**3. Flow Problem Formulation**

A two-dimensional electrically-conducting viscoelastic Walter's B fluid flow in the presence of viscous dissipation over a vertical nonlinear stretching sheet at *y* = 0 is considered in this problem. The sheet was assumed to vary nonlinearly with the velocity *Uw* = *axn*, where *n* represents the nonlinear stretching parameter and *a*(> 0) is the stretching rate constant. This fluid flow obeys the constitutive Equation (6). The surface of the sheet was held at power law surface temperature *Tw* <sup>=</sup> *<sup>T</sup>*<sup>∞</sup> <sup>+</sup> *bx*(2*n*−1) and power law heat flux <sup>−</sup>*<sup>k</sup> <sup>∂</sup><sup>T</sup> <sup>∂</sup><sup>y</sup>* <sup>=</sup> *cx*2*n*−1, where *<sup>n</sup>* is the parameter for surface temperature, *b*(> 0) and *c*(> 0) are constants, and *T*<sup>∞</sup> is the ambient temperature of the viscoelastic fluid. The variable magnetic field *B*(*x*) = *B*0*x*(2*n*−1) was applied normal to the sheet as shown in Figure 1 with *xy*-Cartesian coordinates in the horizontal and vertical direction. The steady two-dimensional continuity and Cauchy momentum equations are [36,37,39,40]:

$$
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \tag{9}
$$

$$\rho \left( u \frac{\partial \mu}{\partial \mathbf{x}} + v \frac{\partial \mu}{\partial y} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \frac{\partial \tau\_{\mathbf{xx}}}{\partial \mathbf{x}} + \frac{\partial \tau\_{\mathbf{xy}}}{\partial y} + F\_{\mathbf{x}} \tag{10}$$

$$
\rho \left( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = -\frac{\partial p}{\partial y} + \frac{\partial \tau\_{yx}}{\partial x} + \frac{\partial \tau\_{yy}}{\partial y} + F\_y \tag{11}
$$

where τ*xx*, τ*xy*, τ*yx*, and τ*yy* are the components of the stress matrix, *<sup>∂</sup>*τ*xx <sup>∂</sup><sup>y</sup>* and *<sup>∂</sup>*τ*yy <sup>∂</sup><sup>y</sup>* are elastic terms, while *<sup>∂</sup>*τ*xy <sup>∂</sup><sup>y</sup>* and *<sup>∂</sup>*τ*yy <sup>∂</sup><sup>y</sup>* are viscous terms.

**Figure 1.** A schematic diagram showing the flow geometry.

To simplify Equations (10) and (11), we need to find *<sup>∂</sup>*τ*xx <sup>∂</sup><sup>y</sup>* , *<sup>∂</sup>*τ*yy <sup>∂</sup><sup>y</sup>* , *<sup>∂</sup>*τ*xy <sup>∂</sup><sup>y</sup>* , and *<sup>∂</sup>*τ*yy <sup>∂</sup><sup>y</sup>* through Equation (6) and substitute it back into (10) and (11) to get:

The *x* momentum equation:

$$\rho \left( u \frac{\partial u}{\partial \mathbf{x}} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p}{\partial \mathbf{x}} + \mu\_0 \left( \frac{\partial^2 u}{\partial \mathbf{x}^2} + \frac{\partial u^2}{\partial y^2} \right) - k\_0 \left( u \frac{\partial^3 u}{\partial \mathbf{x}^3} + u \frac{\partial^3 u}{\partial x \partial y^2} + v \frac{\partial^3 u}{\partial x^2 \partial y} + v \frac{\partial^3 u}{\partial y^3} + 3 \frac{\partial u}{\partial x} \frac{\partial^2 v}{\partial x \partial y} \right)$$

$$+ k\_0 \left( \frac{\partial u}{\partial \mathbf{x}} \frac{\partial^2 u}{\partial y^2} - \frac{\partial u}{\partial y} \frac{\partial^2 u}{\partial x \partial y} - \frac{\partial u}{\partial y} \frac{\partial^2 v}{\partial x^2} - 2 \frac{\partial v}{\partial x} \frac{\partial^2 u}{\partial y \partial x} + F\_x \right) \tag{12}$$

The *y* momentum equation:

$$\rho \left( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = -\frac{\partial p}{\partial y} + \mu\_0 \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial v^2}{\partial y^2} \right) - k\_0 \left( u \frac{\partial^3 v}{\partial x^3} + u \frac{\partial^3 v}{\partial x \partial y^2} + v \frac{\partial^3 v}{\partial y \partial x^2} + v \frac{\partial^3 v}{\partial y^3} + 3 \frac{\partial v}{\partial y} \frac{\partial^2 u}{\partial x \partial y} \right)$$

$$k\_0 \left( + \frac{\partial v}{\partial y} \frac{\partial^2 v}{\partial x^2} - \frac{\partial v}{\partial x} \frac{\partial^2 v}{\partial y \partial x} - \frac{\partial v}{\partial x} \frac{\partial^2 u}{\partial y^2} - 2 \frac{\partial u}{\partial y} \frac{\partial^2 v}{\partial x \partial y} + F\_y \right) \tag{13}$$

Following boundary layer theory, we can assume an order of magnitude approach on each term of Equations (12) and (13) as (see Beard and Walter [35]):

$$O(\mathfrak{u}) = 1, O(\mathfrak{x}) = 1, O(\mathfrak{v}) = \delta, O(\mathfrak{v}) = \delta, O(\mathfrak{p}) = 1, O(k\_0) = \delta^2, O(\mathfrak{u}\_0) = \delta^2, \text{ and } O(B^2) = \delta^2 \tag{14}$$

Simplify boundary layer Equations (12) and (13) with these orders of magnitude to obtain:

$$
\rho \left( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = -\frac{\partial p}{\partial x} + \nu \frac{\partial^2 u}{\partial y^2} - k\_0 \left\{ u \frac{\partial^3 u}{\partial x \partial y^2} + \frac{\partial u}{\partial x} \frac{\partial^3 u}{\partial y^2} - \frac{\partial u}{\partial y} \frac{\partial^2 u}{\partial x \partial y} + v \frac{\partial^3 u}{\partial y^3} \right\} + \sigma \mathcal{B}^2 u \tag{15}
$$

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

Equation (15) gives the momentum boundary layer equation for the viscoelastic fluid with the magnetic field. Since the vertical plate is directed along the *x*-axis, the pressure gradient *<sup>∂</sup><sup>p</sup> <sup>∂</sup><sup>x</sup>* = 0. Hence, the momentum boundary layer governing equations for two-dimensional electrically-conducting viscoelastic fluid are given by:

$$u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2} - \frac{k\_0}{\rho} \left\{ u \frac{\partial^3 u}{\partial x \partial y^2} + \frac{\partial u}{\partial x} \frac{\partial^3 u}{\partial y^2} - \frac{\partial u}{\partial y} \frac{\partial^2 u}{\partial x \partial y} + v \frac{\partial^3 u}{\partial y^3} \right\} + \frac{\sigma}{\rho} B^2(x) u \tag{17}$$

subject to the following flow boundary conditions:

$$\mu = \mathcal{U}\_w(\mathfrak{x}) = a\mathfrak{x}^n, \quad \upsilon = 0, \qquad \text{at } \mathcal{y} = 0$$

and:

$$
\mu \to 0, \quad \frac{\partial u}{\partial y} \to 0, \qquad \text{as} \quad y \to \infty \tag{18}
$$

where *x* is parallel along the sheet, *y* is the direction perpendicular to the sheet, *u* and *v* are the horizontal and vertical velocity in the *xy*-direction, respectively, *ν* is the kinematic viscosity, and *ko* is the coefficient of the viscoelasticity, while all other physical parameters are as defined above.

To reduce the complexity of the system of the governing equations into the system of ordinary differential equations, a similarity transformation in the following form is introduced as (see Vajravelu [7]),

$$
\eta = \sqrt{\frac{(n+1)u}{2vx}} y \qquad \text{and} \qquad \psi(x,y) = \sqrt{\frac{2vux}{(n+1)}} f(\eta) \tag{19}
$$

where ψ(*x*, *y*) denotes the stream function and is defined by:

$$u = \frac{\partial \psi}{\partial y} \qquad \text{and} \qquad v = -\frac{\partial \psi}{\partial x} \tag{20}$$

Substituting Equations (19) and (20) into Equations (17) and (18) yields dimensionless ordinary differential equations:

$$\left(f'' + ff'' - \left(\frac{2u}{n+1}\right)f'^2 f' - K\left\{(3u-1)f'f'' - \left(\frac{3u-1}{2}\right)f'^2 - \left(\frac{n+1}{2}\right)ff^{ip}\right\} - M\left(\frac{2}{n+1}\right) \\ = 0 \quad \text{(21)}$$

subject to the following dimensionless boundary conditions:

$$f(\eta) = 0, \qquad f'(\eta) = 1 \qquad \text{at} \qquad \eta = 0$$

$$f'(\eta) \to 0, \qquad f''(\eta) \to 0 \qquad \text{as} \qquad \eta \to \infty \tag{22}$$

where the prime represents the derivative of *f* with respect to η and the dimensionless quantities in these equations are nonlinear sheet parameter *n*, viscoelastic parameter *K*, and magnetic parameter *M*, which are defined as:

$$K = \frac{a x^{n-1}}{\rho v} \qquad \text{and} \qquad M = \frac{\sigma B\_0^2}{\rho a} \tag{23}$$

The physical quantity of interest is the coefficient of skin friction *Cf* at the stretched surface and defined as:

$$\mathcal{C}\_f = \frac{\tau\_w}{\rho u\_w^2} \tag{24}$$

where τ*<sup>w</sup>* is the wall shear stress from the plate and is given by:

$$\tau \pi\_w = \mu\_0 \left( \frac{\partial u}{\partial y} \right)\_{y=0} - k\_0 \left( u \frac{\partial^2 u}{\partial u \partial y} + v \frac{\partial^2 u}{\partial y^2} - 2 \frac{\partial u}{\partial y} \frac{\partial v}{\partial y} \right)\_{y=0} \tag{25}$$

#### **4. Heat Transfer Analysis**

For the analysis of heat transfer, two cases of the heating process are considered, i.e.,


These non-isothermal conditions are applicable in industrial and engineering processes where the temperature is not constant. The energy equation with the viscous dissipation term is given by (see Cortell [9]):

$$
\mu \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2} + \frac{\nu}{C\_p} \left(\frac{\partial u}{\partial y}\right)^2 \tag{26}
$$

where *T*, α, and ρ*Cp* are respectively the temperature, thermal diffusivity, and specific heat capacity of the fluid at constant pressure, while all other physical parameters are as defined above. The thermal boundary conditions depend on the type of heating process under consideration. The momentum Equation (15) and the energy Equation (26) are decoupled and thereby solved sequentially.
