Peristaltic flow of Williamson nanofluid on a rough surface

Abstract

The aim of this current research is to investigate the peristaltic flow of Williamson nanofluid across a rough surface in a non-uniform channel under the influence of inclined magnetic field. The Joule heating and viscous dissipation effects are also retained in the current scrutiny. The objective of studying peristaltic flow of Williamson nanofluid on a rough surface is to gain insights into the complex fluid dynamics and heat transfer phenomena occurring in such systems. This knowledge can be used to design more efficient and effective nanofluid-based devices and processes. In the context of mathematical modeling, the appropriate dimensional nonlinear equations for momentum, heat and mass transport are simplified into dimensionless equation by applying the essential estimation of long wavelength and low Reynolds number. The equations subjected to boundary conditions have solved numerically by the Mathematica software built-in numerical Solver ND_solve method. Various essential physical characteristics on velocity, temperature and concentration are presented graphically in the end. It can be seen that fluid velocity decreases at the central part of the channel for the escalting values Hartman number M. As Darcy number Da increases then velocity profile increases at the core part of the channel and the walls of the channel experiencing an opposite behavior. It is noticed that Higher value of Eckert number Ec enhances the temperature profile. When Weissenberg number We gets stronger then temperature profile decreases. It is observed that the temperature and concentration profiles show an opposite behavior for the rising values of thermophoresis parameter $N_t$ .

Introduction

A fluid containing nanometer sized particles, is known as nano fluid. These fluids create colloidal suspensions in a base fluid. Usually, carbides, metals and oxides are employed as the basis for the nanoparticles in nanofluid. The minimum obstruction, long-term stability and greater thermal conductivity enhance heat transfer capabilities of nanoparticles due to their small size and large surface area. Nanofluid has many industrial applications such as in electronics, heat exchangers, food industry as well as transportation, biomedicine, antibacterial therapeutics, drug delivery and diagnosis of different diseases. Rekha et al. ¹ studied the impact of thermophoretic particle deposition on heat transfer and nanofluid flow through different geometries. Influence of thermophoretic particle deposition on the 3D flow of Casson nanofluid over a stretching sheet is investigated by Shankaralingappa et al. ² Naveen Kumar et al. ³ explored the Carbon nanotubes suspended dusty nanofluid flow over stretching porous rotating disk with non-uniform heat source/sink. Madhukesh et al. ⁴ studied the development for thermal and solutal transport analysis of non-Newtonian nanofluid flow over a riga surface. The assessment of Arrhenius activation energy in stretched flow of nanofluid over a rotating disk is examined by Kotresh et al. ⁵ Madhukesh et al. ⁶ analyzed of ternary nanofluid flow in a microchannel with nonuniform heat source/sink. Biomechanics of cilia-assisted flow with hybrid nanofluid by convective conditions is examined by Ijaz et al. ⁷ Biomechanics of swimming microbes in atherosclerotic region with infusion of nanoparticles is explored by Ijaz et al. et l. ⁸ Song et al. ⁹ investigated mixed convection flow of magneto-Williamson nanofluid due to stretched cylinder with non-uniform heat. Nonlinear mixed convective Williamson nanofluid flow with the suspension of gyrotactic microorganisms is examined by Zhou et al. ¹⁰

Peristalsis is a mechanism in which bio fluid carried along the channel by sinusoidal waves that flow axially. Due to its inclusive use in biomechanics, medicine and engineering Peristalsis is one of the most significant pumping mechanisms. Hayat et al. ¹¹ observed the peristalsis of Williamson nanofluid in an endoscopic to determine various underlying aspects. Butt et al. ¹² investigated the thermally conductive electro-osmotic propulsive pressure-driven peristaltic streaming flow study with a suspended nanomaterial in a micro-ciliated tube. Rashid et al. ¹³ examined how a magnetic field applied to a curved channel effects the peristaltic movement of Williamson fluid. Bhatti and Zeeshan ¹⁴ studied the slip impacts of particle-fluid suspension on Peristalsis through mass and heat transmission.

Rough surface keeps unique elements of surface texture. However, the perceived roughness is slight, but cannot be ignored since in the production process surface texture variation appears continuously. Patil et al. ¹⁵ analyzed the rough surface impacts of a continually stretching permeable surface on mixed convective nanofluid flow. Patil et al. ¹⁶ studied the rough vertical cone with mixed bio convective flow of Williamson nanofluid. Patil et al. ¹⁷ observed the diffusion of liquid hydrogen across a rough spherical surface in a nonlinear mixed convective nanofluid flow.

When fluid particles are moving, the fluid viscosity converts some of its kinetic energy into thermal energy. Due to viscosity irreversible conversion of kinetic energy, the heat generation occur this is known as viscous dissipation. The process of dissipation involves transforming the mechanical energy of downwardly flowing fluid into thermal and sound energy. When a non-linear stretch sheet is inserted through Magnetohydrodynamics (MHD) Williamson fluid, the effects of heat generation and viscous dissipation on the flow of fluid and the transfer of heat in a porous medium were investigated by Abbas et al. ¹⁸ In Newtonian and non-Newtonian fluids, the combined impacts of a viscous dissipation and magnetic field has been the major topic of interest in literature up till now. Shaw et al. ¹⁹ investigated a permeable sphere implanted in a porous medium through nanofluid containing gyrotactic microorganisms under the influence of a magnetic field and viscous dissipation.

Thermal energy is produced when current flows through an electrical conductor is known as Joule heating. The term “heating” refers to the indication of this thermal energy as a temperature increase of the conductor material. In particularly, resistive loses inside the material causes electrical energy is transformed to heat when an electric current passes through a solid or liquid with finite conductivity. Yaseen Khan et al. ²⁰ studied the hydromagnetic Williamson nanofluid flow with entropy approach and Joule heating. Kumar et al. ²¹ investigated the Williamson nanofluid flow with Joule heating, variable thermal conductivity and entropy generation. Bouslimi et al. ²² examined the influences of joule heating, nonlinear thermal radiation, heat generation and chemical reaction on MHD Williamson nanofluid flow across a stretching surface via porous medium.

The main intention of this investigation is to establish a mathematical framework for evaluating the impact of inclined magnetic field, joule heating and viscous dissipation on the peristaltic flow of a Williamson nanofluid. The fundamental assumptions of a long wavelength and low Reynolds number are applied in the relevant nonlinear equations for momentum, heat and mass transfer as part of mathematical modeling. The equations subjected to boundary conditions have solved numerically by the Mathematica software built-in numerical Solver ND solve method. This analysis specifically contributes to explore how velocity, temperature and concentration behave under the considered effects in porous channel with rough walls surface. The physical characteristics of evolving components are explored by plotting their graphs in order to investigate the quantitative effects. The peristaltic flow of Williamson nanofluid on a rough surface has significant implications in biomedicine, particularly in the areas of drug delivery and cancer treatment. The use of nanofluid in peristaltic pumps can provide targeted drug delivery and improve the efficacy of cancer treatment. Recently many researchers considered the Williamson nanofluid with different effects. Al-Khafajy and Al-Delfi ²³ investigated the Peristaltic flow of Williamson nanofluid through a flexible channel. Williamson nanofluid in a convectively heated peristaltic channel and magnetic field examined by Alharbi et al. ²⁴ But the novelty of this study lies in the comprehensive analysis of multiple complex phenomena in a single framework. The interplay between inclined MHD, Williamson nanofluid, Peristalsis, Non-uniform channel, rough surface walls, Viscous dissipation, Joule heating has not been explored in previous research yet.

Mathematical formulation

We began our investigation by taking into account the peristaltic movement Williamson nanofluid in two-dimensional non-uniform channel. The peristaltic waves are traveling with constant speed c and wavelength λ along the walls of channel. The uniform magnetic field $(0,B_0\mathrm{Sin}\alpha ,0)$ is also applied. The walls of the channel possess sinusoidal wave that propagates along the walls enclosing porous medium. $T_0,C_0$ indicates the temperature and concentration at upper wall while $T_{1,}{C}_1$ indicates the temperature and concentration at lower wall of the channel. Geometry of the wall is characterized by the subsequent form ²⁵ : (see Figure 1).

$$\begin{array}{c}\multicolumn{1}{c}{H(\overline{X},\overline{t})=d(\overline{X})+b\sin (\frac{2\pi }{\lambda }(\overline{X}-c\overline{t}))-b_1\mathrm{co}{\mathrm{s}}^4\left(\frac{\pi \overline{X}}{{\lambda }_1}\right),}\\ \mathrm{with}d(\overline{X})=a+K\overline{X.}\hfill \end{array}$$

(1)

Here $b$ denotes the amplitude of the wave, λ the wavelength, $c$ the velocity propagation, $a$ half channel width, t the time, $K$ denotes the non-uniformity parameter, $b_1$ the height of roughness and ${\lambda }_1$ represents the pitch.

Figure 1.

Physical configuration.

We enforce uniform magnetic field which is.

$$\overline{\mathrm{B}}=(0,B_0\mathrm{Sin}\alpha ,0),$$

(2)

By generalized Ohm’s law ²⁶ :

$$\overline{J}=\sigma [(\overline{E}+\overline{V}\times \overline{B})-\frac{1}{e{n}_e}(\overline{J}\times \overline{B})],$$

(3)

where $\overline{V}$ is fluid velocity, $\overline{J}$ the current density, $\sigma$ the electrical conductivity and we disregarded the impact of the electric force, $\overline{E}$ =0. By using equations (2) and (3), we obtain

$$\overline{J}\times \overline{B}=\frac{\sigma {B_0}^2\mathrm{si}{n}^2\alpha }{1+m^2\mathrm{si}{n}^2\alpha }[(\overline{U}\mathrm{msin}\mathrm{\alpha }-\overline{W})i-(\overline{U}+\overline{\mathrm{W}}\mathrm{msin}\alpha )k].$$

(4)

Where $m=\frac{\sigma B_0}{e{n}_e}$ represents the Hall current parameter.

Joule heating is stated as below ²⁶ :

$$\frac{\overline{J}.\overline{J}}{\sigma }=\frac{\sigma {B_0}^2\mathrm{si}{n}^2\alpha }{1+m^2\mathrm{si}{n}^2\alpha }[\overline{U^2}+\overline{W^2}].$$

(5)

The extra stress tensor S has the following constitutive equation ²⁷ :

$$S=({\mu }_{\infty }+({\mu }_0-{\mu }_{\infty })(1-\Gamma \stackrel{·}{\gamma }{)}^{-1})\stackrel{·}{\gamma },$$

(6)

where, ${\mu }_0$ is the zero-shear rate viscosity, ${\mu }_{\infty }$ the infinite shear rate viscosity and $\Gamma$ is the time constant.

$$\stackrel{·}{\gamma }=\sqrt{\frac{1}{2}\Pi },\mathrm{and}\Pi =\mathrm{trace}(\mathrm{grdV}+{\left(\mathrm{gradv}\right)}^T){)}^2.$$

(7)

The following equations describe the flow of an incompressible nanofluid in a fixed frame ²⁶ :

$$\frac{\partial \overline{U}}{\partial \overline{X}}+\frac{\partial \overline{W}}{\partial \overline{Z}}=0,$$

(8)

$$\begin{array}{c}\rho [\frac{\partial \overline{U}}{\partial \overline{t}}+\overline{U}\frac{\partial \overline{U}}{\partial \overline{X}}+\overline{W}\frac{\partial \overline{U}}{\partial \overline{Z}}]=-\frac{\partial \overline{P}}{\partial \overline{X}}+\frac{\partial }{\partial \overline{X}}(S_{\mathrm{xx}})+\frac{\partial }{\partial \overline{Z}}(S_{\mathrm{xz}})\hfill \\ -\frac{\sigma {B_0}^2\mathrm{si}{n}^2\alpha }{1+m^2\mathrm{si}{n}^2\alpha }(\overline{U}\mathrm{msin}\mathrm{\alpha }-\overline{W})-\frac{\mu }{K_1}\overline{U},\hfill \end{array}$$

(9)

$$\begin{array}{c}\rho [\frac{\partial \overline{W}}{\partial \overline{t}}+\overline{U}\frac{\partial \overline{W}}{\partial \overline{X}}+\overline{W}\frac{\partial \overline{W}}{\partial \overline{Z}}]=-\frac{\partial \overline{P}}{\partial \overline{Z}}+\frac{\partial }{\partial \overline{X}}(S_{\mathrm{zx}})+\frac{\partial }{\partial \overline{Z}}(S_{\mathrm{zz}})\hfill \\ -\frac{\sigma {B_0}^2\mathrm{si}{n}^2\alpha }{1+m^2\mathrm{si}{n}^2\alpha }(\overline{U}+\overline{\mathrm{W}}\mathrm{msin}\alpha )-\frac{\mu }{K_1}\overline{W},\hfill \end{array}$$

(10)

$$\begin{array}{c}\rho c_p[\frac{\partial \overline{T}}{\partial \overline{t}}+\overline{U}\frac{\partial \overline{T}}{\partial \overline{X}}+\overline{W}\frac{\partial \overline{T}}{\partial \overline{Z}}]=K(\frac{{\partial }^2\overline{T}}{\partial {\overline{X}}^2}+\frac{{\partial }^2\overline{T}}{\partial {\overline{Z}}^2})+\varphi \hfill \\ +{\left(\rho c\right)}_p[D_B(\frac{\partial \overline{C}}{\partial \overline{X}}\frac{\partial \overline{T}}{\partial \overline{X}}+\frac{\partial \overline{C}}{\partial \overline{z}}\frac{\partial \overline{T}}{\partial \overline{z}})+\frac{D_T}{T_m}({\left(\frac{\partial \overline{T}}{\partial \overline{X}}\right)}^2+{\left(\frac{\partial \overline{T}}{\partial \overline{z}}\right)}^2)]\hfill \\ +\frac{\sigma {B_0}^2\mathrm{si}{n}^2\alpha }{1+m^2\mathrm{si}{n}^2\alpha }[\overline{U^2}+\overline{W^2}],\hfill \end{array}$$

(11)

$$\begin{array}{c}[\frac{\partial \overline{C}}{\partial \overline{t}}+\overline{U}\frac{\partial \overline{C}}{\partial \overline{X}}+\overline{W}\frac{\partial \overline{C}}{\partial \overline{Z}}]=D_B(\frac{{\partial }^2\overline{C}}{\partial {\overline{X}}^2}+\frac{{\partial }^2\overline{C}}{\partial {\overline{Z}}^2})\hfill \\ +\frac{{\mathrm{D}}_{\mathrm{T}}}{T_m}(\frac{{\partial }^2\overline{T}}{\partial {\overline{X}}^2}+\frac{{\partial }^2\overline{T}}{\partial {\overline{Z}}^2}).\hfill \end{array}$$

(12)

The transformation between the wave frame and laboratory frame are

$$\overline{x}=\overline{X}-c\overline{t},\overline{z}=\overline{Z},\overline{u}=\overline{\mathrm{U}}-c,\overline{p}(\overline{x})=\overline{P}(\overline{X},\overline{t}),$$

(13)

with dimensionless quantities:

$$\begin{array}{cc}\multicolumn{1}{c}{x}& =\frac{\overline{x}}{\lambda },z=\frac{\overline{z}}{a},u=\frac{\overline{u}}{c},w=\frac{\overline{w}}{c},t=\frac{c\overline{t}}{\lambda },\overline{\stackrel{·}{\gamma }}=\stackrel{·}{\gamma }\frac{a}{c},p=\frac{a^2\overline{p}}{\mu c\lambda },\hfill \\ \multicolumn{1}{c}{h}& =\frac{H}{a},{\lambda }_1=\frac{{\overline{\lambda }}_1}{\lambda },\theta =\frac{\overline{T}-T_0}{T_1-T_0},C=\frac{\overline{C}-C_0}{C_{1,}-C_{0,}},\mathrm{Re}=\frac{\rho c\mathrm{a}}{\mu },\hfill \\ \multicolumn{1}{c}{M}& =\sqrt{\frac{\sigma }{\mu }}{B}_{0}a,\mathrm{Da}=\frac{K_1}{a^2},\mathrm{Pr}=\frac{\mu c_p}{k},\mathrm{Ec}=\frac{c^2}{c_p(T_1-T_{0)}},\hfill \\ \multicolumn{1}{c}{\mathrm{We}}& =\frac{\Gamma c}{a},\mathrm{Br}=\mathrm{Pr}\times \mathrm{Ec},N_b=\frac{{\left(\rho c\right)}_p{\mathrm{D}}_{\mathrm{B}}(C_{1,}-C_{0,})}{\mathrm{K}},\hfill \\ \multicolumn{1}{c}{N_t}& =\frac{{\left(\rho c\right)}_p{\mathrm{D}}_{\mathrm{T}}(T_1-T_0)}{T_0\mathrm{K}},u=\frac{\partial \psi }{\partial z},w=-\delta \frac{\partial \psi }{\partial x}.\hfill \end{array}$$

(14)

Where K the non-uniformity, $T_m$ the mean fluid temperature, $D_{\mathrm{B}}$ the mass diffusivity coefficient, $K_1$ the permeability, ${\mathrm{B}}_0$ the magnetic field, Br the Brinkman number, $\mathrm{Pr}$ the Prandtl number, $\mathrm{Re}$ Reynolds number, We Weissenberg number, $M$ the Hartmann number, $\mathrm{Da}$ Darcy number, $\mathrm{Ec}$ the Eckert number, $\theta$ the non-dimensional temperature, $C$ the dimensionless concentration, $\mu$ dynamic viscosity, $v$ kinematic viscosity, $N_t$ the thermophoresis parameter and $N_b$ Brownian motion parameter.

Dimensionless system

After using dimensionless variables (13–14), equations (9) to (12) become:

$$\begin{array}{c}\mathrm{\delta }\mathrm{Re}\left[(\frac{\partial \psi }{\partial z}\frac{\partial }{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial }{\partial z})\frac{\partial \psi }{\partial z}\right]=-\frac{\partial p}{\partial x}+2{\delta }^2\frac{\partial }{\partial x}\left((1+\mathrm{We}\stackrel{·}{\gamma })\frac{{\partial }^2\psi }{\partial x\partial z}\right)\hfill \\ +\frac{\partial }{\partial z}\left((1+\mathrm{We}\stackrel{·}{\gamma })(\frac{{\partial }^2\psi }{\partial z^2}-{\delta }^2\frac{{\partial }^2\psi }{\partial x^2})\right)+\delta \left[\frac{m{M}^2\mathrm{si}{\mathrm{n}}^3\alpha }{1+m^2\mathrm{si}{\mathrm{n}}^2\alpha }\right]\frac{\partial \psi }{\partial x}\hfill \\ -[\frac{M^2\mathrm{si}{\mathrm{n}}^2\alpha }{1+m^2\mathrm{si}{\mathrm{n}}^2\alpha }+\frac{1}{\mathrm{Da}}](\frac{\partial \psi }{\partial z}+1),\hfill \end{array}$$

(15)

$$\begin{array}{c}-{\delta }^3\mathrm{Re}\left[(\frac{\partial \psi }{\partial z}\frac{\partial }{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial }{\partial z})\frac{\partial \psi }{\partial z}\right]=-\frac{\partial p}{\partial z}+{\delta }^2\frac{\partial }{\partial x}\hfill \\ \left((1+\mathrm{We}\stackrel{·}{\gamma })(\frac{{\partial }^2\psi }{\partial z^2}-{\delta }^2\frac{{\partial }^2\psi }{\partial x^2})\right)-2{\delta }^2\frac{\partial }{\partial z}(1+\mathrm{We}\stackrel{·}{\gamma })\frac{{\partial }^2\psi }{\partial x\partial z})\hfill \\ -\delta \left[\frac{m{M}^2\mathrm{si}{\mathrm{n}}^3\alpha }{1+m^2\mathrm{si}{\mathrm{n}}^2\alpha }\right](\frac{\partial \psi }{\partial z}+1)-{\delta }^2[\frac{M^2\mathrm{si}{\mathrm{n}}^2\alpha }{1+m^2\mathrm{si}{\mathrm{n}}^2\alpha }+\frac{1}{\mathrm{Da}}]\frac{\partial \psi }{\partial x},\hfill \end{array}$$

(16)

$$\begin{array}{c}\delta \mathrm{Re}(\frac{\partial \psi }{\partial z}\frac{\partial \theta }{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial \theta }{\partial z})=\frac{1}{\mathrm{Pr}}({\delta }^2\frac{{\partial }^2\theta }{\partial x^2}+\frac{{\partial }^2\theta }{\partial z^2})\hfill \\ +\mathrm{Ec}(1+\mathrm{We}\stackrel{·}{\gamma })(4{\delta }^2{\left(\frac{{\partial }^2\psi }{\partial x\partial z}\right)}^2+{(\frac{{\partial }^2\psi }{\partial z^2}-{\delta }^2\frac{{\partial }^2\psi }{\partial x^2})}^2)\hfill \\ +\mathrm{Ec}\frac{M^2\mathrm{si}{\mathrm{n}}^2\alpha }{1+m^2\mathrm{si}{\mathrm{n}}^2\alpha }[{\delta }^2{\left(\frac{\partial \psi }{\partial x}\right)}^2+{(\frac{\partial \psi }{\partial z}+1)}^2]\hfill \\ +\frac{N_b}{\mathrm{Pr}}({\delta }^2\frac{\partial c}{\partial x}\frac{\partial \theta }{\partial x}+\frac{\partial c}{\partial z}\frac{\partial \theta }{\partial z})+\frac{N_t}{\mathrm{Pr}}({\delta }^2{\left(\frac{\partial \theta }{\partial x}\right)}^2+{\left(\frac{\partial \theta }{\partial z}\right)}^2),\hfill \end{array}$$

(17)

$$\mathrm{\delta }\mathrm{Re}(\frac{\partial \psi }{\partial z}\frac{\partial c}{\partial x}-\frac{\partial \psi }{\partial x}\frac{\partial c}{\partial z})={\delta }^2\frac{{\partial }^{2}c}{\partial x^2}+\frac{{\partial }^{2}c}{\partial z^2}+\frac{N_t}{N_b}({\delta }^2\frac{{\partial }^2\theta }{\partial x^2}+\frac{{\partial }^2\theta }{\partial z^2}).$$

(18)

The Williamson nanofluid flow equations in dimensionless form under $\delta <<$ 1, Re → 0 give:

$$\begin{array}{c}-\frac{\partial p}{\partial x}+\frac{\partial }{\partial z}\left((1+\mathrm{We}\frac{{\partial }^2\psi }{\partial z^2})\frac{{\partial }^2\psi }{\partial z^2}\right)\hfill \\ -[\frac{M^2\mathrm{si}{\mathrm{n}}^2\alpha }{1+m^2\mathrm{si}{\mathrm{n}}^2\alpha }+\frac{1}{\mathrm{Da}}](\frac{\partial \psi }{\partial z}+1)=0,\hfill \end{array}$$

(19)

$$-\frac{\partial p}{\partial z}=0,$$

(20)

$$\begin{array}{c}\frac{1}{\mathrm{Pr}}\left(\frac{{\partial }^2\theta }{\partial z^2}\right)+\mathrm{Ec}({\left(\frac{{\partial }^2\psi }{\partial z^2}\right)}^2+\mathrm{We}{\left(\frac{{\partial }^2\psi }{\partial z^2}\right)}^3)\hfill \\ +\mathrm{Ec}\frac{M^2\mathrm{si}{\mathrm{n}}^2\alpha }{1+m^2\mathrm{si}{\mathrm{n}}^2\alpha }{(\frac{\partial \psi }{\partial z}+1)}^2+\frac{N_b}{\mathrm{Pr}}\left(\frac{\partial c}{\partial z}\frac{\partial \theta }{\partial z}\right)+\frac{N_t}{\mathrm{Pr}}=0,\hfill \end{array}$$

(21)

$$\frac{{\partial }^{2}C}{\partial z^2}+\frac{N_t}{N_b}\left(\frac{{\partial }^2\theta }{\partial z^2}\right)=0.$$

(22)

Equation (20) suggests that p ≠ p (z). Equation (19) can be written in the form:

$$\frac{{\partial }^2\psi }{\partial z^2}((1+\mathrm{We}\frac{{\partial }^2\psi }{\partial z^2})\frac{{\partial }^2\psi }{\partial z^2}-(\frac{M^2\mathrm{si}{\mathrm{n}}^2\alpha }{1+m^2\mathrm{si}{\mathrm{n}}^2\alpha }+\frac{1}{\mathrm{Da}})\psi )=0,$$

(23)

The non-dimensional boundaries will assume the form ²⁸ :

$$\psi =0,\frac{{\partial }^2\psi }{\partial z^2}=0,\theta =0,C=0,\mathrm{at}\mathrm{z}=0.$$

(24)

$$\psi =q,\frac{\partial \psi }{\partial z}=-1,\theta =1,C=1,\mathrm{at}\mathrm{z}=h,$$

(25)

where,

$$h=1+\frac{K\lambda (x+t)}{a}+ϵ\mathrm{Sin}\left(2\pi x\right)-{ϵ}_1\mathrm{co}{\mathrm{s}}^4\left(\frac{\mathrm{\pi }\mathrm{\lambda }(\mathrm{x}+\mathrm{t})}{{\lambda }_1}\right).$$

The wave and fixed frame’s flow rates are related by ²⁸ :

$$\mathrm{Q}=\mathrm{q}+1,$$

(26)

where

$$\mathrm{q}={\int }_0^h\frac{\partial \psi }{\partial z}\mathrm{dz}.$$

(27)

Numerical method

By using the Mathematica NDSolve technique, the modified dimensionless equations are numerically processed. The benefit of this approach is that it picks the right algorithm and detects any possible errors automatically. Furthermore, this technique delivers excellent computational output with just 3–4 min of CPU time required for each evaluation. In reality, this approach avoids complex solution expressions and directly displays graphical depictions. The equations (21) to (23) subjected to boundary conditions (24) and (25) have been solved numerically by the ND solve method. The commercial software Mathematica includes a built-in feature for validating the results. The governing equations (21) to (23) are coupled and have a significant degree of nonlinearity. It is impossible to find the exact solution. Consequently, the numerical solution has been obtained.

Results and discussion

In this section, the effects of pertinent parameters on common profiles (velocity, temperature and concentration) are discussed. In-depth discussion is held regarding the factors that affect the peristaltic flow of Williamson nanofluids on a rough surface including Hartmann number $M$ ( $1.0\le M\le 3.0$ ), angle of inclination α( $\frac{\pi }{6}\le \alpha \le \frac{\pi }{2}$ ), Eckert number Ec( $1.0\le \mathrm{Ec}\le$ $3.5$ ), Weissenberg number We( $0.1\le \mathrm{We}\le 0.15$ ), hall parameter $m$ ( $1.0\le m\le 3.0$ ), Darcy number Da( $0.1\le \mathrm{Da}\le 0.5$ ), Brownian motion parameter $N_b$ ( $1.0\le N_b\le 4.0$ ), thermophoresis parameter $N_t$ ( $1.0\le N_t\le 4.0$ ), and Prandtl number Pr( $1.0\le \mathrm{Pr}\le 3.0$ ). The numerical computation is performed using the Mathematica built-in numerical ND-Solve method.

Flow analysis

Figure 2(a) to (e) illustrate the impacts on velocity parameters by increasing different parameter including Hartman number (M), Darcy number (Da), hall parameter (m), angle of inclination (α)and Weissenberg number (We). The impact of the Hartman number M on the velocity distribution is captured in Figure 2(a). It can be seen that fluid velocity drops at the centeral part of the channel as Hartmann number M increases. Because a rising value Hartmann number M produces a stronger Lorentz force that leads to the fluid’s velocity to fall.^29,30 It is depicted in Figure 2(b) that the velocity profile increases at the centrel region and the channel walls experiencing an opposite behavior for the increasing value Darcy number Da. Physically, larger Darcy number Da enhances the porous effect in the medium which promotes permeability which in turn lowers the resistance to flow through the porous layers. Therefore, velocity field increases. ³¹ From the Figure 2(c) it oberved that as the value of the hall parameter $m$ rises, the fluid’s effective conductivity drops, which reduces the resistivity of the Lorentz force and enhances the fluid velocity in the central portion. ³² In response to the rising values of angle of inclinaton α the velocity profile exhibits a declining tendency at center of the channel as seen in Figure 2(d). The influence of Weissenberg number We on the velocity distribution is demonstrated in Figure 2(e). It can be seen that the velocity profile drops when the value of Weissenberg number We increases. The physical justification behind such tendency is associated with the dominant of viscous forces.

Figure 2.

(a) Velocity profile for Hartman number M, (b) velocity profile for Darcy number Da, (c) velocity profile for Hall parameter m, (d) velocity profile for angle of inclination α, and (e) velocity profile for Weissenberg number We.

Thermal analysis

The consequences of the following parameters on the temperature profile are illustrated in Figure 3(a) to (g) through Hartman number M, Prantdl number Pr, Weissenberg number We, hall parameter $m$ , Eckert number Ec, thermophoresis parameter Nt and Brownian motion parameter $N_b$ . Figure 3(a) depicts how the Hartman number M impact the temperature profile. An increasing behavior of temperature profile is noted for higher values of Hartman number M. Physically, escalation the Hartman number M causes the Lorentz force which opposes the fluid’s movement and raises the temperature profile. Figure 3(b) illustrates the impact of Prandtl number Pr on the temperature distribution.The temperature of a fluid increases in proportion to the Prandtl number. Due to an increase in Prandtl number Pr, the thermal conductivity of the material decreases, making it less effective at conducting heat. Moreover, the rate of heat transmission is accelerated. The influence of Weissenberg number We on temperature profile is depicted in Figure 3(c). From the figure, it is clear that rise in Weissenberg number We, drops the temperature profile. This is because a higher shear stress results in more efficient heat transfer, causing the liquid to cool down. As the value of hall parameter m rises, the temperature profile decreases which is depicted in Figure 3(d). It is observed in Figure 3(e) that temperature profile growsup as Eckert number Ec increases. Physically, heat dissipation is characterized by Eckert number Ec. In viscous fluid, the increase of the kinetic energy produces internal heat energy which in terms enhances the fluid temperature. The influences of the thermophoresis parameter $N_t$ and Brownian motion parameter $N_b$ on the temperature profile are captured in Figure 3(f) and (g). The thermophoresis parameter $N_t$ raises the fluid temperature as seen in Figure 3(f). This is because as the thermophoresis parameter $N_t$ increases, the fluid particles are moved away from the cold surfae to hot surface then the fluid temperature increases. It is observed in Figure 3(g) that temperature profile upsurges as Brownian motion parameter $N_b$ increases. Because enhancement in Brownian motion parameter $N_b$ cause the random motion of the fluid particles that produce more heat. The temperature eventually rises as a result of more heat generated. Our findings for temperature profile are matched for pertinent perameters with the results gained in refs.^29,33

Figure 3.

(a) Temperature profile for Hartman number M, (b) temperature profile for Prandtl number Pr, (c) temperature profile for Weissenberg number We, (d) temperature profile for Hall parameter m, (e) temperature profile for Eckert number Ec, (f) temperature profile for thermophoresis parameter, and (g) temperature profile for Brownian motion parameter .

Mass distribution analysis

Figure 4(a) to (e) reveal the concentration profile and represent the impact of alteration in various involved parameters. It can be seen in Figure 4(a) that the concentration decays due to elevation in the values of Hartman number M. Figure 4(b) shows the increasing trend in concentation profile as we increase the value Hall parmeter m. The impact of Weissenberg number We on concentreation profile is demonstrated in Figure 4(c). It is observed that owing to increase in Weissenberg number We the concentration profile shows growing tendency. The impacts of the thermophoresis parameter $N_t$ and Brownain motion parameter $N_b$ on the distribution of concentration are shown in Figure 4(d) and (e). It can be seen that the thermophoresis parameter $N_t$ and Brownian motion parameter $N_b$ have an opposite tendency on the concentration of nanoparticles. As the value of thermophoresis parameter $N_t$ rises, the concentation profile decays. Because more nanoparticles are extract from the heated surface to the cold surface when thermophoresis parameter $N_t$ grows (see Figure 4(d)). The concentration profile rises when Brownain motion parameter $N_b$ is enhanced. This is due to fact that larger Brownain motion parameter $N_b$ causes random collisions and movement of the macroscopic material particles, which raises the fluid concentration (see Figure 4(e)). These outcomes for concentration profile are appropriately matched with respect to pertinent parameter with the results gained in.^29,33

Figure 4.

(a) Concentration profile for Harman number M, (b) concentration profile for Hall parameter m, (c) concentration profile for Weissenberg number We, (d) concentration profile for thermophoresis parameter , and (e) concentration profile for Brownian motion parameter.

Conclusions

In this investigaton we taken into account the peristaltic flow of Williamson nanofluid in a non-uniform channel having rough walls surface. Magnetohydrodynamics (MHD), Joule heating, porosity factor, viscous dissipation and roughness effects are also considered. The modified dimensionless equations are numerically processed by using the Mathematica ND solve technique. As the outcome of the research, we arrived at the following conclusions:

• •It can be seen that fluid velocity declines in the channel’s central portion whilst boosted up near the walls under a stronger magnetic field parameter.

• •The velocity profile increases at the central part of the channel for the escalating value of Darcy number Da. It is because greater number of pores aid the flow speed.

• •The velocity profile decreases at center region as the value of Weissenberg number We rises.

• •When hall parameter $m$ increases, the effective conductivity of the fluid decreases, thus decreasing the Lorentz force’s resistivity and increasing the fluid velocity at the central portion.

• •Increasing behavior of temperature profile is noticed for larger values of Hartman number M.

• •Rise in temperature profile is noticed for higher values of Prandtl number Pr.

• •A rise in Brownian motion parameter $N_b$ and thermophoresis parameter $N_t$ represents an increament in temperature profile.

• •It observe that the concentration rises due to increase in Weissenberg number and decays owing to increment in Hartmann number M.

• •The Brownian motion parameter $N_b$ and thermophoresis parameter $N_t$ have an opposite effect on the concentration of nanoparticles.