| $\left(\overline{X},\overline{Y}\right)$ stationary co-ordinates | |
| | $\left(\overline{x},\overline{y}\right)$ moving co-ordinates |
| | $\left(\overline{W},\overline{V}\right)$ velocity components in laboratory frames |
| | $\left(\overline{w},\overline{v}\right)$ velocity components in wave frames |
| | $\overline{T}$ dimensional temperature |
| | $\overline{C}$ dimensional temperature |
| | $\overline{{T}_{0}}$ reference temperature |
| | $\overline{{C}_{0}}$reference concentration |
| | $a$ dimension of the wall |
| | $b$amplitude |
| | $t$time of fluid flow |
| | $m$non-uniformity parameter |
| | $p$ pressure |
| | $g$ gravity |
| | ${k}_{T}$ thermal diffusivity |
| Re Reynolds number | |
| $Ec$ Eckert number | |
| Pr Prandtl number | |
| $N$ Brinkmann number | |
| $Sc$ Schmidt number | |
| $Sr$ Soret number | |
| $Mn$ magnetic field parameter | |
| $k$ thermal conductivity | |
| $F$ body force parameter | |
| $B$ strength of applied magnetic field | |
| ${E}_{1}$ wall tension parameter | |
| ${E}_{2}$ mass characterizing parameter | |
| ${E}_{3}$ wall damping parameter | |
| Greek symbols |
|
| $\varepsilon $ amplitude ratio | |
| $\sigma $ dimensionless concentration | |
| $\theta $ dimensionless temperature | |
| $\psi $ stream function | |
| $\mu $ viscosity | |
| $\nu $ kinematic viscosity | |
| $\rho $ density | |
| $\delta $ specific heat at constant volume | |
| $\zeta $ electrical conductivity | |
| $\alpha $ velocity slip parameter | |
| ${\alpha }_{1}$ temperature slip parameter | |
| ${\alpha }_{2}$ concentration slip parameter | |
| $\beta $ variable viscosity | |
| $\gamma $ variable thermal conductivity | |
| ${\tau }_{0}$ yield stress parameter | |
| ${\tau }_{xx},{\tau }_{xy},{\tau }_{yy}$ extra stress components | |
| ${\eta }_{1}$ mass per unit area | |
| ${\eta }_{2}$ wall damping force coefficient | |
| $\tau $ elastic tension |
Nomenclature
1. Introduction
Peristaltic transport is one of the tombstones for the event of science and engineering research in current years. Peristalsis plays an essential role in transporting several biological liquids such as the flow of urine through the ureter, the movement of chyme through the gastrointestinal tract, and the flow of blood through the vessels. Further, this mechanism has a lot of practical applications in designing heat lung and dialysis machines. The framework of the peristalsis mechanism plays a valuable role in various biological systems like chyme motion in the gastrointestinal tract, circulation of blood, etc. The primary investigation of the peristaltic flow of Newtonian fluid was carried out by Latham [1]. Since then, numerous researchers have examined the peristaltic flow by taking Newtonian fluid with different suppositions and geometries [2]. Most of the biofluids possess non-Newtonian behavior. By considering the nonlinearity of the fluid, Raju and Devanathan [3] have examined the peristaltic flow. Later, various researchers have investigated the peristaltic mechanism by utilizing completely different geometrical conditions [4–6].
Numerous researchers have frequently addressed the significance of flow through saturated porous media due to valuable applications in petroleum engineering, ground material flow, geophysics and geothermal extraction and oil recovery. El Shehawey et al [7, 8] examined the peristaltic flow through a saturated porous medium. The consequence of heat transfer on fluid flow is one amongst the indispensable components, which has extraordinary significance in contemporary environs as it is applicable in biomedical engineering and industries. Convective boundary conditions are wrought to depict a direct convective heat transfer for no less than one geometric affluence. The scrutiny of convective boundary conditions plays a significant role in gas turbines, atomic plants, and thermal energy storage. Mass transfer is one more precarious aspect that encountered the nutrients decomposition in various blood cells, membrane separation applications, and distillation processes. The utilization of combined heat and mass transportation is associated with the driving potentials and fluxes with complex nature. Such valuable significance in biological transport insisted on numerous investigations in recent times [9–11].
In most of the tender problems, the flow form illustrated by the slip boundary conditions plays a remarkable role in understanding the physiological behaviors of intricate techniques. In some issues, the fluid slides over the boundary due to adhesion and makes it difficult to understand the flow characteristics with the help of no-slip conditions. To overwhelm this, the slip effects play a significant role in understanding the flow of fluids in such circumstances. However, in the investigation of the peristaltic flow of non-Newtonian fluid, the slip effects play an influential role in discovering the physiological parameters, precisely, the flow of blood in narrow arteries. In such circumstances, the slip effects ascend when there exists a linear relationship between shear stress and velocity. Keeping this in cognizance, Kwang-Hua and Fang [12] utilized the slip features in the peristaltic flow of viscous material. According to their analysis, the utilization of slip consequences, the retrospective fluid is easily prompted without slip effects inside the channel. An increment in pumping power is observed at the start. The interaction of slip features involved variety of applications in many physical applications like emulsion, polymer industry and artificial heart valves. Such insisted applications lead the investigators in recent years [13–18].
It is often observed that most of the investigations have been performed by keeping constant thermal features. However, the role of these physical properties in isotropic liquids is quite limited. However, in the case of variable thermo-physical properties, the effects of such physical quantities cannot be denied. In such a situation, the thermal conductivity and viscosity of variable nature must be evaluated. Recently, the investigations concerning the appliance of variable properties are recorded in the literature [19–21]. It is speckled from the available information about peristalsis that typically devotion in the past has been focused on the flow in straight channels/uniform applied magnetic fields. Deliberation of the magnetic field is imperative since many materials like corrosive, toxic, saline water, and blood is electrically conducting [22–26].
Due to the applications of biological fluids, numerous researchers have examined the peristaltic mechanism by taking non-Newtonian fluids. Casson model is one amongst the non-Newtonian models that play a significant role in understanding the advanced rheologic behaviors of biological fluids. Assumptive blood as incompatible fluids, Srivastava and Srivastava [27] studied on the peristaltic pumping of blood by considering Casson fluid. Recently numerous researchers have studied impact of various parameters on peristaltic flow by taking Casson fluid into account [28–30].
Eventually, systems are needed to cool any device that carries out energy transport. Such devices are commonly used in power plants, manufacturing, and transportation. The approach involving both magnetic and slip conditions nowadays see considerable interest from many researchers, as well as high demand from industry-dependent thermal systems. Further, most of the investigations on the peristaltic flow deal with constant physical properties. However, there exists a considerable impact on the flow of non-Newtonian fluids, precisely in the flow of blood through arteries, with viscosity and thermal conductivity very concerned with thickness and temperature, respectively. Due to the high importance of peristaltic flow in the permeable walls, authors were inspired to investigate the influence of slip and magnetic field on the peristaltic flow of Casson fluid when subjected to variable fluid properties. Such a dimension of research has not yet been debated. The nonlinear governing equations are solved by using the perturbation technique, and the analytical solutions are obtained for velocity, skin-friction, temperature, Nusselt number, concentration, Sherwood number, and streamlines. The influence of relevant parameters on physiological quantities of interest is analyzed and conferred through graphs.
2. Mathematical formulation
To formulate the physical problem, we assume an incompressible fluid flow in a channel having dimension ‘a’ with porous walls. The walls of the channel are flexible and are maintained at a relentless temperature ${T}_{0}.$ The flow is ruled by non-Newtonian Casson fluid induced by the propagation of infinite sinusoidal wave trains moving with constant speed along the flexible walls of the channel, as shown in figure 1. The geometry of the sinusoidal wave is mathematically expressed as: $ \begin{eqnarray}\overline{h}(\overline{X},\overline{t})=1+m\,\overline{X}+\varepsilon \,\sin \left(2\pi \left(\overline{X}-\overline{t}\right)\right),\end{eqnarray}$ where $\overline{X},\,\varepsilon ,\,\overline{t},\,m$ represents axial direction, amplitude ratio, time of the fluid flow and non-uniform parameter, respectively.
Figure 1. The geometry of the physical model. |
The equations which govern the fluid flow in the laboratory frame are [15, 22] $ \begin{eqnarray}\displaystyle \frac{\partial \overline{W}}{\partial \overline{X}}+\displaystyle \frac{\partial \overline{V}}{\partial \overline{Y}}=0,\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\rho \left(\displaystyle \frac{\partial \overline{W}}{\partial \overline{t}}+\overline{W}\displaystyle \frac{\partial \overline{W}}{\partial \overline{X}}+\overline{V}\displaystyle \frac{\partial \overline{W}}{\partial \overline{Y}}\right)=-\displaystyle \frac{\partial \overline{P}}{\partial \overline{X}}+\displaystyle \frac{\partial {\overline{\tau }}_{\overline{X}\overline{X}}}{\partial \overline{X}}\\ \,+\,\displaystyle \frac{\partial {\overline{\tau }}_{\overline{X}\overline{X}}}{\partial \overline{Y}}+\rho \,g\,\sin (\varphi )-\varsigma {B}^{2}\overline{W},\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\rho \left(\displaystyle \frac{\partial \overline{V}}{\partial \overline{t}}+\overline{W}\displaystyle \frac{\partial \overline{V}}{\partial \overline{X}}+\overline{V}\displaystyle \frac{\partial \overline{V}}{\partial \overline{Y}}\right)=-\displaystyle \frac{\partial \overline{P}}{\partial \overline{Y}}+\displaystyle \frac{\partial {\overline{\tau }}_{\overline{X}\overline{X}}}{\partial \overline{X}}+\displaystyle \frac{\partial {\overline{\tau }}_{\overline{X}\overline{Y}}}{\partial \overline{Y}}\\ \,+\,\rho \,g\,\cos (\varphi ),\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\delta \rho \left(\displaystyle \frac{\partial \overline{T}}{\partial \overline{t}}+\overline{W}\displaystyle \frac{\partial \overline{T}}{\partial \overline{X}}+\overline{V}\displaystyle \frac{\partial \overline{T}}{\partial \overline{Y}}\right)=k\left(\displaystyle \frac{{\partial }^{2}\overline{T}}{\partial {\overline{X}}^{2}}+\displaystyle \frac{{\partial }^{2}\overline{T}}{\partial {\overline{Y}}^{2}}\right)\\ \,+\,{\overline{\tau }}_{\overline{X}\overline{X}}\displaystyle \frac{\partial \overline{W}}{\partial \overline{X}}+{\overline{\tau }}_{\overline{Y}\overline{Y}}\displaystyle \frac{\partial \overline{V}}{\partial \overline{Y}}+{\overline{\tau }}_{\overline{X}\overline{Y}}\left(\displaystyle \frac{\partial \overline{V}}{\partial \overline{X}}+\displaystyle \frac{\partial \overline{W}}{\partial \overline{Y}}\right),\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\left(\displaystyle \frac{\partial \overline{C}}{\partial \overline{t}}+\overline{W}\displaystyle \frac{\partial \overline{C}}{\partial \overline{X}}+\overline{V}\displaystyle \frac{\partial \overline{C}}{\partial \overline{Y}}\right)=D\left(\displaystyle \frac{{\partial }^{2}\overline{C}}{\partial {\overline{X}}^{2}}+\displaystyle \frac{{\partial }^{2}\overline{C}}{\partial {\overline{Y}}^{2}}\right)\\ \,+\,\displaystyle \frac{D{k}_{T}}{{T}_{m}}\left(\displaystyle \frac{{\partial }^{2}\overline{T}}{\partial {\overline{X}}^{2}}+\displaystyle \frac{{\partial }^{2}\overline{T}}{\partial {\overline{Y}}^{2}}\right).\end{array}\end{eqnarray}$
In the laboratory frame of reference, the motion of the fluid particles is assumed to be unsteady. On this end, a new wave frame $(x,\,y)$ has been introduced which moves away from the laboratory frame of reference $(\overline{X},\,\overline{Y}).$ The reported collateral transformations are $ \begin{eqnarray}\begin{array}{rll}\overline{x} & = & \overline{X}-c\overline{t},\,\overline{y}=\overline{Y},\,\overline{p}(\overline{x},\overline{y})=\overline{P}(\overline{X},\overline{Y},\overline{t}),\\ \overline{w}(\overline{x},\overline{y}) & = & \overline{W}(\overline{X},\overline{Y},\overline{t})-c,\\ \overline{v}(\overline{x},\overline{y}) & = & \overline{V}(\overline{X},\overline{Y},\overline{t}),\,\overline{T}(\overline{x},\overline{y})=\overline{T}(\overline{X},\overline{Y},\overline{t}),\\ \overline{C}(\overline{x},\overline{y}) & = & \overline{C}(\overline{X},\overline{Y},\overline{t}).\end{array}\end{eqnarray}$
On transforming from laboratory frame to wave frame using equation (7 ) the set of equations (3 )–(6 ) are formulated in the following form: $ \begin{eqnarray}\displaystyle \frac{\partial \overline{w}}{\partial \overline{x}}+\displaystyle \frac{\partial \overline{v}}{\partial \overline{y}}=0,\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\rho \left(\displaystyle \frac{\partial \overline{w}}{\partial \overline{t}}+\overline{w}\displaystyle \frac{\partial \overline{w}}{\partial \overline{x}}+\overline{v}\displaystyle \frac{\partial \overline{w}}{\partial \overline{y}}\right)=-\displaystyle \frac{\partial \overline{p}}{\partial \overline{x}}+\displaystyle \frac{\partial {\overline{\tau }}_{\overline{X}\overline{X}}}{\partial \overline{x}}+\displaystyle \frac{\partial {\overline{\tau }}_{\overline{X}\overline{Y}}}{\partial \overline{y}}\\ +\rho g\,\sin (\varphi )-\varsigma {B}^{2}\overline{w},\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\rho \left(\displaystyle \frac{\partial \overline{v}}{\partial \overline{t}}+\overline{w}\displaystyle \frac{\partial \overline{v}}{\partial \overline{x}}+\overline{v}\displaystyle \frac{\partial \overline{v}}{\partial \overline{y}}\right)=-\displaystyle \frac{\partial \overline{p}}{\partial \overline{y}}+\displaystyle \frac{\partial {\overline{\tau }}_{\overline{x}\overline{x}}}{\partial \overline{x}}+\displaystyle \frac{\partial {\overline{\tau }}_{\overline{x}\overline{y}}}{\partial \overline{y}}\\ \,+\rho g\,\cos (\varphi ),\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\delta \rho \left(\displaystyle \frac{\partial \overline{T}}{\partial \overline{t}}+\overline{w}\displaystyle \frac{\partial \overline{T}}{\partial \overline{x}}+\overline{v}\displaystyle \frac{\partial \overline{T}}{\partial \overline{y}}\right)=k\left(\displaystyle \frac{{\partial }^{2}\overline{T}}{\partial {\overline{x}}^{2}}+\displaystyle \frac{{\partial }^{2}\overline{T}}{\partial {\overline{y}}^{2}}\right)\\ \,+\,{\overline{\tau }}_{\overline{x}\overline{x}}\displaystyle \frac{\partial \overline{w}}{\partial \overline{x}}+{\overline{\tau }}_{\overline{x}\overline{y}}\displaystyle \frac{\partial \overline{v}}{\partial \overline{y}}+{\overline{\tau }}_{\overline{x}\overline{y}}\left(\displaystyle \frac{\partial \overline{v}}{\partial \overline{x}}+\displaystyle \frac{\partial \overline{w}}{\partial \overline{y}}\right),\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\left(\displaystyle \frac{\partial \overline{C}}{\partial \overline{t}}+\overline{w}\displaystyle \frac{\partial \overline{C}}{\partial \overline{x}}+\overline{v}\displaystyle \frac{\partial \overline{C}}{\partial \overline{y}}\right)=D\left(\displaystyle \frac{{\partial }^{2}\overline{C}}{\partial {\overline{x}}^{2}}+\displaystyle \frac{{\partial }^{2}\overline{C}}{\partial {\overline{y}}^{2}}\right)\\ \,+\displaystyle \frac{D{k}_{T}}{{T}_{m}}\left(\displaystyle \frac{{\partial }^{2}\overline{T}}{\partial {\overline{x}}^{2}}+\displaystyle \frac{{\partial }^{2}\overline{T}}{\partial {\overline{y}}^{2}}\right).\end{array}\end{eqnarray}$
The dimensional boundary conditions are given by [15] $ \begin{eqnarray}\overline{w}+\alpha \displaystyle \frac{\partial \overline{w}}{\partial \overline{y}}=-c\,{\rm{at}}\,\overline{y}=\overline{h},\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial \overline{w}}{\partial \overline{y}}=0\,{\rm{at}}\,\overline{y}=0,\end{eqnarray}$ $ \begin{eqnarray}\overline{T}+{\alpha }_{1}\displaystyle \frac{\partial \overline{T}}{\partial \overline{y}}={\overline{T}}_{1}\,{\rm{at}}\,\overline{y}=\overline{h},\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial \overline{T}}{\partial \overline{y}}=0\,{\rm{at}}\,\overline{y}=0,\end{eqnarray}$ $ \begin{eqnarray}\overline{C}+{\alpha }_{2}\displaystyle \frac{\partial \overline{C}}{\partial \overline{y}}={\overline{C}}_{1}\,{\rm{at}}\,\overline{y}=\overline{h},\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial \overline{C}}{\partial \overline{y}}=0\,{\rm{at}}\,\overline{y}=0.\end{eqnarray}$
The non-dimensional quantities of interest are given by $ \begin{eqnarray}\begin{array}{l}x=\displaystyle \frac{\overline{x}}{\lambda },y=\displaystyle \frac{\overline{y}}{a},t=\displaystyle \frac{c\overline{t}}{\lambda },{\tau }_{xx}=\displaystyle \frac{a{\overline{\tau }}_{\overline{x}\overline{y}}}{{\mu }_{0}c},{\tau }_{xy}=\displaystyle \frac{a{\overline{\tau }}_{\overline{x}\overline{y}}}{{\mu }_{0}c},\\ {\tau }_{yy}=\displaystyle \frac{a{\overline{\tau }}_{\overline{x}\overline{y}}}{{\mu }_{0}c},p=\displaystyle \frac{\bar{p}{a}^{2}}{\lambda {\mu }_{0}c},\delta =\displaystyle \frac{a}{\lambda },h=\displaystyle \frac{\overline{h}}{a},\\ \varepsilon =\displaystyle \frac{b}{a},{Re}=\displaystyle \frac{ca}{\upsilon },{\Pr }=\displaystyle \frac{{\mu }_{0}\delta }{k},Mn=\sqrt{\displaystyle \frac{\varsigma }{{\mu }_{0}}}Ba,\\ \theta =\displaystyle \frac{\overline{T}-{\overline{T}}_{0}}{{\overline{T}}_{0}},w=\displaystyle \frac{\bar{w}}{c},v=\displaystyle \frac{\bar{v}}{c\delta },\psi =\displaystyle \frac{\overline{\chi }}{c\delta },\end{array}\end{eqnarray}$ $ \begin{eqnarray*}\begin{array}{l}\upsilon =\displaystyle \frac{{\mu }_{0}}{\rho },{\mu }_{0}=\displaystyle \frac{\overline{{\mu }_{0}}}{\mu },Ec=\displaystyle \frac{{c}^{2}}{\delta {T}_{0}},\sigma =\displaystyle \frac{\overline{C}-{\overline{C}}_{0}}{{\overline{C}}_{0}},Sc=\displaystyle \frac{{\mu }_{0}}{\rho D},\\ Sr=\displaystyle \frac{\rho D{k}_{T}(\overline{T}-{\overline{T}}_{0})}{{\overline{C}}_{0}},N=Ec\,\Pr ,\end{array}\end{eqnarray*}$ $ \begin{eqnarray*}{E}_{1}=\displaystyle \frac{-\tau {a}^{3}}{\lambda {\mu }_{0}^{3}c},{E}_{2}=\displaystyle \frac{{m}_{1}c{a}^{3}}{{\lambda }^{3}{\mu }_{0}},{E}_{3}=\displaystyle \frac{{m}_{2}{a}^{3}}{{\lambda }^{3}{\mu }_{0}}.\end{eqnarray*}$
The expression for Casson fluid is $ \begin{eqnarray}\left.\begin{array}{ll}{\tau }^{1/2}=\mu {\left(-\displaystyle \frac{\partial w}{\partial y}\right)}^{1/2}+{\tau }_{0}^{1/2}, & \tau \geqslant {\tau }_{0}\\ \displaystyle \frac{\partial w}{\partial y}=0, & \tau \leqslant {\tau }_{0}\end{array}\right\},\end{eqnarray}$ where ${\tau }_{0}$ represents the yield stress and $\mu $ is the viscosity of a material. The equation of the flexible wall motion is expressed as [4] $ \begin{eqnarray}L\left(h\right)=p-{p}_{0}.\end{eqnarray}$
The operator $L$ represents the motion of a stretched membrane with viscosity damping forces, such that [4] $ \begin{eqnarray}L=-\tau \displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}+{\eta }_{1}\displaystyle \frac{{\partial }^{2}}{\partial {x}^{2}}+{\eta }_{2}\displaystyle \frac{\partial }{\partial t},\end{eqnarray}$ where ${\eta }_{1}$ determine the mass per unit area, $\tau $ is the elastic tension while ${\eta }_{2}$ wall damping force coefficient. It is remarked that the present flow computations are performed in half-width of the channel for the sake of simplicity. The equation (17 ) in terms of $x$ component of the momentum can be written in the following form $ \begin{eqnarray}\begin{array}{l}\displaystyle \frac{\partial }{\partial x}L\left(h\right)=\displaystyle \frac{\partial p}{\partial t}=\displaystyle \frac{\partial {\tau }_{xx}}{\partial x}+\displaystyle \frac{\partial {\tau }_{xy}}{\partial y}+\rho g\,\sin (\varphi )\\ \,\,\,\,-\,\rho \left(\displaystyle \frac{\partial u}{\partial t}+u\displaystyle \frac{\partial u}{\partial x}+v\displaystyle \frac{\partial u}{\partial y}\right).\end{array}\end{eqnarray}$
Underneath the assumptions of long wavelength and small Reynolds number, and using dimensionless parameter, equations (8 )–(12 ) takes the following form $ \begin{eqnarray}\displaystyle \frac{\partial p}{\partial x}=\displaystyle \frac{\partial {\tau }_{xy}}{\partial y}+\displaystyle \frac{\sin (\varphi )}{F}-M{n}^{2}(w+1),\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial p}{\partial y}=0,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial }{\partial y}\left[k(\theta )\displaystyle \frac{\partial \theta }{\partial y}\right]+N{\tau }_{xy}\displaystyle \frac{\partial w}{\partial y}=0,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{{\partial }^{2}\sigma }{\partial {y}^{2}}+ScSr\displaystyle \frac{{\partial }^{2}\theta }{\partial {y}^{2}}=0.\end{eqnarray}$
The boundary assumptions in dimensionless forms are expressed in the following from $ \begin{eqnarray}w+\alpha \displaystyle \frac{\partial w}{\partial y}=-1\,{\rm{a}}{\rm{t}}\,y=h,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial w}{\partial y}=0\,{\rm{at}}\,y=0,\end{eqnarray}$ $ \begin{eqnarray}\theta +{\alpha }_{1}\displaystyle \frac{\partial \theta }{\partial y}=1\,{\rm{at}}\,y=h,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial \theta }{\partial y}=0\,{\rm{at}}\,y=0,\end{eqnarray}$ $ \begin{eqnarray}\sigma +{\alpha }_{2}\displaystyle \frac{\partial \sigma }{\partial y}=1\,{\rm{at}}\,y=h,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial \sigma }{\partial y}=0\,{\rm{at}}\,y=0.\end{eqnarray}$
The viscosity truncates across the wall thickness of the channel while variation in thermal conductivity is associated with following temperature relation $ \begin{eqnarray}\mu (y)=1-\beta y\,{\rm{for}}\,\beta \ll 1,\end{eqnarray}$ $ \begin{eqnarray}k(\theta )=1+\gamma \theta \,{\rm{for}}\,\gamma \ll 1,\end{eqnarray}$ where $\beta $ and $\gamma $ are the coefficient of viscosity and thermal conductivity respectively.
3. Methodology of the solution
Since the equations (19a )–(19c ) are coupled with nonlinear equations, the perturbation technique is employed to solve and get an appropriate solution for them. However, the values of $\beta \,$ and $\gamma $ are small in almost all practical problems; hence it can be used for perturbation solution.
3.1. Perturbation solution
The analytical solution of equations (19a )–(19c ) has been suggested with expansion of following relations $ \begin{eqnarray}w=\displaystyle \sum _{n=0}^{\infty }{\beta }^{n}{w}_{n},\end{eqnarray}$ $ \begin{eqnarray}\theta =\displaystyle \sum _{n=0}^{\infty }{\gamma }^{n}{\theta }_{n}.\end{eqnarray}$
3.1.1. Zeroth order system
The appropriate boundary conditions are $ \begin{eqnarray}{w}_{0}+\alpha \displaystyle \frac{\partial {w}_{0}}{\partial y}=-1\,{\rm{at}}\,y=h,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial {w}_{0}}{\partial y}=0\,{\rm{at}}\,y=0,\end{eqnarray}$ $ \begin{eqnarray}{\theta }_{0}+{\alpha }_{1}\displaystyle \frac{\partial {\theta }_{0}}{\partial y}=1\,{\rm{at}}\,y=h,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial {\theta }_{0}}{\partial y}=0\,{\rm{at}}\,y=0.\end{eqnarray}$
3.1.2. First order system
The appropriate boundary conditions are $ \begin{eqnarray}{w}_{1}+\alpha \displaystyle \frac{\partial {w}_{1}}{\partial y}=0\,{\rm{a}}{\rm{t}}\,y=h,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial {w}_{1}}{\partial y}=0\,{\rm{at}}\,y=0,\end{eqnarray}$ $ \begin{eqnarray}{\theta }_{1}+{\alpha }_{1}\displaystyle \frac{\partial {\theta }_{1}}{\partial y}=0\,{\rm{at}}\,y=h,\end{eqnarray}$ $ \begin{eqnarray}\displaystyle \frac{\partial {\theta }_{1}}{\partial y}=0\,{\rm{at}}\,y=0.\end{eqnarray}$
The expression for concentration is obtained by equation (19d ) using boundary conditions (20e ) and (20f ). The expressions for velocity, temperature and concentration are obtained on solving zeroth and first order systems. $ \begin{eqnarray}{w}_{0}=\displaystyle \frac{-A+\tfrac{\left(A-{m}_{1}\right)\,\cosh \left[\sqrt{{m}_{1}}y\right]}{\cosh \left[h\sqrt{{m}_{1}}\right]+\alpha \sqrt{{m}_{1}}\,\sinh \left[h\sqrt{{m}_{1}}\right]}}{{m}_{1}},\end{eqnarray}$ $ \begin{eqnarray}{w}_{1}={A}_{8}\left(\begin{array}{l}\left(\cosh \left[\sqrt{{m}_{2}}\left(h+y\right)\right]-\,\sinh \left[\sqrt{{m}_{2}}\left(h+y\right)\right]\right)\\ \left({A}_{9}+{A}_{3}{A}_{4}\left(\cosh \left[2\sqrt{{m}_{2}}y\right]+\,\sinh \left[2\sqrt{{m}_{2}}y\right]\right)\right.\\ +\left.{A}_{6}{m}_{9}\left(\cosh \left[\sqrt{{m}_{2}}\left(h+2y\right)\right]+\,\sinh \left[\sqrt{{m}_{2}}\left(h+2y\right)\right]\right)\right)\\ -{A}_{5}\left({m}_{5}y\,\cosh \left[\sqrt{{m}_{1}}y\right]+{m}_{6}\,\sinh \left[\sqrt{{m}_{1}}y\right]\right)\end{array}\right),\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{c}{\theta }_{0}=\displaystyle \frac{N}{{m}_{1}}\left(\begin{array}{l}{\alpha }_{1}\left({A}_{11}\left({\rm{\cosh }}\left[h\sqrt{{m}_{1}}\right]-1\right)-{A}_{12}\,{\rm{\sinh }}\left[h\sqrt{{m}_{1}}\right]\right)\\ \left(-,{A}_{11}\left(h-y\right)+{A}_{13}\left({\rm{\cosh }}\left[h\sqrt{{m}_{1}}\right]-\,{\rm{\cosh }}\left[\sqrt{{m}_{1}}y\right]\right)\right.\\ \left.-{A}_{14}\left({\rm{\sinh }}\left[h\sqrt{{m}_{1}}\right]-\,{\rm{\sinh }}\left[\sqrt{{m}_{1}}y\right]\right)\right)\end{array}\right)+1,\end{array}\,\end{eqnarray}$ $ \begin{eqnarray}{\theta }_{1}=\displaystyle \frac{1}{24{{m}_{1}}^{2}}\left(\left.\begin{array}{l}{\alpha }_{1}\left(-24{A}_{26}{{m}_{1}}^{2}+{A}_{27}N\right)+{A}_{34}\left({y}^{3}-{h}^{3}\right)\\ +\,{A}_{35}\left(1+12{{A}_{11}}^{2}{N}^{2}\left({h}^{2}-{y}^{2}\right)-\,\cosh \left[\sqrt{{m}_{2}}\left(y-h\right)\right]+\,\sinh \left[\sqrt{{m}_{2}}\left(y-h\right)\right]\right)\\ +\,N\left({A}_{36}-{A}_{38}-{A}_{39}+{A}_{40}-6{A}_{16}N\right.\,\cosh \left[2\sqrt{{m}_{1}}y\right]+{A}_{18}{m}_{12}{m}_{14}{m}_{15}y\,\cosh \left[2\sqrt{{m}_{1}}y\right]\\ -\,{A}_{1}{A}_{21}{m}_{12}{m}_{14}{m}_{15}\,\sinh \left[2\sqrt{{m}_{1}}y\right]+12{A}_{13}{A}_{14}N\,\sinh \left[2\sqrt{{m}_{1}}y\right]\\ +\,\sinh \left[\sqrt{{m}_{1}}y\right]\left(3\left({A}_{19}{m}_{12}{m}_{14}{m}_{15}y+8{A}_{14}\left({A}_{25}+{A}_{11}N\left(y-h\right)\right)\right)\right.\\ \left.+\,{A}_{37}\left(\cosh \left[\sqrt{{m}_{2}}\left(y-h\right)\right]+\,\sinh \left[\sqrt{{m}_{2}}\left(y-h\right)\right]\right)\right)\\ -\,24\,\cosh \left[\sqrt{{m}_{1}}y\right]\left.\left({A}_{13}{A}_{25}-{A}_{8}{m}_{12}{m}_{13}-{A}_{11}{A}_{13}hN+{A}_{11}{A}_{13}Ny\right.\right)\\ \left.+\,{A}_{22}{A}_{8}{m}_{12}\left(\cosh \left[\sqrt{{m}_{2}}\left(y-h\right)\right]+\,\sinh \left[\sqrt{{m}_{2}}\left(y-h\right)\right]\right)\right)\end{array}\right)\right.,\end{eqnarray}$ $ \begin{eqnarray}\sigma =1+\displaystyle \frac{Sc\,Sr}{24{{m}_{1}}^{2}}\left(\begin{array}{l}\left(1+{\alpha }_{2}\right)\left(3{A}_{34}{h}^{2}+{A}_{35}\left(\sqrt{{m}_{2}}-24{{A}_{11}}^{2}h\,{N}^{2}\right)\right)-24{{A}_{11}}^{2}{A}_{35}{N}^{2}y+3{A}_{34}{y}^{2}\\ +\,\cosh \left[\sqrt{{m}_{2}}\left(y-h\right)\right]\left({A}_{35}\sqrt{{m}_{2}}+N\,\cos \,{\rm{h}}\left[\sqrt{{m}_{1}}y\right]\left(\begin{array}{l}{A}_{37}\sqrt{{m}_{1}}-\\ 24{A}_{22}{A}_{8}{m}_{12}\sqrt{{m}_{2}}\end{array}\right)\right.\\ \left.-\left(24{A}_{22}{A}_{8}\sqrt{{m}_{1}}{m}_{12}-{A}_{37}\sqrt{{m}_{2}}\right)N\,\sinh \left[\sqrt{{m}_{1}}y\right]\right)+N{A}_{46}\left(\left(1+{\alpha }_{2}\right)\right.\\ -3\left(8{A}_{11}{A}_{13}N-\sqrt{{m}_{1}}\left({A}_{19}{m}_{12}{m}_{14}{m}_{15}y+8{A}_{14}\left({A}_{25}+{A}_{11}N\left(y-h\right)\right)\right)\right)\\ \cosh \left[\sqrt{{m}_{1}}y\right]+{A}_{41}\,\cosh \left[2\sqrt{{m}_{1}}y\right]+3\left(8{A}_{8}\sqrt{{m}_{1}}{m}_{12}{m}_{13}+{A}_{19}{m}_{12}{m}_{14}{m}_{15}\right.\\ \left.+8{A}_{11}{A}_{14}N-8{A}_{13}\sqrt{{m}_{1}}\left({A}_{25}+{A}_{11}N\left(y-h\right)\right)\right)\sinh \left[\sqrt{{m}_{1}}y\right]\\ -2\sqrt{{m}_{1}}\,\sinh \left.\left[2\sqrt{{m}_{1}}y\right]\right)\left(6{A}_{16}N-{A}_{18}{m}_{12}{m}_{14}{m}_{15}y\right)\\ -\,\sinh \left.\left[\sqrt{{m}_{2}}\left(y-h\right)\right]\right)\left({A}_{35}\sqrt{{m}_{2}}-{A}_{47}N\,\cosh \left[\sqrt{{m}_{1}}y\right]-{A}_{48}N\,\sinh \left[\sqrt{{m}_{1}}y\right]\right)\end{array}\right).\end{eqnarray}$
The expressions for skin friction coefficient, Nusselt number and Sherwood number is obtained as follows $ \begin{eqnarray}{C}_{f}={\left.h^{\prime} \displaystyle \frac{{\partial }^{2}\psi }{\partial {y}^{2}}\right|}_{y=h},\end{eqnarray}$ $ \begin{eqnarray}Nu={\left.h^{\prime} \displaystyle \frac{\partial \theta }{\partial y}\right|}_{y=h},\end{eqnarray}$ $ \begin{eqnarray}Sh={\left.h^{\prime} \displaystyle \frac{\partial \sigma }{\partial y}\right|}_{y=h},\end{eqnarray}$ where $h^{\prime} =\varepsilon \,\,\cos \left[2\pi \left(x-t\right)\right]2\pi $ and $\psi $ is the stream function obtained by integrating equation (27 ) at plug flow region and core region by using the conditions ${\psi }_{p}=0\,{\rm{at}}\,y=0\,{\rm{and}}\,\psi \,={\psi }_{p}\,{\rm{at}}\,y={y}_{0}.$
4. Results and discussions
In the present section, the foremost objective is to analyze the impact of slip and variable transport properties on velocity, skin-friction, temperature, Nusselt number, concentration, Sherwood number, and streamlines and deliberate the results through graphs.
4.1. Velocity profiles
Figures 2(a)–(e) portray the variation of pertinent parameters on velocity profiles. Figure 2(a) shows the variation of variable viscosity on the velocity profile. From the figure, it is noticed that an increase in the value of variable viscosity declines the velocity profile. This shows that variable viscosity plays an imperative role in monitoring the velocity profile. Figure 2(b) characterizes the variation of yield stress parameter on the velocity profile. Since the yield stress parameter present in the model requires an extra quantity of energy for the flow to initiate, the velocity profile shrinks for large values of yield stress parameter. Figure 2(c) explains the velocity slip parameter on the velocity profile. As the velocity slip parameter decreases, the velocity profile shows an increment behavior. Figure 2(d) notices that the decrease in the magnetic parameter hikes the velocity profile. The effect of the angle of inclination on the velocity profile is illustrated in figure 2(e). It is noticed from the figure that a decrease in the angle of inclination increases the velocity profile.
Figure 2. Velocity profile for different values of (a) |
4.2. Temperature profiles
To explore the physical aspects of associated flow parameters, figures 3(a)–(g) are prepared. The resulted temperature distribution of parabolic nature has been obtained clearly due to the involvement of viscous dissipation forces. Physically, such dissipation forces are associated with the kinetic energy of liquids, which enables them to truncate the internal thermal energy due to the involvement of fluid viscosity. The change in temperature distribution due to variable thermal conductivity has been inspected in figure 3(a). An arising distribution of temperature has been reported with a variety of variable thermal conductivity parameters. In fact, due to a deviation in temperature distribution between fluid and wall surface, a higher temperature distribution in the whole domain is noted. Moreover, the more significant thermal conductivity permits the fluid particles to absorb or dissipate the heat to its surroundings. The behavior of Brinkman number on the temperature profile shows concordant expression as that of variable thermal conductivity parameter on temperature profile, which is rendered in figure 3(b). Figure 3(c) narrates the temperature slip parameter on the temperature profile. From this figure, it is concluded that a decrease in the temperature slip shows a decline in the temperature profile. The demeanor of the velocity slip on the temperature profile is illustrated in figure 3(d). Here, it is deduced that a decrease in the velocity slip parameter decreases the temperature profile. Figure 3(e) is plotted to examine the impact of the yield stress parameter on the temperature profiles. A boost up temperature distribution has been observed when yield stress parameter assigned maximum values. The effect of the magnetic parameter on the temperature profile is characterized by figure 3(f) from which one can notice that the decline in the magnetic parameter hikes the temperature profile. Figure 3(g) depicts the impact of the angle of inclination on the temperature profile. It is noted from the figure that a decline in the angle of inclination increases the temperature profile of the fluid flow.
Figure 3. Temperature profile for different values of, (a) γ, (b) N, (c) α 1, (d) α, (e) τ 0, (f) Mn (g) φ with β = 0.1, F = 0.2, E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6, φ = π/6. |
4.3. Concentration profiles
Figures 4(a)–(i) are plotted to demonstrate the impact of relevant parameters on the concentration profile. Alike temperature profiles, concentration profiles as well show parabolic nature. Figures 4(a)–(g) show the effect of concentration slip parameter, Schmidt number, Soret number, velocity slip parameter, magnetic parameter, temperature slip parameter, and Brinkmann number on the concentration profile. From these figures, it is grasped that increment in all these parameters shows an increase in the concentration profile. But whereas in figure 4(h), which illustrates the effect of yield stress parameter on the concentration profile, it is seen that the decline in the yield stress parameter increases the concentration profile. The effect of the angle of inclination on the concentration profile is depicted in figure 4(i). It is noticed clearly from the figure that enhancing the value of the angle of inclination, hikes the concentration profile.
Figure 4. Concentration profile for different values of (a) α 2, (b) Sc, (c) Sr, (d) α, (e) Mn, (f) α 1, (g) N, (h) τ 0 (i) φ with γ = 0.1, β = 0.1, F = 0.2, E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6. |
4.4. Skin-friction coefficient
Figures 5(a)–(e) are sketched to grasp the effects of relevant parameters on the skin friction coefficient. Figure 5(a) shows the impact of the velocity slip parameter on the skin friction coefficient. It is noticed from the figure that the skin friction coefficient slightly enhances with expanding approximation of the velocity slip parameter. The effect of the magnetic parameter on skin friction coefficient is remarked by figure 5(b). From the figure, an increment in the magnetic parameter increases the skin friction coefficient. The effect of yield stress parameter on the skin friction coefficient is characterized in figure 5(c). From the figure, it is noticed that the skin friction coefficient diminishes for a large estimation of the yield stress parameter. Figure 5(d) is the graph of the skin friction coefficient for expanding estimations of variable viscosity. It is noticed from the figure that an expansion in the estimate of variable viscosity enhances the skin friction coefficient. Henceforth, it is concluded that the impact of variable viscosity plays a prominent role in monitoring the skin friction coefficient. The impact of the body force parameter on the skin friction coefficient is examined in figure 5(e). It is illustrated from the figure that an increment in body force constant reduces the wall shear stress.
Figure 5. Skin-friction coefficient for different values of, (a) α, (b) Mn, (c) τ 0, (d) β, (e) F with E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6, φ = π/6. |
4.5. Nusselt number
Figures 6(a)–(e) portrays the essential role of different flow parameters on Nusselt number. Figure 6(a) specified the impact of the magnetic parameter on the Nusselt number. The Nusselt number oscillates periodically, and the magnitude of oscillation enhanced with the magnetic parameter. Figures 6(b)–(d) are drawn to illustrate the significance of Brinkmann number, variable thermal conductivity, and yield stress parameter on the Nusselt number. Here, an expansion in the estimate of the above-mentioned parameters enhances the value of Nusselt number. Figure 6(e) shows the effect of the body force parameter on the Nusselt number. The observation-based from this curve reveals that a decline in the value of the body force parameter hikes the Nusselt number.
Figure 6. Nusselt number for different values of, (a) Mn, (b) N, (c) γ, (d) τ 0, (e) F, with α 1 = 0.1, α = 0.01, β = 0.1, E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6, φ = π/6. |
4.6. Sherwood number
Figures 7(a)–(f) are plotted to grasp the effects of relevant parameters on Sherwood number. The impact of the magnetic parameter on the Sherwood number is shown in figure 7(a). From the figure, it is characterized that an increment in the magnetic parameter increases the Sherwood number. Figures 7(b)–(e) are plotted to illustrate the impact of Schmidt number, Soret number, yield stress parameter, and Brinkmann number on Nusselt number. At this moment, it is learned that the decrease in the value of Schmidt number, Soret number, yield stress parameter, and Brinkmann number shows an increment in Nusselt number. The impact of the body force parameter on the Nusselt number is concordant as that of the effect on the magnetic parameter, which is sketched in figure 7(f).
Figure 7. Sherwood number for different values of (a) Mn, (b) Sc, (c) Sr, (d) τ 0, (e) N, (f) F with γ = 0.1, α 2 = 0.01, α = 0.01, α 1 = 0.1, β = 0.1, E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6, φ = π/6. |
4.7. Trapping phenomena
The trapping phenomenon in peristalsis motion is considered as the most exciting phenomenon due to interesting physical significance. It represents the arrangements of bolus associated with the flow. The evaluation of flowing bolus in a fluid so-called trapping phenomenon is related to the stream functions. The boluses followed the sinusoidal movement for the peristaltic wave. The gastrointestinal tract and blood formulation encountered the applications of this phenomenon. Figures 8–11 are drawn to demonstrate the impact of relevant parameters on the trapped bolus. Figure 8 is constituted to explore the change in trapped bolus due to variable viscosity. From the figure, it is found that an increase in the value of variable viscosity enhances the size of the tapped bolus and thereby increases the formation of the bolus. Figure 9 is drawn to illustrate the effects of the velocity slip parameter on the trapped bolus. It is noticed from the figures that an increment in the value of the velocity slip parameter enhances the trapped bolus. The impact of the magnetic parameter on the tapered bolus is demonstrated in figure 10. From the figure, it is seen that an increase in the magnetic parameter enhances the size of the trapped bolus. The impact of the yield stress parameter on the trapped bolus is depicted in figure 11. Here, an increase in the value of the yield stress parameter diminishes the volume of the trapped bolus.
Figure 8. Streamline for different values of β with α = 0.01, τ 0 = 0.1, Mn = 1, F = 2, E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6, φ = π/6. |
Figure 9. Streamline for different values of α with τ 0 = 0.1, β = 0.1, Mn = 1, F = 2, E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6, φ = π/6. |
Figure 10. Streamline for different values of Mn with α = 0.01, τ 0 = 0.1, β = 0.4, F = 2, E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6, φ = π/6. |
Figure 11. Streamline for different values of τ 0 with α = 0.01, β = 0.1, Mn = 1, F = 2, E 1 = 0.1, E 2 = 0.04, E 3 = 0.4, m = 0.2, x = 0.2, t 1 = 0.1, ε = 0.6, φ = π/6. |
5. Conclusion
The existing model explores the combined effects of slip and wall features in the peristaltic flow of Casson fluid configured by an inclined channel. The findings of the model help in understanding the flow of blood in an artery. Specifically, this helps the doctors during the surgery to take control over the flow of blood by adjusting the magnetic field intensity. Further, one can use the Casson nanofluid, which will help you understand a few more aspects related to composition of biological fluids. Some of the thought-provoking outcomes are as follows:
| • | An increase in the value of the yield stress parameter diminishes the velocity of the flow while decrease in variable viscosity hikes the fluid flow. |
| • | The temperature of the fluid increases for a significant value of variable viscosity, thermal conductivity, temperature slip parameter. |
| • | The concentration is an increasing function of the magnetic parameter and Brinkmann’s number. |
| • | The consideration of variable viscosity and velocity slip parameter enhances the skin friction parameter. |
| • | The presence of convective conditions at the boundary diminishes the Nusselt number. |
| • | TAn increment in the magnetic parameter increases the Sherwood number. |
| • | The presence of magnetic field and variable viscosity increases the trapped bolus volume efficiently. |