymmetric channel under the lubrication hypothesis. Furthermore, Chemical reactions are classified as homogeneous and heterogeneous reaction effects. During a uniform flow across a flat surface, homogeneous reactions were represented by cubic autocatalysis, and heterogeneous reactions were represented by a first-order chemical reaction. Chemical reactions between acid and metal, reactions in batteries, electrolytic cells, and corrosion are all examples of heterogeneous reactions. When compared to heterogeneous reactions, homogeneous reactions are theoretically simpler because the reacting product is determined solely based on the species characteristics. However, the heterogeneous reactions are still important in practice because they relate the product’s dependence on the nature of two or more different reactant molecules. The peristaltic motion of Prandtl nanofluid by considering the homogeneous–heterogeneous reaction under lubrication approximation theory was discussed by Hayat et al.³⁷ The comportments of homogeneous-heterogeneous reactions on the peristalsis transportation mechanism of the Rabinowitsch fluid model inside a flexible channel were reported by Imran et al.³⁸ Javed et al.³⁹ investigated the effect of homogeneous-heterogeneous reactions on the MHD peristaltic mechanism of Ellis fluid in a curved channel. Recently, the peristaltic flow of Johnson-Segalman fluid in a curved flow channel in the presence of a homogeneous-heterogeneous reaction was inspected by Zahir et al.⁴⁰

The research gap in existing studies lies in the insufficient consideration of slip effects on the walls of larger peristaltic channels, where velocity slip might typically be neglected. Most current models assume no-slip conditions, as these were generally deemed irrelevant for larger channels, especially in biomedical or industrial settings. However, in specific scenarios such as microfluidic systems, drug delivery channels, and biomedical devices like endoscopes, slip effects become crucial due to the fine scale of these systems and the precise control required over fluid movement. Additionally, while homogeneous and heterogeneous reactions have been studied independently in peristaltic flows, their combined effects under slip conditions remain underexplored, particularly in MHD flows of non-Newtonian fluids like the hyperbolic tangent fluid. The present study, therefore, addresses these overlooked aspects by investigating the combined influence of velocity slip and reaction effects on the peristaltic flow of hyperbolic tangent fluids in an asymmetric channel under magnetic fields. This model is relevant for applications requiring precise fluid control under electromagnetic influence, such as microfluidic drug delivery systems, where chemical reactions between drug carriers and biological fluids might significantly impact drug dispersion. In such systems, slip conditions on channel walls can influence drug distribution rates and flow efficiency, affecting treatment efficacy. Similarly, in chemical reactors or biomedical devices that handle viscous or shear-sensitive materials, both slip effects and reaction kinetics play a role in optimizing flow and ensuring efficient transport without excessive shear stress on the fluid. This study not only fills the research gap but also has significant implications for designing biomedical devices that rely on controlled peristaltic flow, enabling enhanced precision in therapeutic applications. The novelty of the problem is given in Table 1. The resulting nonlinear equations are solved using the lubrication theory approximation, which is particularly relevant for predicting physiological characteristics, such as blood flow dynamics. Analytical solutions for velocity, pressure gradient, streamlines, and concentration distributions are obtained using the regular perturbation technique. The influence of various parameters on these distributions is presented through graphical analysis. Additionally, the pressure rise per wavelength along the channel walls is computed numerically.

Table 1.

The novelty of the problem.

Ref. no.	[17]	[18]	[19]	[22]	Present
Peristaltic flow	Yes	Yes	Yes	Yes	Yes
Tangent fluid	Yes	Yes	Yes	Yes	Yes
Magnetic field	No	Yes	No	No	Yes
Lubrication theory	Yes	Yes	Yes	Yes	Yes
Asymmetric channel	Yes	Yes	Yes	Yes	Yes
Velocity slip conditions	No	No	No	No	Yes
Homogeneous and heterogeneous reactions	No	No	No	No	Yes

Mathematical modeling

Let us consider the peristaltic transport of electrically conducting tangent hyperbolic fluid in an asymmetric channel with systematically relaxing and contracting sinusoidal walls. The flow model is sketched in Figure 1. The mathematical flow problem is modeled under the following assumptions:

• The peristaltic flow of tangent hyperbolic fluid is considered.

• The width of the channel is taken $d_{1} + d_{2}$ and flow is produced by a sinusoidal wave travel with speed c along the walls of the channel.

• The flow analysis is formulated using a Cartesian frame. Here $\bar{X}$ and $\bar{Y}$ are considered parallel and perpendicular to the walls of the tapered channel respectively.

• The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase differences.

• The electrically conducting flow is referred to as the impact of magnetic force along $Y$ -direction.

• Low magnetic Reynolds is considered. Induced magnetic and electric field contributions are neglected.

• The velocity slip conditions and homogeneous and heterogeneous reactions are deliberated.

• Long wavelength and small Reynolds number assumptions are taken for the simplification of the problem.

Figure 1.

Flow configuration of the problem.

The wave geometry is characterized by the following expressions¹⁷:

H_{1} (\bar{X}, \bar{t}) = d_{1} + a_{1} \cos (\frac{2 π (\bar{X} - c \bar{t})}{λ})

(1)

H_{2} (\bar{X}, \bar{t}) = - d_{2} - b_{1} \cos (ϕ + \frac{2 π (\bar{X} - c \bar{t})}{λ})

(2)

In equations (1) and (2), the symbols $a_{1}$ and $b_{1}$ represent the wave amplitudes, $λ$ is the wavelength, channel width is indicated with $d_{1} + d_{2}$ , wave propagation velocity is depicted with $c$ , $t$ be the time, and X indicates the sinusoidal wave propagation direction. However, the phase difference $ϕ \in [0, π]$ and the quantity $ϕ = 0$ signify the symmetric channel with waves out of phase mode and $ϕ = π$ indicate the waves are in phase. As a result, these variables must satisfy the condition ${a_{1}}^{2} + {b^{2}}_{1} + 2 a_{1} b_{1} \cos ϕ \leq {(d_{1} + d_{2})}^{2} .$

The fundamental equations of non-Newtonian tangent hyperbolic fluid are considered as follows^19,22:

div V = 0

(3)

ρ \frac{d V}{dt} = - \nabla P + div τ + J \times B

(4)

\frac{da}{dt} = D_{A} \nabla^{2} a - k_{c} a b^{2}

(5)

\frac{db}{dt} = D_{B} \nabla^{2} b + k_{c} a b^{2}

(6)

Where $ρ$ denotes the fluid density, $V$ signifies the velocity vector, $τ$ represents the Cauchy stress tensor, $P$ symbolizes the pressure, $D_{A}$ and $D_{B}$ the coefficient of mass diffusivity for homogeneous and heterogeneous reactions respectively, $a$ and $b$ the concentrations of $A$ and $B$ respectively and $d / dt$ is the material times derivative. The magnetic field is applied by the following definition:

B = (0, B_{0}, 0)

(7)

The induced magnetic field is neglected by assuming a very small magnetic Reynolds number $(R e_{m} << 1)$ . Now Ohm’s law leads to the given form

J = σ (E + V \times B)

(8)

where the effect of the electric field is absent (E = 0). Thus, we get

J \times B = (0, - σ {B_{0}}^{2} u, 0)

(9)

where, $B_{0}$ is the strength of the magnetic field, $J$ denotes the current density, and $σ$ symbolizes the electrical conductivity of the fluid.

The model for homogeneous-heterogeneous interaction is defined as follows³⁸:

A + 2 B \to 3 B, rate = k_{c} a b^{2}

(10)

Here, we emphasize a single, isothermal, and first-order chemical reaction on the surface of the catalyst. Therefore, for such considerations

A \to B, rate = k_{s} a

(11)

where $k_{c}$ and $k_{s}$ embody the rate parameters. Note that both reactions are assumed to occur at the same temperature.

The Cauchy stress and extra stress tensors for the tangent hyperbolic fluid are given by^23,24:

τ = - PI + S

(12)

S = [μ_{\infty} + (μ_{0} + μ_{\infty}) \tanh {(Γ γ)}^{n}] A_{1},

(13)

Here $I$ represents the identity vector, $S$ identifies the extra stress tensor, $μ_{\infty}$ is the infinite shear rate viscosity, $μ_{0}$ is the zero shear rate viscosity, $Γ$ is the time constant $(> 0)$ , $A_{1}$ denotes the first Rivlin Erickson tensor and $γ$ symbolizes shear rate defined as follows:

γ = \sqrt{\frac{1}{2} \sum_{i} \sum_{j} γ_{ij} γ_{ji}} = \sqrt{\frac{1}{2} Π}

(14)

In which the second invariant strain tensor is designated by $Π = trace {(grad A + {(grad A)}^{T})}^{2}$ . Considered the case where $μ_{\infty} = 0$ and $Γ γ < 1$ , equation (15) can be expressed as

S = μ_{0} [1 + n (Γ γ - 1) A_{1}]

(15)

The velocity field for an unsteady two-dimensional flow is considered to be:

V = [\bar{U} (\bar{t}, \bar{X}, \bar{Y}), \bar{V} (\bar{t}, \bar{X}, \bar{Y}), 0]

(16)

Using equation (18), the component form of equations (4)–(8) are given as follows:

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

(17)

ρ [\frac{\partial \bar{U}}{\partial \bar{t}} + \bar{U} \frac{\partial \bar{U}}{\partial \bar{X}} + \bar{V} \frac{\partial \bar{U}}{\partial \bar{Y}}] = - \frac{\partial \bar{P}}{\partial \bar{X}} + \frac{\partial {\bar{S}}_{XX}}{\partial \bar{X}} + \frac{\partial {\bar{S}}_{XY}}{\partial \bar{Y}} - σ B_{0}^{2} \bar{U}

(18)

ρ [\frac{\partial \bar{V}}{\partial \bar{t}} + \bar{U} \frac{\partial \bar{V}}{\partial \bar{X}} + \bar{V} \frac{\partial \bar{V}}{\partial \bar{Y}}] = - \frac{\partial \bar{P}}{\partial \bar{Y}} + \frac{\partial {\bar{S}}_{XY}}{\partial \bar{X}} + \frac{\partial {\bar{S}}_{YY}}{\partial \bar{Y}}

(19)

\frac{d \bar{a}}{d \bar{t}} = D_{A} (\frac{\partial^{2} \bar{a}}{\partial {\bar{X}}^{2}} + \frac{\partial^{2} \bar{a}}{\partial {\bar{Y}}^{2}}) - k_{c} \bar{a} {\bar{b}}^{2}

(20)

\frac{d \bar{b}}{dt} = D_{B} (\frac{\partial^{2} \bar{b}}{\partial {\bar{X}}^{2}} + \frac{\partial^{2} \bar{b}}{\partial {\bar{Y}}^{2}}) + k_{c} \bar{a} {\bar{b}}^{2}

(21)

The slip condition allows for a non-zero fluid velocity at the boundary, meaning the fluid can slip along the surface. This condition is particularly relevant in microfluidic systems or situations where the boundary surface has special properties, such as low friction or porous structures. In the context of this study, incorporating slip at the sinusoidal walls reflects the realistic behavior of fluid flow in permeable or low-friction boundaries, common in biological systems and industrial processes. The slip condition allows for more complex and accurate modeling of flow near boundaries, especially in the presence of deformable or permeable walls. The slip velocity boundary conditions are^32,33:

\begin{matrix} \bar{U} - β^{*} {\bar{S}}_{XY} = 0, at \bar{Y} = {\bar{H}}_{1}, \\ \bar{U} + β^{*} {\bar{S}}_{XY} = 0, at \bar{Y} = {\bar{H}}_{2} . \end{matrix}

(22)

The conditions for homogeneous–heterogeneous reactions are represented by the following expressions¹³:

\begin{matrix} \bar{a} = α_{0}, \bar{b} = 0, at \bar{Y} = {\bar{H}}_{1}, \\ D_{A} \frac{\partial \bar{a}}{\partial \bar{Y}} = k_{s} \bar{a}, D_{B} \frac{\partial \bar{b}}{\partial \bar{Y}} = - k_{s} \bar{b}, at \bar{Y} = {\bar{H}}_{2} . \end{matrix}

(23)

The relation between fixed and moving reference frames is designated as:

\bar{x} = \bar{X} - c \bar{t}, \bar{y} = \bar{Y}, \bar{u} = \bar{U} - c, \bar{v} = \bar{V}, \bar{p} = \bar{P}

(24)

The physical relations utilized to convert the set of governing equations from (17) to (23) into their corresponding dimensionless form are listed below:

\begin{matrix} x = \frac{\bar{x}}{λ}, y = \frac{\bar{y}}{d_{1}}, u = \frac{\bar{u}}{c}, t = \frac{c}{λ} \bar{t}, v = \frac{\bar{v}}{c δ}, h_{1} = \frac{{\bar{h}}_{1}}{d_{1}}, \\ h_{2} = \frac{{\bar{h}}_{2}}{d_{2}}, S_{xx} = \frac{λ}{μ c} {\bar{S}}_{\bar{x} \bar{x}}, S_{xy} = \frac{d_{1}}{μ c} {\bar{S}}_{\bar{x} \bar{y}}, S_{c} = \frac{v}{D_{A}}, a = \frac{f}{α_{0}}, \\ b = \frac{g}{α_{0}}, L = \frac{k_{c} α_{0}^{2} d_{1}^{2}}{ν}, S_{yy} = \frac{d_{1}}{μ c} {\bar{S}}_{\bar{y} \bar{y}}, δ = \frac{d_{1}}{λ}, Re = \frac{ρ c d_{1}}{μ}, \\ We = \frac{Γ c}{d_{1}}, p = \frac{{d_{1}}^{2}}{c λ μ} \bar{p}, ξ = \frac{D_{B}}{D_{A}}, M = \sqrt{\frac{σ}{μ} B_{0} d_{1}}, W = \frac{k_{s} d_{1}}{D_{A}}, \\ β = \frac{β^{*}}{d_{1}}, γ = \frac{γ d}{c} . \end{matrix}

(25)

Where $Re$ is the Reynolds number which designates the impact of inertia as equated with viscous liquid impact, $θ$ is the dimensionless temperature, $We$ is the Weissenberg number, $M$ is the magnetic parameter which measures the influence of Lorentz force impact on liquid flow, $δ$ is the wave number which is the ratio that agrees to the width of the channel over the length of the wave and $S_{c}$ the Schmidt number, $ξ$ the diffusion coefficients ratio, $L$ and $W$ the strength measuring parameters for homogeneous and heterogeneous reactions respectively.

It is assumed that diffusion coefficients $D_{A}$ and $D_{B}$ of both chemical species are equal that is, ξ = 1. This assumption leads us to the given relation:

f + g = 1

(26)

Utilizing equations (24) and (25) with stream functions $(u = \partial ψ / \partial y, v = - \partial ψ / \partial x)$ along with long wavelength approximation $δ << 1$ and small Reynolds number $Re$ , we get the following form of equations

\frac{\partial P}{\partial x} = \frac{\partial}{\partial y} [1 + n (We \frac{\partial^{2} ψ}{\partial y^{2}} - 1)] \frac{\partial^{2} ψ}{\partial y^{2}} - M^{2} \frac{\partial ψ}{\partial y},

(27)

\frac{\partial p}{\partial y} = 0

(28)

\frac{1}{S_{c}} \frac{\partial^{2} f}{\partial y^{2}} + Lf {(1 - f)}^{2} = 0

(29)

Removing the term of pressure gradient from equations (27) and (28), one obtains

\frac{\partial^{2}}{\partial y^{2}} [1 + n (We \frac{\partial^{2} ψ}{\partial y^{2}} - 1)] \frac{\partial^{2} ψ}{\partial y^{2}} - M^{2} \frac{\partial^{2} ψ}{\partial y^{2}} = 0

(30)

The appropriate boundary constraints are:

\begin{matrix} ψ = - \frac{F}{2}, \frac{\partial ψ}{\partial y} - β S_{xy} = 0, f = 1 at y = h_{1}, \\ ψ = \frac{F}{2}, \frac{\partial ψ}{\partial y} + β S_{xy} = 0, \frac{df}{dy} - Wf = 0 at y = h_{2} . \end{matrix}

(31)

Where $h_{1} (x) = 1 + a \cos (2 π x)$ and $h_{2} (x) = - d - b \cos (2 π x + ϕ)$

The flow rate can be calculated as

Q = 1 + F + d

(32)

where $F = \int_{h_{2} (x)}^{h_{1} (x)} \frac{\partial ψ}{\partial y} dy = ψ (h_{1} (x)) - ψ (h_{2} (x))$ .

Method of solution

The significance of the perturbation method is well established in the literature for its ability to provide an approximate analytical solution to nonlinear problems. The outcomes reported for the other comparatively sophisticated approximate analytical methods to nonlinear problems have decent accuracy, but they are more difficult in applications and analysis than perturbation methods. Thus, for many years, the relative simplicity and high precision in the limitation of small parameters have made perturbation methods fascinating tools among the most frequently utilized approximate analytical procedures. Perturbation methods usually use convenient mathematical formulations to provide accurate outcomes for small perturbation parameters. Therefore, the method of perturbation has been implemented to acquire the expressions for velocity, pressure gradient, and concentration for small values of the Weissenberg number for equation (27) and homogeneous parameter $L$ for equation (29). Hence the flow quantities are expressed as

\begin{matrix} ψ = ψ_{0} + We ψ_{1} + O (W e^{2}), \\ p = p_{0} + We p_{1} + O (W e^{2}), \\ F = F_{0} + We F_{1} + O (W e^{2}), \\ f = f_{0} + L f_{1} + O (L^{2}) . \end{matrix}

(33)

Substituting equation (33) in equations (27)–(31) and repositioning the exponents of $We$ and $L$ , the following systems are obtained.

Zeroth order system

\frac{\partial^{4} ψ_{0}}{\partial y^{4}} - M^{2} \frac{\partial^{2} ψ_{0}}{\partial y^{2}} = 0

(32)

\frac{\partial p_{0}}{\partial x} = (1 - n) \frac{\partial^{3} ψ_{0}}{\partial y^{3}} - M^{2} \frac{\partial ψ_{0}}{\partial y}

(33)

\frac{1}{S_{c}} \frac{\partial^{2} f_{0}}{\partial y^{2}} = 0

(34)

with boundary conditions

\begin{matrix} ψ_{0} = - \frac{F_{0}}{2}, \frac{\partial ψ_{0}}{\partial y} - β S_{xy} = 0, f_{0} = 1 at y = h_{1}, \\ ψ_{0} = \frac{F_{0}}{2}, \frac{\partial ψ_{0}}{\partial y} + β S_{xy} = 0, \frac{\partial f_{0}}{\partial y} = W f_{0} at y = h_{2} . \end{matrix}

(35)

First order system

(1 - n) \frac{\partial^{4} ψ_{1}}{\partial y^{4}} - M^{2} \frac{\partial^{2} ψ_{1}}{\partial y^{2}} = - n \frac{\partial^{2}}{\partial y^{2}} {(\frac{\partial^{2} ψ_{0}}{\partial y^{2}})}^{2}

(36)

\frac{\partial p_{1}}{\partial x} = (1 - n) \frac{\partial^{3} ψ_{1}}{\partial y^{3}} + n \frac{\partial}{\partial y} {(\frac{\partial^{2} ψ_{0}}{\partial y^{2}})}^{2} - M^{2} \frac{\partial ψ_{1}}{\partial y}

(37)

\frac{1}{S_{c}} \frac{\partial^{2} f_{1}}{\partial y^{2}} + f_{0} {(1 - f_{0})}^{2} = 0

(38)

with boundary conditions

\begin{matrix} ψ_{1} = - \frac{F_{1}}{2}, \frac{\partial ψ_{1}}{\partial y} - β S_{xy} = 0, f_{1} = 1 at y = h_{1}, \\ ψ_{1} = \frac{F_{1}}{2}, \frac{\partial ψ_{1}}{\partial y} + β S_{xy} = 0, \frac{\partial f_{1}}{\partial y} = W f_{1} at y = h_{2} . \end{matrix}

(39)

Solutions for the system of zeroth order

The solutions of equations (32)–(34) with surface conditions (35) are given by

ψ_{0} = \frac{e^{- \frac{My}{\sqrt{1 - n}}} (1 - n) (C_{1} e^{\frac{2 My}{\sqrt{1 - n}}} + C_{2})}{M^{2}} + C_{3} + y C_{4}

(40)

\frac{d p_{0}}{dx} = - (1 + C_{4}) M^{2}

(41)

f_{0} = \frac{1 - W (h_{2} - y)}{1 + W (h_{1} - h_{2})}

(42)

Solutions for the system of first-order

The solutions of equations (32)–(34) with surface conditions (35) are given by

\begin{matrix} ψ_{1} = \\ - \frac{e^{- \frac{2 My}{\sqrt{1 - n}}} (C_{2}^{2} n + C_{1}^{2} e^{\frac{4 My}{\sqrt{1 - n}}} n + 3 e^{\frac{My}{\sqrt{1 - n}}} (- 1 + n) (e^{\frac{2 My}{\sqrt{1 - n}}} C_{5} + C_{6}))}{3 M \sqrt{1 - n}} + C_{7} + y C_{8} \end{matrix}

(43)

\frac{d p_{1}}{dx} = C_{8} M^{2}

(44)

\begin{matrix} f_{1} = \\ \frac{W^{2} S_{c} y (10 h^{4} W 20 h^{3} - 2 h^{5} W^{2} + 5 h^{2} W^{2} y^{3} + y^{3} (5 + 3 Wy) - hW y^{3} (10 + 3 Wy))}{60 {(hW - 1)}^{4}} \end{matrix}

(45)

The non-dimensional pressure rise over one wavelength $Δ p$ for the axial velocity is given by the relation.

Δ p = \int_{0}^{1} \frac{dp}{dx} dx

(46)

The values of the constants in zeroth and first-order solutions can be found via MATHEMATICA software.

Results and discussion

This portion characterizes the impacts of sundry parameters on the fluid velocity, pumping characteristics, and fluid temperature and streamlines through graphical outcomes. The expression $Δ p$ contains the integration of $dp / dx$ which is difficult to integrate analytically. Therefore, the integral occurring in equation (46) has been estimated numerically utilizing MATHEMATICA through composite Simpson’s rule. In this analysis, the following default parameter values are adopted for computations: $a = 0.1,$ $We = 0.03,$ $W = 0.3,$ $b = 0.1$ , $M = 1,$ $L = 0.2,$ $n = 0.3,$ $ϕ = 0.06,$ $x = 0.3,$ and $d = 1$ . The velocity of the present study in the absence of a magnetic field, and velocity slip conditions is compared with the velocity of Naduvinamani and Guttedar²² in the absence of a porous medium, for similar values of non-dimensional parameters, and the same is shown in Figure 2. It can be seen in Figure 2 that the velocity of the present study in absence of magnetic field, and velocity slip conditions is in good agreement with the velocity of Naduvinamani and Guttedar.²²

Figure 2.

Comparison of the limiting case of the present study with the results of Naduvinamani and Guttedar.²²

Figure 3(a) to (c) displays the deviations in velocity profile $u$ for several important dimensionless factors. Figure 3(b) describes that the liquid velocity decreases with increasing values of the magnetic parameter $M$ . The increase in the magnetic parameter M indicates an increase in Lorentz force relative to the viscous force. Thus, the velocity of the fluid decreases due to the increase in the Lorentz force which physically retards the flow. This idea is of huge implication in the field of medicine where the occurrence of a regulated exterior magnetic field upsurges the local energy which benefits in abolishing the tumor cells. Furthermore, the liquid velocity is observed to be higher in the case of the slip parameter compared to that of a no-slip condition. The impact of the Weissenberg number on axial velocity is observed in Figure 3(b). It increases in the region (−0.75, 0) with an enhancement in Weissenberg number, and the converse in behavior is noticed in the region (0, 0.75) of the channel. The effect of the power law index parameter $n$ on the velocity profile $u$ is shown in Figure 3(c) and depicts that the velocity of the fluid increases near the wall of the channel and decreases in the central part of the channel.

Figure 3.

Velocity profile $u$ for distinct values of (a) magnetic parameter $M$ , (b) Weissenberg number $We$ , and (c) power law index $n$ for no-slip (dashed lines) and slip (solid lines) conditions.

Figure 4(a) to (c) shows the deviation in pressure gradient $dp / dx$ versus $x$ for dissimilar of important parameters. The impression of magnetic parameters on $dp / dx$ is shown in Figure 4(a). It can be seen that the pressure gradient increases by increasing the values of the magnetic parameter $M$ . Furthermore, the pressure gradient is observed to be higher in the case of the slip parameter compared to that of a no-slip condition. The pressure gradient is also enhanced by enhancing the values of the Weissenberg number $We$ as shown in Figure 4(b). The influence of the power law index $n$ on $dp / dx$ is depicted in Figure 4(c). This graph shows that an enhancement in the values of the power law index decreases the pressure gradient. Figure 5(a) to (c) shows the influence of several control parameters on the average rise in pressure $Δ p$ against time average flux $Θ$ . Figure 5(a) shows that pressure rise increases for the larger values of the magnetic parameter in the pumping region and decreases in the co-pumping region. Similar behavior is exposed in Figure 5(b) for the values of the Weissenberg number $We$ . It is noticed from Figure 5(c) that as the power law index parameter $n$ increases the pressure rise increases.

Figure 4.

Pressure gradient $dp / dx$ for distinct values of (a) magnetic parameter $M$ , (b) Weissenberg number $We$ , and (c) power law index $n$ for no-slip (dashed lines) and slip (solid lines) conditions.

Figure 5.

Pressure rise $Δ p$ for distinct values of (a) magnetic parameter $M$ , (b) Weissenberg number $We$ , and (c) power law index $n$ for no-slip (dashed lines) and slip (solid lines) conditions.

Figure 6(a) and (b) depicts concentration profile deviations concerning the homogeneous parameter $L$ , Schmidt number $S_{c}$ , and heterogeneous reaction parameter $W .$ The impact of the heterogeneous reaction parameter $W$ is shown in Figure 6(a). This graph indicates that the concentration profile increases by increasing the values of the heterogeneous parameter $W$ . In Figure 6(b), we explore the consequences of $S_{c}$ and $L$ on concentration. This graph shows that concentration exhibits a rising trend by enhancing the values of $S_{c}$ and $L$ .

Figure 6.

Concentration profile $f$ for distinct values of (a) heterogeneous parameter $W$ and (b) homogeneous parameter $L$ and Schmidt number $S_{c}$ .

Trapping is an intriguing phenomenon in peristaltic flow, where a fluid bolus forms due to the contraction and expansion of flexible walls, carried along by peristaltic waves. Figures 7(a) and (b) to 9(a) and (b) illustrate the effect of various parameters on trapping through contour plots in the no-slip condition. Figure 7(a) and (b) illustrates the effects of varying the magnetic parameter M on trapping, represented through streamlines. As the magnetic parameter M increases, the Lorentz force exerted on the fluid intensifies, opposing the flow motion and consequently reducing the number of trapped boluses. This suppression effect reflects the damping influence of the magnetic field on fluid flow, diminishing the extent of trapping zones. The findings suggest that magnetic fields can effectively control trapping phenomena in peristaltic systems, which has potential implications for biomedical applications where precise fluid transport is needed. In Figure 8(a) and (b), the Weissenberg number We is varies to observe its impact on streamlines. With higher We values, there is an observable increase in the number of trapped boluses, which indicates that fluids with higher elasticity are more susceptible to trapping. This elasticity-induced enhancement in trapping suggests that in applications where bolus formation is essential, selecting a fluid with suitable elasticity properties could be beneficial. This also emphasizes the role of the Weissenberg number in controlling fluid behavior within peristaltic devices, particularly for non-Newtonian fluids. Figure 9(a) and (b) demonstrates the effects of the power law index parameter n on trapping. Here, as n increases, representing a shift toward shear-thickening behavior, the number of trapped boluses decreases. This trend suggests that shear-thickening fluids are less prone to trapping, which could inform the design of peristaltic systems where minimizing bolus formation is desired. Figures 10(a) and (b) to 12(a) and (b) further explore the impact of the velocity slip parameter on trapping by examining variations in magnetic parameter M, Weissenberg number We, and power law index parameter n under slip conditions. Across these figures, a consistent trend emerges: the introduction of slip diminishes the trapping effect across all parameters. This implies that allowing slip at the boundaries can be a strategy to mitigate trapping, which may be particularly valuable in peristaltic flows where unidirectional fluid transport is required without forming significant trapped boluses. This additional analysis provides a comprehensive view of how each parameter influences trapping dynamics in peristaltic systems.

Figure 7.

Contour plots for distinct values of the magnetic parameter $M$ : (a) $M = 0.5$ and (b) $M = 1$ for no-slip condition.

Figure 8.

Contour plots for distinct values of the Weissenberg number $We$ : (a) $We = 0.01$ and (b) $We = 0.05$ for no-slip condition.

Figure 9.

Contour plots for distinct values of the power law index parameter $n$ : (a) $n = 0.01$ and (b) $n = 0.05$ for no-slip condition.

Figure 10.

Contour plots for distinct values of the magnetic parameter $M$ : (a) $M = 0.5$ and (b) $M = 1$ for slip condition.

Figure 11.

Contour plots for distinct values of the Weissenberg number $We$ : (a) $We = 0.01$ and (b) $We = 0.05$ for slip condition.

Figure 12.

Contour plots for distinct values of the power law index parameter $n$ : (a) $n = 0.01$ and (b) $n = 0.05$ for slip condition.

Conclusions

This study presents an in-depth analysis of the peristaltic flow of an electrically conducting hyperbolic tangent fluid through an asymmetric channel, emphasizing the unique impacts of slip and reaction effects. The lubrication approximation theory is utilized to simplify the normalized equations. The perturbation solutions for the flow fields are obtained and impacts of dissimilar parameters on the flow fields are inspected through graphs. In the future, we can extend this work by adding various models like the Casson fluid model, the Jeffrey model, and the Buongiorno fluid model with more effects like mixed convection, bioconvection, and joule heating. We are assured that our current investigations are significantly applied to numerous mechanisms like cancer therapy, solar physics, the chemical industry, hyperthermia, the petroleum industry, and the recovery of targeted human cells of drugs. The summarized achievements of the current contribution are:

• The velocity profile increases in the upper half of the channel and decreases in the lower half of the channel with an increase in Weissenberg number. The increase in velocity in the upper half of the channel and the decrease in the lower half provide insight into asymmetric flow behavior, which is critical for accurately modeling non-Newtonian fluids in biological systems, such as blood or other physiological fluids.

• Slip conditions significantly influence the velocity profile and enhance overall flow efficiency, particularly in applications where no-slip conditions are typically assumed. Results demonstrate that under slip conditions, the fluid velocity is consistently higher compared to no-slip cases. This finding is relevant for applications that involve high-viscosity fluids, where channel wall adherence often impacts flow efficiency. Incorporating slip conditions in peristaltic flow designs could, therefore, optimize fluid transport processes in both medical devices and industrial applications.

• The pumping rate increases in the retrograde pumping region and decreases in the augmented pumping region by increasing the values of the magnetic parameter. The effects of the magnetic parameter on the pumping rate underscore the importance of electromagnetic fields in controlling flow dynamics, which could be valuable for designing more precise control mechanisms in magnetohydrodynamic (MHD) flows.

• Both homogeneous and heterogeneous reactions exhibit pronounced impacts on concentration profiles. The homogeneous reaction parameter is shown to increase concentration distribution, which is critical for applications requiring controlled substance diffusion, such as in drug delivery systems. Additionally, the opposing effects of homogeneous and heterogeneous reactions on the concentration field highlight an effective mechanism for fine-tuning peristaltic flow in biomedical contexts, where targeted delivery and precise fluid manipulation are essential.

• The trapped bolus number is enhanced by enhancing the value of the Weissenberg number. The discovery that the trapped bolus number increases with the Weissenberg number has potential implications for designing medical devices, particularly for managing conditions involving bolus trapping in the gastrointestinal or cardiovascular systems.

• The interaction between slip conditions, reaction effects, and the Weissenberg number offers a new perspective for designing peristaltic pumps. By carefully adjusting these parameters, engineers can optimize fluid movement and concentration distributions within channels. This knowledge contributes to improved models for peristaltic systems, supporting enhanced performance in applications ranging from drug delivery to gastrointestinal tract simulations.

Limitations of the study

• One of the primary limitations of this study is the use of the lubrication theory approximation, which, while suitable for predicting certain physiological flows like blood dynamics, restricts the analysis to low Reynolds number flows and long-wavelength assumptions. This limits the applicability of the results to higher Reynolds number scenarios, where inertial effects become more significant.

• Additionally, the study assumes an electrically conducting hyperbolic tangent fluid, which, although useful for capturing non-Newtonian fluid behavior, simplifies the complex rheology of actual biological fluids such as blood. The chosen model may not account for all aspects of real-world non-Newtonian behavior, particularly in more extreme conditions, which could affect the generalizability of the findings.

• Another limitation is the assumption of steady-state flow conditions. In practice, physiological flows are often transient, and this study does not capture time-dependent variations that could provide a more comprehensive understanding of flow behavior.

• Moreover, the homogeneous and heterogeneous reactions considered in the study are modeled under ideal conditions, without accounting for possible variations in reaction kinetics or external influences, which may affect the accuracy of the concentration distributions predicted.

• These limitations suggest that while the results provide valuable insights, they should be interpreted with caution when applying them to more complex or dynamic systems. Future studies could address these limitations by incorporating more detailed fluid models, transient analysis, and broader parameter ranges.

Footnotes

Appendix

Notation

$(X, Y)$	fixed frame coordinate axes	$(x, y)$	wave frame coordinate axes
$(U, V)$	fixed frame velocity constituents	$(u, v)$	wave frame velocity constituents
$\bar{t}$	Time	$(a_{1}, b_{1})$	wave amplitudes
$V$	velocity vector	$B_{0}$	uniform magnetic field
$S$	Cauchy stress tensor	$Q$	mean flow rate
$S_{c}$	Schmidt number	$M$	magnetic parameter (magnetic field vector)
$\frac{d}{dt}$	time-dependent material derivative	$d$	channel width
$n$	Power law index	$H_{1}$	superior wall surfaces of the channel
$Re$	Reynolds number	$H_{2}$	inferior wall surfaces of the channel
$P$	pressure	$a$	concentration of $A$
$We$	Weissenberg number	$b$	concentration of $B$
$c$	wave propagation velocity	$L, W$	homogeneous and heterogeneous reaction parameters
$d_{1} + d_{2}$	channel width	$k_{c}, k_{s}$	rate coefficients
$D_{A}, D_{B}$	mass diffusivity coefficients
Greek Letters
$λ$	Wavelength	$Π$	second invariant tensor
$ρ$	liquid density	$μ_{0}$	viscosity at zero shear rate
$μ_{\infty}$	viscosity at an infinite shear rate	$σ$	electrical conductivity of fluid
$ξ$	diffusion coefficient ratio	$Γ$	time constant
$Δ$	dimensionless wave number	$ϕ$	phase difference
$ψ$	stream function	$β^{*}$	velocity slip length
$β$	velocity slip parameter

Acknowledgements

We are thankful to the reviewers for their encouraging comments and constructive suggestions to improve the quality of the manuscript.

Handling Editor: Sharmili Pandian

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Mariam Sheikh

Jafar Hasnain

Zaheer Abbas

Muhammad Yousuf Rafiq

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