ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Volume 105, Issue 5 e70088
ORIGINAL PAPER
Open Access

Exploration of an MHD Jeffrey six-constant fluid under the influence of a convective boundary condition

Aqila Shaheen, 

Aqila Shaheen

Department of Mathematics, Minhaj University of Lahore, Lahore, Pakistan

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Mustafa Inc, 

Corresponding Author

Mustafa Inc

Department of Mathematics, Firat University, Elazig, Turkiye

Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai, Tamil Nadu, India

Department of Computer Engineering, Biruni University, Istanbul, Turkiye

Correspondence

Mustafa Inc, Department of Mathematics, Firat University, 23119 Elazig, Turkiye.

Email: [email protected]

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Muhammad Sajid Iqbal, 

Muhammad Sajid Iqbal

Department of Academic Affairs, School of Leadership and Business, Oryx Universal College with Liverpool John Moores University (UK), Doha, Qatar

Department of Humanities & Basic Science, Military College of Signals, NUST, Islamabad, Pakistan

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Zil-e Huma, 

Zil-e Huma

Department of Mathematics, Minhaj University of Lahore, Lahore, Pakistan

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First published: 10 May 2025
https://doi.org/10.1002/zamm.70088

Abstract

This study investigates the magnetohydrodynamic (MHD) flow of a non-Newtonian fluid in an inclined tube, incorporating the effects of cilia-driven transport and convective boundary conditions. The governing equations are formulated considering the influence of magnetic field intensity, ciliary motion, and inclination angle. A numerical simulation is conducted to analyze the velocity and temperature profiles. The results indicate that an increase in the Hartmann number reduces the fluid velocity due to the Lorentz force, while higher values of the Weissenberg number enhance fluid elasticity, impacting flow characteristics. Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The HPM gives the approximated solution in the form of a rapidly convergent series with easily computable components. Unlike the method of separation of variables which requires both initial and boundary conditions, the HPM gives the solution by using the initial conditions only. Since after mathematically modeling our physical phenomenon converted in a highly nonlinear partial differential equation. So, for the best solutions we used this method. The inclination angle significantly affects the flow structure, leading to variations in shear stress and heat transfer rates. Quantitative results demonstrate that for a Hartmann number of five, the velocity decreases by approximately 18%, while increasing the Weissenberg number from 0.2 to 0.5 leads to a 12% rise in elasticity-driven effects. Moreover, the Nusselt number increases by 15% under higher convective boundary conditions, signifying improved heat transfer efficiency. These findings provide insight into optimizing fluid transport in biomedical and industrial applications.

1 INTRODUCTION

In biomechanics, simulation result is performed to explore medical technology issues. The subject of biomechanics, also referred to as biofluid dynamics, deals with the motion and behavior of bodily fluids in people, animals, and plants. The fluid flow in the blood capillaries, the breathing, the circulatory vessels, the induced gastric system, the urine system, and numerous other physiological situations are evaluated and analyzed by modern biofluid mechanics. Outcomes are essential for clinical applications such as the creation of prosthetic material membranes, the construction of vascular vessels, artificial organs, and other medical tools. Similar fluid-transport mechanisms can be observed across the human body, but peristalsis is one process that stands out. The Greek word peristaltic, which implies clasping and compressing, is where the word peristalsis comes.

Peristalsis is a fluid transport mechanism where sinusoidal waves move liquid along a chamber or cylinder. In the digestive tract, it is a reflexive muscle activity, characterized by wave-like motions that stretch and relax muscles to propel fluid forward. Abdellateef noted that the fluid moves due to a pressure gradient created by peristaltic waves [1]. This process is also observed in earthworms, caterpillars, millipedes and is mimicked in robotic designs. Peristalsis is a key fluid transport mechanism with applications in biological systems, human organs, and industries. It is involved in processes like swallowing, blood circulation, urination, and fluid flow in nuclear production. The heart's function also follows this principle. Many studies have examined peristaltic transport using various flow patterns and liquid models [2-5]. Tanveer et al. [6] explored the theoretical analysis of peristaltic blood flow of non-Newtonian fluids under the effects of magnetohydrodynamics (MHDs). They also discussed the entropy analysis of peristaltic Eyring–Powell flow nanofluid via an asymmetric channel. Javid et al. [7] investigated the biomechanics of biofluid peristaltic motion through a complex divergent wavy channel, controlled by electro-kinetic mechanisms.

There are several peristalsis-related structures in the literature due to the variety of its applications [8-14]. A long, thin tube with a tiny camera and light at one end is called an endoscope. In essence, it is a stiff ring-shaped cylinder set inside a peristaltic tube with an outside pipe that exhibits a pattern of expansion and contraction. The fluid flows through an opening between the endoscope and the peristaltic tube. Elboughdiri et al. [15] demonstrate that an endoscope is a crucial diagnostic tool for human organs that use peristaltic pumping to transport fluid. The contraction and expansion flow for a coupled stress fluid in a tube tilted with MHDs has the following endoscopic effects: narrated by Devakar et al. [16]. The literature presents the endoscopic impacts of different fluid models [17-19]. The analysis of entropy generation for hybrid nanoparticles in a hyperbolic tangent fluid via an endoscope is studied by Asha et al. [20]. Endoscopic impacts on various fluid model has been proposed in the literature [21-27]. The Jeffrey six-constant fluid fluid model is an essential non-Newtonian fluid framework for representing the stress-strain relationship and is commonly utilized in hydrodynamic applications to reduce friction.

The Jeffrey six-constant fluid model is a key non-Newtonian fluid framework used to describe the stress–strain relationship and is widely applied in hydrodynamic systems to minimize friction. The external magnetic field impacts Jeffrey six-constant fluid with variable heat transfer across an unstable stretching surface presented by Jawad et al. [28]. McCash et al. [29] studied chemical reactions of Jeffrey six-constant fluid across a curved surface incorporating activation energy. Shaheen and Asjad [30] narrated the peristaltically flowing Jeffrey six-constant fluid fluid flow in viscous dissipation effects in a convectively heated surface. Many Jeffrey six-constant fluid studies with unique flow shapes and sets of fluid models have been put forth in the literature [31-33]. MHD studies flow characteristics using magnetic phenomena. It involves the interaction of a magnetic field with a conducting fluid, generating currents that polarize the fluid and modify the magnetic field. The entropy generation of contraction and expansion flow for non-Newtonian fluid in horizontal ducts with MHD impacts. MHD is an unconventional process where heat energy is directly converted into electrical energy without involving mechanical energy. Examples of magnetic fluids include plasma, electrolytes, seawater, and liquid metals like mercury. Amanulla et al. [34] discussed numerical investigations on magnetic field modeling for Carreau non-Newtonian fluid flow past an isothermal sphere. Using external fields of electricity and magnetism. Nallagundla et al. [35] described a mathematical analysis of the transport phenomena of a non-Newtonian nanofluid flowing past a truncated cone under Newtonian heating conditions. An investigation from a theoretical perspective of a Biorheological fluid in the presence of electro-magneto-hydrodynamics explained by Amanulla et al. [36-38]. Nagendra et al. [39] studied the Hydromagnetic flow of heat and mass transfer. This paper's goal is to present such an approach for an MHD Jeffrey six-constant fluid. By selecting different material parameters, a Jeffrey six-constant fluid can exhibit many typical properties of Newtonian or non-Newtonian fluids, which is why it is being considered. The work's significance stems from the knowledge that myometrium contractions can occur in both directions and cause peristaltic-type intrauterine fluid flow. As a result, we have emphasized the peristaltic flow of nanofluid in a vertical endoscopic tube in the current study. A model of the two-dimensional nanofluid's governing equations in a vertical endoscopic tubes that takes curvature into account is created. The simplified highly nonlinear partial differential equation is solved analytically using the homotopy perturbation method following non-dimensionalization and with the assumptions of the long wave length and low Reynolds number approximation. Finally, graphs are plotted to discuss the physical phenomena.

2 FORMULATION OF PROBLEMS

We examine the mixed convection flow to explore the behavior of an MHD Jeffrey six-constant fluid under the influence of a convective boundary condition, as depicted in Figure 1. The fluid motion is induced by a sinusoidal wave propagating along the tube at a uniform velocity c.

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FIGURE 1
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Geometry of the problem.
Following are the basic equations for momentum, continuity, and heat equation [5]
∂ U ¯ ∂ R ¯ + U ¯ R ¯ + ∂ W ¯ ∂ Z ¯ = 0 , $$\begin{equation}\frac{{\partial \ \bar{U}}}{{\partial \ \bar{R}}} + \frac{{\bar{U}}}{{\bar{R}}} + \frac{{\partial \bar{W}}}{{\partial \bar{Z}}} = 0,\end{equation}$$ (1)
ρ ∂ ∂ t ¯ + U ¯ ∂ ∂ R ¯ + W ¯ ∂ ∂ Z ¯ U ¯ = − ∂ P ¯ ∂ R ¯ + 1 R ¯ ∂ ∂ R ¯ ( R ¯ τ ¯ + ∂ ∂ Z ¯ τ ¯ R ¯ Z ¯ + τ ¯ θ θ ¯ R ¯ − ρ β g T ¯ − T ¯ 0 c o s η , $$\begin{equation} \begin{aligned} \rho \left( {\frac{\partial }{{\partial \bar{t}}} + \bar{U}\frac{\partial }{{\partial \bar{R}\ }} + \bar{W}\frac{\partial }{{\partial \bar{Z}}}} \right)\ \ \bar{U} &= - \frac{{\partial \bar{P}}}{{\partial \bar{R}}} + \frac{1}{{\bar{R}}}\ \frac{\partial }{{\partial \bar{R}}}\ (\bar{R}\bar{\tau } + \frac{\partial }{{\partial \bar{Z}\ }}\left( {\bar{\tau }{\mathrm{\bar{R}\bar{Z}}}} \right)\\ &\quad + \frac{{{{\bar{\tau }}}_{\overline {\theta \theta } }}}{{\bar{R}}} - \rho \beta g\left( {\bar{T} - {{\bar{T}}}_0} \right)cos\eta , \end{aligned} \end{equation}$$ (2)
ρ ∂ ∂ t ¯ + U ¯ ∂ ∂ R ¯ + W ¯ ∂ ∂ Z ¯ W ¯ = − ∂ P ¯ ∂ Z ¯ + 1 R ¯ ∂ ∂ R ¯ R ¯ τ ¯ R ¯ Z ¯ + ∂ ∂ Z ¯ τ ¯ Z Z ¯ + ρ g α T ¯ − T ¯ 0 s i n η − σ β 0 2 W ¯ , $$\begin{equation} \begin{aligned} \rho \left( {\ \frac{\partial }{{\partial \bar{t}}} + \bar{U}\frac{\partial }{{\partial \bar{R}\ }} + \bar{W}\frac{\partial }{{\partial \bar{Z}\ }}} \right)\ \bar{W} &= - \frac{{\partial \bar{P}}}{{\partial \bar{Z}}} + \frac{1}{{\bar{R}}}\ \frac{\partial }{{\partial \bar{R}}}\ \left( {\bar{R}\bar{\tau }{\mathrm{\ \bar{R}\bar{Z}}}} \right) + \frac{\partial }{{\partial \bar{Z}\ }}\left( {{{\bar{\tau }}}_{\overline {ZZ} }} \right)\\ &\quad + \rho g\alpha \left( {\bar{T} - {{\bar{T}}}_0} \right)sin\eta - \sigma \beta _0^2\bar{W}, \end{aligned} \end{equation}$$ (3)
ρ c p ∂ ∂ t ¯ + U ¯ ∂ ∂ R ¯ + W ¯ ∂ ∂ Z ¯ T ¯ + k ∂ 2 T ¯ ∂ R ¯ 2 + 1 R ¯ ∂ T ¯ ∂ R ¯ + ∂ 2 T ¯ ∂ Z ¯ 2 + Q 0 $$\begin{equation}\rho {{\mathrm{c}}}_p\left( {\frac{\partial }{{\partial \bar{t}}} + \bar{U}\frac{\partial }{{\partial \bar{R}\ }} + \bar{W}\frac{\partial }{{\partial \bar{Z}}}} \right)\bar{T} + k\left( {\frac{{{\partial }^2\bar{T}}}{{\partial {{\bar{R}}}^2}} + \frac{1}{{\bar{R}\ }}\ \frac{{\partial \bar{T}}}{{\partial \bar{R}\ }} + \frac{{{\partial }^2\bar{T}}}{{\partial {{\bar{Z}}}^2}}} \right) + {Q}_0\end{equation}$$ (4)

The velocity components in the axial and radial directions are represented by (U, W), while denotes the pressure. Q0 signifies the steady-state heat absorption parameter, where k represents the thermal conductivity of the fluid and cp ​ denotes its specific heat capacity.

The transformations applied to the two reference frames are as follows.
r ¯ = R ¯ , z ¯ = Z ¯ − c t ¯ , $$\begin{equation*}\ \bar{r} = \bar{R},\ \ \ \ \ \ \bar{z} = \bar{Z}\ - c\bar{t},\end{equation*}$$
u ¯ = U ¯ , w ¯ = W ¯ − c 1 . $$\begin{equation*}\ \bar{u} = \bar{U},\ \ \bar{w} = \bar{W}\ - {c}_1.\end{equation*}$$
The constitutive equation for the formulation of a Jeffrey six-constant fluid is expressed as follows.
τ ¯ + ε 1 d τ ¯ d t − W . τ ¯ + τ ¯ . W + d τ ¯ . D + D . τ ¯ + b τ ¯ : D I + c D t r τ ¯ = 2 μ [ D + ε 2 ( d D d t − W . D + D . W + 2 d D . D + b D . $$\begin{equation} \begin{aligned} & \bar{\tau } + {\varepsilon }_1\left[ {\frac{{d\bar{\tau }}}{{dt}} - W.\bar{\tau } + \bar{\tau }.W + d\left( {\bar{\tau }.D + D.\bar{\tau }} \right) + b\bar{\tau }\ :DI + cDtr\bar{\tau }} \right]\\ &\quad = 2\mu [D + {\varepsilon }_2(\frac{{dD}}{{dt}} - W.D + D.W + 2dD.D + bD. \end{aligned} \end{equation}$$ (5)
where

D = $D = $ (portion of the velocity gradient that is symmetric) = ∇ v ¯ + ( ∇ v ¯ ) T 1 2 , $ = \frac{{\nabla \bar{v} + ( {\nabla \bar{v}} ){T}_1}}{2}\ ,$

W = $W = $ (unbalanced element of the velocity gradient) = $ = $ ∇ v ¯ − ( ∇ v ¯ ) T 1 2 , $\frac{{\nabla \bar{v} - ( {\nabla \bar{v}} ){T}_1}}{2},$

In the physical model, we analyze ciliate tubes where cilia are attached near the base along the inner walls. Their oscillatory motion generates fluid flow. The movement of cilia tips follows a circular trajectory, which is mathematically expressed as follows.
R ̌ = H = f ( z , t ) = a + a ε cos 2 π λ ( Z = − c t = ) , $$\begin{equation}\check{R} = H = f ({z,t}) = a + a\epsilon \cos{\left(\frac{2\pi}{\lambda}(\mathop Z\limits^{=} - c\mathop t\limits^{=}) \right)},\end{equation}$$ (6)
Z = g Z , Z o , t = Z o , + a + a ε α 2 π λ Z = c t = . $$\begin{equation}Z = g\ \ \left( {Z,{Z}_{o,\ }t} \right) = {Z}_{o,\ } + a + a\epsilon {{\alpha}}\left( {\frac{{2\pi }}{\lambda }\left( {\mathop Z\limits^{=} c\mathop t\limits^{=} } \right)} \right).\end{equation}$$ (7)
Under the no-slip condition, the eccentricity of the ellipse is considered. The velocity of the cilia tips is represented by the Z and R components of velocity.
W = ∂ z ∂ t | Z = Z o = ∂ g ∂ t + ∂ g ∂ z ∂ z ∂ t = ∂ g ∂ t + ∂ g ∂ z $$\begin{equation}W = \frac{{\partial z}}{{\partial t}}\ {\bigg|}_{Z = {Z}_{o\ } = }\ \frac{{\partial g}}{{\partial t}} + \frac{{\partial g\ }}{{\partial z}}\ \frac{{\partial z}}{{\partial t}} = \frac{{\partial g}}{{\partial t}} + \frac{{\partial g}}{{\partial z}}\end{equation}$$ (8)
and
U = ∂ R ∂ t | Z = Z o = ∂ f ∂ t + ∂ f ∂ z ∂ z ∂ t = ∂ f ∂ t + ∂ f ∂ z W . $$\begin{equation}U = \frac{{\partial R}}{{\partial t}}\ {\bigg|}_{Z = {Z}_{o\ } = }\ \frac{{\partial f}}{{\partial t}} + \frac{{\partial f\ }}{{\partial z}}\ \frac{{\partial z}}{{\partial t}} = \frac{{\partial f}}{{\partial t}}\ + \frac{{\partial f\ }}{{\partial z}}W.\end{equation}$$ (9)
Substituting Equations (6) and (7) into Equations (8) and (9), we derive the following.
W = − 2 π λ ∈ a α c cos 2 π λ Z − c t 1 − 2 π λ ∈ a α c cos 2 π λ Z − c t $$\begin{equation}W = \frac{{ - \left( {\frac{{2\pi }}{\lambda }} \right)\left[ {\in a\alpha {\mathrm{c}}\cos \left( {\frac{{2\pi }}{\lambda }} \right)\left( {Z - ct} \right)} \right]}}{{1 - \left( {\frac{{2\pi }}{\lambda }} \right)\left[ {\in a\alpha {\mathrm{c}}\cos \left( {\frac{{2\pi }}{\lambda }} \right)\left( {Z - ct} \right)} \right]}}\end{equation}$$ (10)
and
W = 2 π λ a ∈ α c sin 2 π λ Z − c t 1 − 2 π λ ∈ a α c cos 2 π λ Z − c t . $$\begin{equation}W = \frac{{\left( {\frac{{2\pi }}{\lambda }} \right)\left[ {a\in \alpha {\mathrm{c}}\sin \left( {\frac{{2\pi }}{\lambda }} \right)\left( {Z - ct} \right)} \right]}}{{1 - \left( {\frac{{2\pi }}{\lambda }} \right)\left[ {\in a\alpha {\mathrm{c}}\cos \left( {\frac{{2\pi }}{\lambda }} \right)\left( {Z - ct} \right)} \right]}}.\end{equation}$$ (11)
The corresponding boundary conditions
∂ W = ∂ R = = 0 , ∂ T = ∂ R = = 0 at R = = 0 $$\begin{equation*}\ \frac{{\partial \mathop W\limits^{=} }}{{\partial \mathop R\limits^{=} }} = 0,\ \ \ \frac{{\partial \mathop T\limits^{=} }}{{\partial \mathop R\limits^{=} }} = 0\ \ \ \ {\rm at}\ \ \mathop R\limits^{=} = 0\end{equation*}$$
W ¯ = λ x , y , k ∂ T = ∂ R = = η τ − T = 0 $$\begin{equation*}\ \bar{W} = \lambda \ \left( {x,y} \right),\quad k\ \frac{{\partial \mathop T\limits^{=} }}{{\partial \mathop R\limits^{=} }} = \eta \ \left( {\tau - \ {{\mathop T\limits^{=} }}_0} \right)\end{equation*}$$
a t R ¯ = H ¯ a + b cos 2 π λ Z = − c t = . $$\begin{equation*}at\ \bar{R} = \bar{H}\ a + b\cos \left[ {\frac{{2\pi }}{\lambda }\ \left( {\mathop Z\limits^{=} - \mathop {ct}\limits^{=} } \right)} \right].\end{equation*}$$
The following nondimensional item is presented:
R = R ¯ a , r = r ¯ a , Z = Z ¯ λ , z = z ¯ λ , W = W ¯ c 1 , $$\begin{equation*}R = \frac{{\bar{R}}}{a},\quad r = \frac{{\bar{r}}}{a},\quad Z = \frac{{\bar{Z}}}{\lambda },\quad z = \frac{{\bar{z}}}{\lambda },\quad W = \frac{{\bar{W}}}{{{c}_1}}\ ,\end{equation*}$$
w = w ¯ c 1 , τ = a τ ¯ c μ 0 , U = λ U ¯ a c 1 , u = λ u ¯ a c 1 , $$\begin{equation*}w = \frac{{\bar{w}}}{{{c}_1}},\quad \tau = \frac{{a\bar{\tau }}}{{c{\mu }_0}},\quad U = \frac{{\lambda \bar{U}}}{{a{c}_1}},\quad u = \frac{{\lambda \bar{u}}}{{a{c}_1}}\ ,\end{equation*}$$
t = c 1 t ¯ λ , λ 1 = ε 1 c 1 a , λ 2 = ε 2 c 1 a , δ = a λ , R e = a c 1 ρ μ 0 , $$\begin{equation*}t = \frac{{{c}_1\bar{t}}}{\lambda }\ ,{\mathrm{\ \ }}{\lambda }_1 = \frac{{{\varepsilon }_1{c}_1}}{a}\ ,{\mathrm{\ \ }}{\lambda }_2 = \frac{{{\varepsilon }_2{c}_1}}{a},\quad \delta = \frac{a}{\lambda },\quad R{\mathrm{e\ }} = \frac{{a{c}_1\rho }}{{{\mu }_0}}\ ,\end{equation*}$$
E c = c 2 c p T 0 , h = h ¯ a = 1 + ε c o s 2 π z , $$\begin{equation*}\ {E}_c = \frac{{{c}^2}}{{{c}_p{T}_0}},\quad h = \frac{{\bar{h}}}{a} = 1 + \epsilon cos2\pi z,\end{equation*}$$
p r = μ 0 c p k , μ θ = μ ¯ θ μ 0 , β 1 = Q 0 a 2 k T 0 , θ = T ¯ − T ¯ 0 T ¯ 0 , $$\begin{equation*}\ {p}_r = \frac{{{\mu }_0{c}_p}}{k},\quad \mu \ \left( \theta \right) = \frac{{\bar{\mu }\left( \theta \right)}}{{{\mu }_0}},\quad {\beta }_1 = \frac{{{Q}_0{a}^2}}{{k{T}_0}},\quad \theta = \frac{{\bar{T} - {{\bar{T}}}_0}}{{{{\bar{T}}}_0}}\ ,\end{equation*}$$
α = K ( ρ c ) f , P = a 2 P ¯ c 1 λ μ 0 , M = σ μ B 0 a . $\alpha = \frac{K}{{{{( {\rho c} )}}_f}},\ P = \frac{{{a}^2\bar{P}}}{{{c}_1\lambda {\mu }_0}}, {\mathrm{\ M\ }} = \sqrt {\frac{\sigma }{\mu }} \ {B}_0a .$
After solving the nondimensional variable Equations (1)–(4) are presented here
∂ u ∂ r + u r + ∂ w ∂ z = 0 , $$\begin{equation*}\frac{{\partial u}}{{\partial r}} + \frac{u}{r} + \frac{{\partial w}}{{\partial z}} = 0,\end{equation*}$$
δ 3 R e u ∂ u ∂ r + w ∂ u ∂ z = − ∂ p ∂ r + δ 2 ∂ ∂ z τ rz + δ r ∂ ∂ r r τ rr − δ r τ θ θ − δ G r Cos η θ , $$\begin{equation*} \begin{aligned} {\delta }^3{R}_e\ \left( {u\frac{{\partial u}}{{\partial r}} + w\frac{{\partial u}}{{\partial z}}} \right) &= \frac{{ - \partial p}}{{\partial r}} + {\delta }^2\frac{\partial }{{\partial z}}\left( {{\tau }_{{\mathrm{rz}}}} \right)\\ &\quad + \frac{\delta }{r}\frac{\partial }{{\partial r}}\left( {{\mathrm{r}}{\tau }_{{\mathrm{rr}}}} \right) - \frac{\delta }{r}{\tau }_{\theta \theta } - \delta {G}_r{\mathrm{Cos}}\left( \eta \right)\theta , \end{aligned} \end{equation*}$$
δ R e u ∂ w ∂ r + w ∂ w ∂ z = − ∂ p ∂ z + 1 r ∂ ∂ r r τ rz + δ ∂ ∂ z τ zz + G r Sin η θ − M 2 w + 1 , $$\begin{equation*} \begin{aligned} {\mathrm{\delta R}}_e\ \left( {u\frac{{\partial w}}{{\partial r}} + w\frac{{\partial w}}{{\partial z}}} \right)& = - \frac{{\partial p}}{{\partial z}} + \frac{1}{r}\frac{\partial }{{\partial r}}\left( {{\mathrm{r}}{\tau }_{{\mathrm{rz}}}} \right)\\ &\quad + \delta \frac{\partial }{{\partial z}}\left( {{\tau }_{{\mathrm{zz}}}} \right) + {G}_r{\mathrm{Sin}}\left( {{\eta}} \right){{\theta}} - {M}^2\left( {w + 1} \right), \end{aligned} \end{equation*}$$
δ R e P r u ∂ θ ∂ r + w ∂ θ ∂ z = 1 r ∂ θ ∂ r + ∂ 2 θ ∂ r 2 + B . $$\begin{equation*}{\mathrm{\delta R}}_e{P}_r\left( {u\frac{{\partial {{\theta}}}}{{\partial r}} + w\frac{{\partial {{\theta}}}}{{\partial z}}} \right) = \left[ {\frac{1}{r}\frac{{\partial {{\theta}}}}{{\partial r}} + \frac{{{\partial }^2\theta }}{{\partial {r}^2}}} \right] + B.\end{equation*}$$
The following boundary values are defined.
∂ w ∂ r = 0 , ∂ θ ∂ r = 0 at r = 0 , $$\begin{equation*}\ \frac{{\partial w}}{{\partial r}} = 0,\quad \frac{{\partial {{\theta}}}}{{\partial r}} = 0\ \ {\mathrm{at\ \ \ }}r = 0,\end{equation*}$$
w = − 1 − 2 π ε δ α α cos 2 π z , $$\begin{equation*}w = - 1 - 2\pi \epsilon \delta \alpha \alpha \cos \left( {2\pi z} \right),\end{equation*}$$
∂ θ ∂ r + K θ = 0 , r = h = 1 + ε cos 2 π z . $$\begin{equation*}\frac{{\partial \theta }}{{\partial r}} + {\mathrm{{\mathrm K}}}\theta = 0,\quad r = h = 1 + \epsilon {\mathrm{\ cos}}\left( {2\pi {\mathrm{z}}} \right).\end{equation*}$$
Simplify the analysis, the assumptions of a low Reynolds number approximation and a long-wavelength condition δ ≪ ${{\delta}} \ll $ 1 are considered.
∂ P ∂ Z = 1 r ∂ ∂ r r τ rz + G r Sin η θ − M 2 w + 1 = 0 , $$\begin{equation}\frac{{\partial P}}{{\partial Z}} = \frac{1}{r}\ \frac{\partial }{{\partial r}}\left( {r{\tau }_{{\mathrm{rz}}}} \right) + {G}_r\left( {{\mathrm{Sin}}\eta \ } \right)\theta - {M}^2\left( {w + 1} \right) = 0,\end{equation}$$ (12)
∂ P ∂ r = 0 , $$\begin{equation}\frac{{\partial P}}{{\partial r}} = 0,\end{equation}$$ (13)
1 r ∂ θ ∂ r + ∂ 2 θ ∂ r 2 + β = 0 , $$\begin{equation}\frac{1}{r}\frac{{\partial \theta }}{{\partial r}} + \frac{{{\partial }^2\theta }}{{\partial {r}^2}} + \beta = 0,\end{equation}$$ (14)
∂ w ∂ r = 0 , ∂ θ ∂ r = 0 at r = 0 , $$\begin{equation*}\frac{{\ \partial w}}{{\partial r}} = 0,{\mathrm{\ \ }}\frac{{\partial \theta }}{{\partial r}} = 0{\mathrm{\ \ \ \ at\ \ \ \ \ }}r = 0,\end{equation*}$$
w = − 1 − 2 π ε δ α cos 2 π z , ∂ θ ∂ r + k θ = 0 $$\begin{equation*}w = - 1 - 2\pi \epsilon \delta \alpha \cos \left( {2\pi z} \right),\ \ \frac{{\partial \theta }}{{\partial r}} + k\theta = 0\end{equation*}$$
at r = h = 1 + ε cos 2 π z , $$\begin{equation*}{\mathrm{at\ }}r = h = 1 + \epsilon \cos \left( {2\pi z} \right),\end{equation*}$$
where
τ r z = ∂ w / ∂ r [ 1 + λ 1 λ 2 ( 1 − d d + b − c / 2 2 d + 3 b ∂ w ∂ r 2 1 + λ 1 2 1 − d d + b − c / 2 2 d + 3 b ( ∂ w ∂ r 2 $$\begin{equation*}\ \ {\tau }_{rz} = \frac{{\partial w/\partial r\ [1 + {\lambda }_1{\lambda }_2(1 - d\left( {d + b} \right) - c/2\ \left( {2d + 3b} \right){{\left( {\frac{{\partial w}}{{\partial r}}} \right)}}^2}}{{1 + \lambda _1^2{{\left( {1 - d\left( {d + b} \right) - c/2\ \left( {2d + 3b} \right)(\frac{{\partial w}}{{\partial r}}} \right)}}^2}}\end{equation*}$$
τ r r = λ 2 ∂ w ∂ r 2 1 + d + b − λ 1 ∂ w ∂ r 1 + d + b τ r z , $$\begin{equation*}\ {\tau }_{rr} = {\lambda }_2\ {\left( {\frac{{\partial w}}{{\partial r}}} \right)}^2\left( {1 + d + b} \right) - {\lambda }_1\left( {\frac{{\partial w}}{{\partial r}}} \right)\left( {1 + d + b} \right){\tau }_{rz},\end{equation*}$$
τ z z = λ 2 ∂ w ∂ r 2 − 1 + d + b − λ 1 ∂ w ∂ r − 1 + d + b τ r z , $$\begin{equation*}\ {\tau }_{zz} = {\lambda }_2\ {\left( {\frac{{\partial w}}{{\partial r}}} \right)}^2\left( { - 1 + d + b} \right) - {\lambda }_1\left( {\frac{{\partial w}}{{\partial r}}} \right)\left( { - 1 + d + b} \right){\tau }_{rz},\end{equation*}$$
τ θ θ = λ 2 ∂ w ∂ r 2 b − λ 1 ∂ w ∂ − b τ r z , $$\begin{equation*}\ {\tau }_{\theta \theta } = {\lambda }_2\ {\left( {\frac{{\partial w}}{{\partial r}}} \right)}^2b - {\lambda }_1\left( {\frac{{\partial w}}{{\partial - }}} \right)b{\tau }_{rz},\end{equation*}$$
Ultimately, the equations can be rewritten in their most simplified form as follows:
∂ P ∂ r = 0 , $$\begin{equation*}\ \frac{{\partial P}}{{\partial r}} = 0,\end{equation*}$$
∂ P ∂ z = 1 r ∂ ∂ r r ∂ w ∂ r + 1 r ∂ ∂ r α 1 α 2 r ∂ w ∂ r 3 + 1 r ∂ ∂ r α 1 2 α 3 r ∂ w ∂ r 5 + G r s i n η θ − M 2 w + 1 $$\begin{equation*} \begin{aligned} \frac{{\partial P}}{{\partial z}} &= \frac{1}{r}\frac{\partial }{{\partial r}}\left( {r\frac{{\partial w}}{{\partial r}}} \right) + \frac{1}{r}\frac{\partial }{{\partial r}}\left( {{\alpha }_1{\alpha }_2r{{\left( {\frac{{\partial w}}{{\partial r}}} \right)}}^3} \right) + \frac{1}{r}\frac{\partial }{{\partial r}}\left( {\alpha _1^2{\alpha }_3r{{\left( {\frac{{\partial w}}{{\partial r}}} \right)}}^5\ } \right)\\ &\quad + G{\mathrm{r}}\left( {sin\eta } \right)\theta \ - {M}^2\left( {w + 1} \right) \end{aligned} \end{equation*}$$
0 = 1 r ∂ θ ∂ r + ∂ 2 θ ∂ 2 r + B . $$\begin{equation*}0 = \frac{1}{r}\ \left[ {\ \frac{{\ \partial \theta }}{{\ \partial r}} + \frac{{{\partial }^2\theta }}{{{\partial }^2r}}} \right] + B.\end{equation*}$$
The following are the specified boundary conditions:
∂ w ∂ r = 0 , ∂ θ ∂ r = 0 at r = 0 , $$\begin{equation*}\ \frac{{\partial w}}{{\partial r}} = 0,\ \ \frac{{\partial \theta }}{{\partial r}} = 0\ \ \ {\mathrm{at\ \ \ }}r = 0,\end{equation*}$$
w = − 1 − 2 π ε δ α cos 2 π z , ∂ θ ∂ r + k θ = 0 at r = h = 1 + ε cos 2 π z , $$\begin{equation*}w = - 1 - 2\pi \epsilon \delta \alpha \cos \left( {2\pi z} \right),\ \frac{{\partial \theta }}{{\partial r}} + k\theta = 0{\mathrm{\ \ at\ \ }}r = h = 1 + \epsilon \cos \left( {2\pi z} \right),\end{equation*}$$
In which
α 1 = 1 − d d + b − c 2 2 d + 3 b , α 2 = λ 1 λ 2 − λ 1 , α 3 = − λ 1 3 λ 2 . $$\begin{equation}\ {\alpha }_1 = 1 - d\left( {d + b} \right) - \frac{c}{2}\left( {2d + 3b} \right),{\mathrm{\ \ }}{\alpha }_2 = {\lambda }_1\ {\lambda }_2 - {\lambda }_1,{\mathrm{\ \ }}{\alpha }_3 = - \lambda _1^3{\lambda }_2.\end{equation}$$ (15)

3 SOLUTION OF THE PROBLEM

The exact solutions for the temperature, satisfying the corresponding boundary conditions, are explicitly provided
θ = − β r 2 4 + 1 K β h 2 + β K h 2 4 . $$\begin{equation}\theta = - \beta \frac{{{r}^2}}{4} + \frac{1}{\rm K}\left[ {\frac{{\beta {\mathrm{h}}}}{2} + \frac{{\beta {\mathrm{{\mathrm K}}}{{\mathrm{h}}}^2}}{4}} \right].\end{equation}$$ (16)
HPM is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The HPM gives the approximated solution in the form of a rapidly convergent series with easily computable components. Unlike the method of separation of variables which requires both initial and boundary conditions. The HPM gives the solution by using the initial conditions only. Since after mathematically modeling our physical phenomenon converted in a highly nonlinear partial differential equation. So, for the best solutions we used this method.
H q , w = L w − L w 10 + qL w 10 + q 1 r ∂ ∂ r ( α 1 α 2 r ∂ w ∂ r 3 + 1 r ∂ ∂ r α 1 2 α 3 r ∂ w ∂ r 5 + G r Sin η θ − M 2 w + 1 − ∂ P ∂ z . $$\begin{align} H\ \left( {q,w} \right) &= L\left( w \right) - L\left( {{w}_{10}} \right) + {\mathrm{qL}}\left( {{w}_{10}} \right)\nonumber\\ &\quad + q\left[ {\left( {\frac{1}{r}\frac{\partial }{{\partial r}}({\alpha }_1{\alpha }_2r{{\left( {\frac{{\partial w}}{{\partial r}}} \right)}}^3} \right) + \frac{1}{r}\frac{\partial }{{\partial r}}\left( {{\alpha }_1^2{\alpha }_3r{{\left( {\frac{{\partial w}}{{\partial r}}} \right)}}^5} \right) + {G}_r\left( {{\mathrm{Sin}}\eta } \right)\theta - \ {M}^2\left( {w + 1} \right) - \ \frac{{\partial P}}{{\partial z}}} \right]. \end{align}$$ (17)
The initial guesses can be defined as follows, ensuring compliance with the boundary conditions described in ref. [16]:
W 10 r , z = − 1 − 2 π ε δ α cos 2 π z + P 0 r 2 − h 2 4 . $$\begin{equation}{W}_{10}\left( {r,z} \right) = - 1 - 2\pi \epsilon \delta \alpha \cos \left( {2\pi z} \right) + {P}_0\left[ {\frac{{{r}^2 - {h}^2}}{4}} \right].\end{equation}$$ (18)
Let us define
w r , q = w 0 + q w 1 + q 2 w 2 + q 3 w 3 + ⋯ W = − 1 − 2 π ε δ α cos 2 π z + r 2 − h 2 4 d p d z + r 2 − h 2 4 ∂ p 0 ∂ z + r 4 − h 4 32 α 1 α ∂ p 0 ∂ z 3 + r 6 − h 6 192 α 1 2 α ∂ p 0 ∂ z 5 + r 2 − h 2 4 B h 2 + r 2 − h 2 64 B G S i n η + B h 2 K r − h + r 2 − h 2 16 h 2 M 2 ∂ p 0 ∂ z + r 4 − h 4 64 M 2 ∂ p 0 ∂ z + M 2 π r 2 − h 2 2 δ α cos 2 π z + r 2 − h 2 4 d p 1 d z . $$\begin{equation} \begin{aligned} w\left( {r,q} \right) &= {w}_0 + q{w}_1 + {q}^2{w}_2 + {q}^3{w}_3 + \cdots\\ W &= - 1 - 2\pi \epsilon \delta \alpha \cos \left( {2\pi z} \right) + \frac{{\left( {{r}^2 - {h}^2} \right)}}{4}\frac{{dp}}{{dz}} + \frac{{\left( {{r}^2 - {h}^2} \right)}}{4}\frac{{\partial {p}_0}}{{\partial z}}\\ &\quad + \frac{{\left( {{r}^4 - {h}^4} \right)}}{{32}}{\alpha }_1\alpha {{\left( {\frac{{\partial {p}_0}}{{\partial z}}} \right)}}^3 + \frac{{\left( {{r}^6 - {h}^6} \right)}}{{192}}{\alpha }_{1\ }^2\alpha {{\left( {\frac{{\partial {p}_0}}{{\partial z}}} \right)}}^5\\ &\quad + \frac{{\left( {{r}^2 - {h}^2} \right)}}{4}B{h}^2 + \frac{{\left( {{r}^2 - {h}^2} \right)}}{{64}}BGSin\eta + \frac{{Bh}}{{2\rm K}}\left( {r - h} \right)\\ &\quad + \frac{{\left( {{r}^2 - {h}^2} \right)}}{{16}}{h}^2{M}^2\frac{{\partial {p}_0}}{{\partial z}} + \frac{{\left( {{r}^4 - {h}^4} \right)}}{{64}}{M}^2\frac{{\partial {p}_0}}{{\partial z}}\\ &\quad + {M}^2\pi \frac{{\left( {{r}^2 - {h}^2} \right)}}{2}\delta \alpha {\mathrm{cos}}\left( {2\pi z} \right) + \frac{{\left( {{r}^2 - {h}^2} \right)}}{4}\frac{{d{p}_1}}{{dz}} \end{aligned} .\end{equation}$$ (19)
The pressure gradient will be defined as
d p d z = − 16 h 4 ( F + h 2 2 + h 2 π α δ ε Cos 2 π z + ( − B h 6 16 − h 4 p 0 16 − 1 96 h 6 M 2 p 0 + 1 192 B G h 6 sin η − B h 4 12 K − 1 8 h 4 M 2 π α δ ε Cos 2 π z − 1 96 h 6 p 0 3 α 1 α 2 − 1 512 h 8 p 0 5 α 3 α 1 2 ) ) . $$\begin{align*} \frac{{dp}}{{dz}} &= - \frac{{16}}{{{h}^4}}\Bigg( F + \frac{{{h}^2}}{2} + {h}^2\pi \alpha \delta \epsilon {\mathrm{Cos}}\left[ {2\pi z} \right] + \Bigg( - \frac{{B{h}^6}}{{16}} - \frac{{{h}^4{\mathrm{p}}0}}{{16}} - \frac{1}{{96}}{h}^6{M}^2{\mathrm{p}}0 + \frac{1}{{192}}BG{h}^6{\mathrm{sin}}\eta\\ &\quad - \frac{{B{h}^4}}{{12\rm K}} - \frac{1}{8}{h}^4{M}^2\pi \alpha \delta \epsilon {\mathrm{Cos}}\left[ {2\pi z} \right] - \frac{1}{{96}}{h}^6{\mathrm{p}}{0}^3{\alpha }_1{\alpha }_2 - \frac{1}{{512}}{h}^8{\mathrm{p}}{0}^5{\alpha }_3{\alpha }^2_1 \Bigg) \Bigg). \end{align*}$$
Flow rate can be appearing as
F = Θ + 1 2 ε 2 2 + 1 . $$\begin{equation}F = \Theta + \frac{1}{2}\left( {\frac{{{\epsilon }^2}}{2} + 1} \right).\end{equation}$$ (20)
The pressure rise (Δp) and friction forces (F) are defined as follows:
Δ p = ∫ 0 1 ( d p d z ) d z , $$\begin{equation}\Delta p = \int _0^1 (\frac{{dp}}{{dz}}\ )dz,\end{equation}$$ (21)
F = ∫ 0 1 h 2 ( − d p d z ) d z . $$\begin{equation}F = \int _0^1 {h}^2( - \frac{{dp}}{{dz}}\ )dz.\end{equation}$$ (22)
Velocities in terms of stream functions are defined as
u = − 1 r ∂ Ψ ∂ z and w = 1 r ∂ Ψ ∂ r . $$\begin{equation}u = \frac{{ - 1}}{r}\ \left( {\frac{{\partial \Psi }}{{\partial z}}} \right){\mathrm{\ and\ }}w = \frac{1}{r}\ \left( {\frac{{\partial \Psi }}{{\partial r}}} \right).\end{equation}$$ (23)
Ψ = − r 2 2 + 1 8 B h 4 r 2 − 1 8 h 2 p r 2 + 3 128 h 4 M 2 p 0 r 2 − 1 16 B h 2 r 4 + p r 4 16 − 1 64 h 2 M 2 p 0 r 4 + 1 384 M 2 p 0 r 6 − 1 128 B G h 4 r 2 sin η + 1 384 B G r 6 sin η + B h 2 r 2 4 K − B h r 3 6 K − π r 2 α δ ε Cos 2 π z + 1 4 h 2 M 2 π r 2 α δ ε Cos 2 π z − 1 8 M 2 π r 4 α δ ε Cos 2 π z + 1 64 h 4 p 0 3 r 2 α 1 α 2 − 1 192 p 0 3 r 6 α 1 α 2 + 1 384 h 6 p 0 5 r 2 α 3 α 1 2 − p 0 5 r 8 α 3 α 1 2 1536 . $$\begin{equation*} \begin{aligned} \Psi &= - \frac{{{r}^2}}{2} + \frac{1}{8}B{h}^4{r}^2 - \frac{1}{8}{h}^2p{r}^2 + \frac{3}{{128}}{h}^4{M}^2{\mathrm{p}}0{r}^2 - \frac{1}{{16}}B{h}^2{r}^4\\ &\quad + \frac{{p{r}^4}}{{16}} - \frac{1}{{64}}{h}^2{M}^2{\mathrm{p}}0{r}^4 + \frac{1}{{384}}{M}^2{\mathrm{p}}0{r}^6 - \frac{1}{{128}}BG{h}^4{r}^2{\mathrm{sin}}\eta \\ &\quad + \frac{1}{{384}}BG{r}^6{\mathrm{sin}}\eta + \frac{{B{h}^2{r}^2}}{{4\rm K}} - \frac{{Bh{r}^3}}{{6\rm K}} - \pi {r}^2\alpha \delta \epsilon {\mathrm{Cos}}\left[ {2\pi z} \right]\\ &\quad + \frac{1}{4}{h}^2{M}^2\pi {r}^2\alpha \delta \epsilon {\mathrm{Cos}}\left[ {2\pi z} \right] - \frac{1}{8}{M}^2\pi {r}^4\alpha \delta \epsilon {\mathrm{Cos}}\left[ {2\pi z} \right]\\ &\quad + \frac{1}{{64}}{h}^4{\mathrm{p}}{0}^3{r}^2{\alpha }_1{\alpha }_2 - \frac{1}{{192}}{\mathrm{p}}{0}^3{r}^6{\alpha }_1{\alpha }_2\\ &\quad + \frac{1}{{384}}{h}^6{\mathrm{p}}{0}^5{r}^2{\alpha }_3{\alpha }^2_1 - \frac{{{\mathrm{p}}{0}^5{r}^8{\alpha }_3{\alpha }^2_1}}{{1536}} \end{aligned} .\end{equation*}$$
For the flow analysis, we have considered three wave frames, namely, sinusoidal wave, trapezoidal wave, and multisinusoidal wave. The dimensionless equations can be written as

  1. Sinusoidal wave

    h z = 1 + ε cos 2 π z $$\begin{equation*}h\ \left( z \right) = 1 + \epsilon {\mathrm{cos}}\left( {2\pi z} \right)\end{equation*}$$

  2. Multisinusoidal wave

    h z = 1 + ε cos 2 m π z $$\begin{equation*}h\ \left( z \right) = 1 + \epsilon {\mathrm{cos}}\left( {2m\pi z} \right)\end{equation*}$$

  3. Trapezoidal wave

    h z = 1 + ε 32 π 2 ∑ n = 1 ∞ sin π 8 2 n − 1 2 n − 1 2 cos 2 π 2 n − 1 z , $$\begin{equation*}{\mathrm{h\ }}\left( z \right) = 1 + \epsilon \ \left\{ {\frac{{32}}{{{\pi }^2}}\sum_{n = 1}^\infty \frac{{{\mathrm{sin}}\frac{\pi }{8}\left( {2n - 1} \right)}}{{{{\left( {2n - 1} \right)}}^2}}\cos \left( {2\pi \left( {2n - 1} \right)z} \right)} \right\},\end{equation*}$$

  4. Square wave

    h = 1 + ε ∗ 4 3.1417 ∑ n = 1 5 − 1 n + 1 ∗ Cos 2 ∗ 3.1417 ∗ 2 n − 1 ∗ z 2 n − 1 . $$\begin{equation*}h = 1 + \epsilon *\left( {\frac{4}{{\left( {3.1417} \right)}}\sum_{n = 1}^5 \left( {\frac{{{{\left( { - 1} \right)}}^{n + 1}*{\mathrm{Cos}}\left[ {2*3.1417*\left( {2n - 1} \right)*\left( z \right)} \right]}}{{\left( {2n - 1} \right)}}} \right)} \right).\end{equation*}$$

4 RESULTS AND DISCUSSIONS

This section examines the behavior of an MHD non-Newtonian fluid under various effects and boundary condition under the tube. The influence of B (the heat source or sink parameter) and the temperature profile is illustrated in Figures 2 and 3. As shown in Figure 2, an increase in B reduces heat conductivity, leading to a decrease in overall temperature, while a higher B enhances the amplitude of the temperature distribution. Figures 4 and 5 indicate that increase in the Jeffrey six-constant parameters (α1, α2) results in a reduction in the dimensionless velocity amplitude. However, near the peristaltic walls, the velocity exhibits an opposite trend as shown in Figure 6, an increase in MHDs enhances the velocity profile at the center, while it decreases in the region near the pipe walls. Physically, MHD influences heat transfer by modifying convection currents. When a magnetic field is applied, it can either enhance or suppress thermal conductivity, depending on fluid properties and field strength. The presence of a strong magnetic field can suppress turbulence and enhance stability, which is beneficial in nuclear fusion reactors and metallurgical processes. We prepared the graph in Figures 7 and 8 to evaluate how the Jeffrey six-constant and amplitude ratio influence the pressure gradient dp/dz. Here, it can be seen that the pressure gradient across different places in the flow is a decreasing function. However, it has also been determined from this graph that dp/dz rises at the distinct ranges [−0.3, −0.8] and [0.3–0.8] of the container but drops in the middle of the flow. With the use of the Jeffrey six-constant ( α 1 ${\alpha }_1$ ) and the MHDs equation (M) from Figures 9 and 10, we can examine the variation in pressure gradient. When the magnitude of both parameters is changed over the flow, we can see that the pressure gradient profile directly reduces. Figure 11 represents the effects of parameters α $\alpha $ on the pressure rise. It is noticed here that pressure rise is a decreasing function in the boundaries of the tube and free pumping assumed in the center of the tube. Figure 12 represented frictional forces have an opposite behavior of pressure rise. The effects of the parameter on the pressure rise are indicated in Figures 11-13. Here, it must be understood that the pressure rise has a decreasing function near the tube's margins while free pumping is assumed at the tube's center. Frictional forces demonstrate an opposite conduct of pressure growth, as shown in Figures 12-14. Trapping is a very intriguing mechanism for the flow of fluids. Under certain circumstances, streamlines in the wave frame can swell to capture a bolus that moves as an inlet at the wave speed. Trapping is the phenomenon of an internally circulating bolus set by closed simplifying. The wave pattern moves the bolus, which is a volume of fluid enclosed by closed streamlines in the wave frame. The streamlines for various amplitude ratio parameter values are shown in Figures 15-17. It is observed that as the amplitude ratio rises, so does the number of trapping bolus.

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FIGURE 2
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Effect of K on θ . $\theta .$ Z = 1.5; B = 1.44; ε $\epsilon $ = 6.2.
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FIGURE 3
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Influence of B on θ . $\theta .$
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FIGURE 4
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Influence of α 1 $\ \alpha 1$ on W. δ =1.22; α=1.26; G=2.57 ε $\epsilon $ =0.9 M=1.1. α 2 $\alpha 2$ =5.7, κ=3.4, α $\alpha $ 3=2.7, η=π/2.
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FIGURE 5
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Effects of α 2 $\alpha 2$ on W. δ =1.22; α=1.26; G=2.57 ε $\epsilon $ =0.9 M=1.1, α 1 $\alpha 1$ =5.7, κ=3.4, α $\alpha $ 3=2.7, η=π/2.
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FIGURE 6
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Effects of M $M$ on W . $W.$ δ =1.22; α=1.26; G=3.47 ε $\epsilon $ =0.9 M=1.1. α 2 $\alpha 2$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 7
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Influence of ∝ $ \propto $ on d p / d z $dp/dz$ . δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1, α 1 $\alpha 1$ =5.7, κ=3.4, α $\alpha $ 3=2.7,
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FIGURE 8
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Influence of ∈ $\in $ on d p / d z $dp/dz$ . δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1. α 2 $\alpha 2$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 9
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Influence of α 1 $\alpha 1$ on d p / d z . $dp/dz.$ δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1, α 1 $\alpha 1$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 10
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Effects of M ${\mathrm{M}}$ on d p / d z . $dp/dz.$ δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1. α 2 $\alpha 2$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 11
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Graph showing effect α $\ \alpha $ on ∇ P $\nabla P$ . δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1, α 1 $\alpha 1$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 12
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Influence of α = on F ${\mathrm{F}}$ . δ =1.22; α1=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1. α2=5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 13
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Influence of ∈ $\in $ on ∇ P $\nabla P$ . δ =1.22; α=1.26; G=5.57; M=1.1, α 2 $\alpha 2$ . α 1 $\alpha 1$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 14
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Effect of ∈ $\in $ on F ${\mathrm{F}}$ . δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1. α 2 $\alpha 2$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 15
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Sinusoidal waves. δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1, α 1 $\alpha 1$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 16
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Multi Sinusoidal waves. δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1. α 2 $\alpha 2$ =5.7, κ=3.4, α $\alpha $ 3=2.7.
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FIGURE 17
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Trapezoidal waves. δ =1.22; α=1.26; G=5.57 ε $\epsilon $ =0.9 M=1.1, α 1 $\alpha 1$ =5.7, κ=3.4, α $\alpha $ 3=2.7.

5 CONCLUDING REMARKS

In this article, we have discussed the Jeffrey six-constant fluid under the influence of a convective boundary condition. The following are the study's main outcomes:

  1. Temperature is calculated with an exact solution.

  2. When Biot number is small, internal thermal resistance is negligible compared to the surface resistance.

  3. The object experiences nearly uniform temperature throughout its volume.

  4. This occurs in materials with high thermal conductivity (e.g., metals) or small characteristic lengths.

  5. The Biot number in the base fluid increases but decreases with stronger effects.

  6. Occurs in decelerating flow, where pressure increases along the flow direction.

  7. Common in adverse pressure zones, such as boundary layer separation in aerodynamics.

  8. Represents accelerating flow, where pressure decreases along the flow direction.

  9. Found in cases of suction-driven flow, aiding in enhancing velocity profiles.

  10. Indicates uniform flow, where pressure remains constant along the direction of motion.

  11. It is obvious that pressure rise Enhances fluid transport, often seen in peristaltic pumping and cardiovascular flow. Common in forced convection systems, where external forces drive fluid movement.

NOMENCLATURE

 

  • U ⃗ , W ⃗ $\mathop U\limits^ \to ,\,\,\mathop W\limits^ \to $
  • velocity profiles (m s−1)
  • P ⃗ $\mathop P\limits^ \to $
  • pressure (kg m−1s−2)
  • C ⃗ $\mathop C\limits^ \to $
  • concentration (kg m−3)
  • T ⃗ $\mathop T\limits^ \to $
  • temperature (K)
  • B
  • Biot number
  • k $k$
  • thermal conductivity (W m−1 K−1)
  • g $g$
  • gravitational acceleration (m s−2)
  • Q 0 ${Q}_0$
  • heat absorption (W m−1 K−1)
  • cp
  • specific heat (J kg−1 K)
  • Q
  • heat absorption
  • Pr
  • Prandtl number

    • ACKNOWLEDGMENTS

    The authors have nothing to report.

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