**2. Mathematical Formulation**

The peristaltic motion of non-Newtonian incompressible fluid in a vertical tapered asymmetric channel, under effects of a constant magnetic field is considered. It is assumed that a wave train is moving with velocity c along non-uniform walls. In addition, we also assume that the channel andmagnetic field are inclined at angles α and Θ, respectively. Further, upper and lower walls of the channel are sustained at temperature *T*<sup>0</sup> and *T*<sup>1</sup> , respectively. For the present flow, *U* and V are velocities in *X* and *Y* directions, respectively, in fixed frame. The upper and lower walls *H*<sup>1</sup> and *H*2, respectively, of tapered asymmetric channel in fixed frame are defined as:

$$\begin{array}{lcl}Y = & H\_1 = & d\_1 + k^\*X + a\_1 \cos\left[\frac{2\pi}{\lambda}(X - ct)\right] \\ Y = & H\_2 = & -d\_2 - k^\*X - b\_1 \cos\left[\frac{2\pi}{\lambda}(X - ct) + \varphi\right] \end{array} \tag{1}$$

where *a*<sup>1</sup> and *b*<sup>1</sup> are amplitudes of waves; λ is wave length; *d*<sup>1</sup> + *d*<sup>2</sup> is width of channel; *k*<sup>∗</sup> (*k*∗ << 1) is non-uniform parameter; *c* is velocity of propagation; *t* is time; phase difference φ varies in range 0 ≤ ϕ ≤ π; ϕ = 0 corresponds to symmetric channel by waves out of phase and φ = π waves are in phase; and further, *a*1, *b*1, *d*1, *d*<sup>2</sup> and φ satisfy the condition [24].

$$a\_1^2 + b\_1^2 + 2a\_1b\_1 \cos \phi \le \left(d\_1 + d\_2\right)^2.$$

An equation that governs flow in the presence of gravity consequences and an inclined magnetic field are defined as [25].

$$\frac{\partial \mathcal{U}}{\partial \mathcal{X}} + \frac{\partial \mathcal{V}}{\partial \mathcal{Y}} = 0 \tag{2}$$

$$\begin{aligned} \rho \left( \frac{\partial \mathcal{U}}{\partial t} + \mathcal{U} \frac{\partial \mathcal{U}}{\partial \mathcal{X}} + V \frac{\partial \mathcal{U}}{\partial \mathcal{Y}} \right) &= -\frac{\partial p}{\partial \mathcal{X}} + \frac{\partial}{\partial \mathcal{X}} (S\_{\mathcal{XX}}) + \frac{\partial}{\partial \mathcal{Y}} (S\_{\mathcal{XY}}) - \sigma B\_0^2 \cos \theta (\mathcal{U} \cos \theta - V \sin \theta) \\ &+ \rho g \sin a \end{aligned} \tag{3}$$

$$\begin{aligned} \rho \left( \frac{\partial V}{\partial \overline{t}} + \mathcal{U} \frac{\partial V}{\partial \overline{X}} + V \frac{\partial V}{\partial \overline{Y}} \right) &= -\frac{\partial p}{\partial \overline{Y}} + \frac{\partial}{\partial \overline{X}} (\mathcal{S}\_{YX}) + \frac{\partial}{\partial \overline{Y}} \mathcal{S}\_{YY} + \sigma B\_0^2 \sin \Theta (\mathcal{U} \cos \Theta - V \sin \Theta) \\ &- \rho g \cos a \end{aligned} \tag{4}$$

$$\begin{array}{l} \mathbf{C}\_{p} \left( \frac{\partial T}{\partial \mathbf{f}} + \mathbf{U} \frac{\partial T}{\partial \mathbf{X}} + \mathbf{V} \frac{\partial T}{\partial \mathbf{Y}} \right) = \frac{K\_{1}}{\rho} \left( \frac{\partial^{2} T}{\partial \mathbf{X}^{2}} + \frac{\partial^{2} T}{\partial \mathbf{Y}^{2}} \right) + \nu \left( \frac{1}{1 + \lambda\_{1}} \left( 1 + \lambda\_{2} \left( \frac{\partial}{\partial \mathbf{t}} + \mathbf{U} \frac{\partial}{\partial \mathbf{X}} + \mathbf{V} \frac{\partial}{\partial \mathbf{Y}} \right) \right) \\\ \left( 2 \left( \frac{\partial \mathbf{U}}{\partial \mathbf{X}} \right)^{2} + 2 \left( \frac{\partial \mathbf{V}}{\partial \mathbf{Y}} \right)^{2} + \left( \frac{\partial \mathbf{U}}{\partial \mathbf{Y}} + \frac{\partial \mathbf{V}}{\partial \mathbf{X}} \right)^{2} \right) \end{array} \tag{5}$$

$$\frac{\partial \mathcal{C}}{\partial t} + \mathcal{U} \frac{\partial \mathcal{C}}{\partial X} + V \frac{\partial \mathcal{C}}{\partial Y} = D\_m \left( \frac{\partial^2 \mathcal{C}}{\partial X^2} + \frac{\partial^2 \mathcal{C}}{\partial Y^2} \right) + \frac{D\_m K\_T}{T\_m} \left( \frac{\partial^2 T}{\partial X^2} + \frac{\partial^2 T}{\partial Y^2} \right) \tag{6}$$

where ρ, *p*, ν, σ, *g*, *K*<sup>1</sup> , *Cp* , *T*, *Dm*, *Tm*, *KT* and *C* represent constant density, pressure, kinematic viscosity, electrical conductivity, acceleration caused by gravity, thermal conductivity, specific heat, temperature, coefficient of mass diffusivity, mean temperature, thermal diffusion ratio and concentration of fluid, respectively.

For the Jeffrey fluid model, extra stress tensor **S** is given as [26].

$$\mathbf{S} = \frac{\mu}{1 + \lambda\_1} (\dot{\mathbf{y}} + \lambda\_2 \ddot{\mathbf{y}}) \tag{7}$$

where <sup>λ</sup><sup>1</sup> is ratio of relaxation to retardation times; . γ is shear rate; μ is viscosity of fluid; λ<sup>2</sup> is retardation time; and dots indicate differentiation with respect to time. Extra stress tensor **S** in component form is defined as:

$$S\_{XX} = \frac{2\mu}{1 + \lambda\_1} \Big( 1 + \lambda\_2 \Big( \frac{\partial}{\partial t} + (II\frac{\partial}{\partial X} + V\frac{\partial}{\partial Y}) \Big) \Big) \frac{\partial II}{\partial X}$$

$$S\_{XY} = \frac{\mu}{1 + \lambda\_1} \Big( 1 + \lambda\_2 \Big( \frac{\partial}{\partial t} + (II\frac{\partial}{\partial X} + V\frac{\partial}{\partial Y}) \Big) \Big) \Big( \frac{\partial U}{\partial Y} + \frac{\partial V}{\partial X} \Big)$$

$$S\_{YY} = \frac{2\mu}{1 + \lambda\_1} \Big( 1 + \lambda\_2 \Big( \frac{\partial}{\partial t} + (II\frac{\partial}{\partial X} + V\frac{\partial}{\partial Y}) \Big) \Big) \frac{\partial V}{\partial Y}$$

Furthermore, we know that the wave frame (*x*,*y*) and fixed frame (*X*,*Y*) are related by the following transformations:

$$\mathbf{x} = X - \mathbf{c}t,\ y = Y,\ u = \mathcal{U} - \mathbf{c},\ v = \mathcal{V},\ \text{and}\ p(\mathbf{x}) = p(\mathbf{X},t). \tag{8}$$

Let us define the following non-dimensional quantities:

− x = <sup>x</sup> λ, − y = <sup>y</sup> d1 , − u = <sup>u</sup> c , − v = <sup>v</sup> <sup>c</sup> , <sup>δ</sup> <sup>=</sup>d1 <sup>λ</sup> , d <sup>=</sup>d2 d1 , − <sup>p</sup> <sup>=</sup> d2 1p <sup>μ</sup>cλ, − t = ct <sup>λ</sup> , h1 <sup>=</sup> H1 d1 , h2 = H2 d2 , a <sup>=</sup> a1 d1 , b =b1 d1 , Re =cd1 v , − Ψ = <sup>Ψ</sup> cd1 , Fr = c2 gd1 , θ = <sup>T</sup>−T0 T1−T0 , Sr =ρDmKt(T1−T0) Tmμ(C1−C0) , Sc <sup>=</sup> <sup>μ</sup> <sup>ρ</sup>Dm , Ec <sup>=</sup> c2 Cp(T1−T0), Pr <sup>=</sup> ρνCp K1 , − S = Sd1 <sup>μ</sup><sup>c</sup> , M = σ <sup>μ</sup> B0d1, <sup>Φ</sup> <sup>=</sup> <sup>C</sup>−C0 C1−C0 (9)

where Re is Reynolds number; *Fr* is Froude number; *Sr* is Soret number; *Sc* is Schmidt number; *Ec* is Eckret number; Pr is Prandtl number; *M* is Hartmann number; θ is temperature of fluid in dimensionless form; and Φ is concentration of fluid in dimensionless form.

With the help of Equations (7) and (8), Equations (2)–(6), in terms of stream function Ψ (dropping the bars, *u* = <sup>∂</sup><sup>Ψ</sup> <sup>∂</sup>*<sup>y</sup>* , *<sup>v</sup>* <sup>=</sup> <sup>−</sup><sup>δ</sup> <sup>∂</sup><sup>Ψ</sup> <sup>∂</sup>*<sup>x</sup>* ), take following form:

$$\begin{array}{l} \text{Re}\delta \big( \mathbb{Y}\_{y}\mathbb{Y}\_{xy} - \mathbb{Y}\_{x}\mathbb{Y}\_{yy} \big) = \ -\frac{\partial p}{\partial x} + \delta \frac{\partial}{\partial x} (\mathbb{S}\_{xx}) + \frac{\partial}{\partial y} \big( \mathbb{S}\_{xy} \big) - \\\ M^{2} \cos \Theta \big( \big( \mathbb{Y}\_{y} + 1 \big) \cos \Theta + \delta \mathbb{Y}\_{x} \sin \Theta \big) + \frac{\text{Re}}{\mathbb{R}^{r}} \sin \alpha \end{array} \tag{10}$$

$$\begin{array}{l} \text{Re}\delta^{3} \{-\Psi\_{y}\Psi\_{xx} + \Psi\_{x}\Psi\_{xy}\} = -\frac{\partial p}{\partial y} + \delta^{2} \frac{\partial}{\partial x} \{S\_{yx}\} + \delta \frac{\partial}{\partial y} \{S\_{yy}\} + \\\ M^{2} \delta \sin\Theta \{ \left(\Psi\_{y} + 1\right) \cos\Theta + \delta \Psi\_{x} \sin\Theta \} - \delta \frac{\partial \pi}{\partial r} \cos\alpha \end{array} \tag{11}$$

$$\begin{aligned} \text{Re}\delta \{\Psi\_y \partial\_\mathbf{x} - \Psi\_x \theta\_y\} &= \frac{1}{\text{Pr}} \Big(\partial\_{\mathcal{Y}\mathcal{Y}} + \delta^2 \theta\_{\text{xx}}\Big) + \frac{\mathbb{E}c}{\left(1 + \lambda\_1\right)} \Big(1 + \frac{\lambda\_2 c \delta}{d\_1} \Big(\Psi\_{\mathcal{Y}} \frac{\partial}{\partial \mathbf{x}} - \Psi\_x \frac{\partial}{\partial y}\Big)\Big) \\ &\quad \left(4\delta^2 \Psi\_{\mathbf{xy}}^2 + \left(\Psi\_{\mathcal{Y}\mathcal{Y}} - \delta^2 \Psi\_{\mathbf{xx}}\right)^2\right) \end{aligned} \tag{12}$$

$$\operatorname{Re}\delta\left(\Psi\_y\Phi\_x - \Psi\_x\Phi\_y\right) = \frac{1}{\operatorname{Sc}}\Big(\delta^2\Phi\_{xx} + \Phi\_{yy}\Big) + \operatorname{Sr}\Big(\delta^2\theta\_{xx} + \theta\_{yy}\Big) \tag{13}$$

where extra stress tensor forJeffrey fluid in component form is defined as:

$$\begin{array}{rcl} S\_{xx} &=& \frac{2\delta}{1+\lambda\_1} \Big( 1 + \frac{\lambda\_2 c\delta}{d\_1} \Big( \Psi\_y \frac{\partial}{\partial x} - \Psi\_x \frac{\partial}{\partial y} \Big) \Big) \Psi\_{xy} \\\ S\_{xy} &=& \frac{1}{1+\lambda\_1} \Big( 1 + \frac{\lambda\_2 c\delta}{d\_1} \Big( \Psi\_y \frac{\partial}{\partial x} - \Psi\_x \frac{\partial}{\partial y} \Big) \Big) \Big( \Psi\_{yy} - \delta^2 \Psi\_{xx} \Big) \\\ S\_{yy} &=& -\frac{2\delta}{1+\lambda\_1} \Big( 1 + \frac{\lambda\_2 c\delta}{d\_1} \Big( \Psi\_y \frac{\partial}{\partial x} - \Psi\_x \frac{\partial}{\partial y} \Big) \Big) \Psi\_{xy} \end{array} \tag{14}$$

Using Equation (14) and assumptions of long wavelength approximation, Equations (10)–(13) reduce in the form:

$$-\frac{\partial p}{\partial x} + \frac{\partial}{\partial y} \left(\frac{1}{1+\lambda\_1} \frac{\partial^2 \Psi}{\partial y^2}\right) - M^2 \cos^2 \Theta \{\Psi\_y + 1\} + \frac{\text{Re}}{Fr} \sin \alpha = 0\tag{15}$$

$$-\frac{\partial p}{\partial y} = 0\tag{16}$$

$$\frac{1}{\text{Pr}} \frac{\partial^2 \theta}{\partial y^2} + \frac{Ec}{(1+\lambda\_1)} \left(\frac{\partial^2 \Psi}{\partial y^2}\right)^2 = 0\tag{17}$$

$$\frac{1}{Sc} \frac{\partial^2 \Phi}{\partial y^2} + Sr \frac{\partial^2 \theta}{\partial y^2} = 0. \tag{18}$$

Elimination of pressure from Equation (15) to (16) gives:

$$\frac{\partial^2}{\partial y^2} \left( \frac{1}{1 + \lambda\_1} \frac{\partial^2 \Psi}{\partial y^2} \right) - M^2 \cos^2 \Theta \frac{\partial^2 \Psi}{\partial y^2} = 0 \tag{19}$$

$$\frac{1}{\Pr} \frac{\partial^2 \theta}{\partial y^2} + \frac{Ec}{(1+\lambda\_1)} \left(\frac{\partial^2 \Psi}{\partial y^2}\right)^2 = 0\tag{20}$$

$$\frac{1}{8c}\frac{\partial^2 \Phi}{\partial y^2} + Sr \frac{\partial^2 \theta}{\partial y^2} = 0.\tag{21}$$

The system of PDEs given above in Equation (19) through (21) is solved subject to the following boundary conditions:

$$\begin{array}{rcl} \Psi &=& \frac{F}{2} \text{ at } y = h\_1 = 1 + k\mathbf{x} + a\cos 2\pi \mathbf{x} \\ \Psi &=& -\frac{F}{2} \text{ at } y = h\_2 = -d - k\mathbf{x} - b\cos(2\pi \mathbf{x} + q) \\ \frac{\partial \Psi}{\partial y} &=& -\frac{\eta\_1^\*}{(1 + \lambda\_1)} \frac{\partial^2 \Psi}{\partial y^2} - \frac{\eta\_2^\*}{\left(1 + \lambda\_1\right)} \frac{\partial^3 \Psi}{\partial y^3} - 1 \text{ at } y = h\_1 \\ \frac{\partial \Psi}{\partial y} &=& \frac{\eta\_1^\*}{\left(1 + \lambda\_1\right)} \frac{\partial^2 \Psi}{\partial y^2} + \frac{\eta\_2^\*}{\left(1 + \lambda\_1\right)} \frac{\partial^3 \Psi}{\partial y^3} - 1 \text{ at } y = h\_2 \end{array} \tag{22}$$

$$\begin{array}{lclcl}\theta + \beta \frac{\partial \theta}{\partial y} & = 0 \text{ at } y = h\_1\\ \text{through} (13) \theta - \beta \frac{\partial \theta}{\partial y} & = 1 \text{ at } y = h\_2 \end{array} \tag{23}$$

$$\begin{aligned} \Phi + \gamma \frac{\partial \Phi}{\partial y} &= 0 \text{ at } y = h\_1 \\ \text{through} (13) \Phi - \gamma \frac{\partial \Phi}{\partial y} &= 1 \text{ at } y = h\_2 \end{aligned} \tag{24}$$

where *F* is flux in wave frame; η∗ 1 ,η∗ <sup>2</sup> ,β and γ represent 1st-order slip parameter, 2nd-order slip parameter, thermal slip parameter, and concentration slip parameter, respectively; *h*<sup>1</sup> and *h*<sup>2</sup> are thedimensionless form of surfaces of peristaltic walls.
