**3. Similarity analysis**

a force in the *x-*direction and the effective stretching sheet rate *a* /(1−*αt*) increases with time. Analogously, the sheet temperature and concentration increase (reduce) if *b* and *c* are positive (negative), respectively, from *T∞* and *C∞* at the sheet in the proportion to *x*. We assume that the radiation effect is significant in this study. The fluid properties are taken to be constant except for density variation with temperature and concentration in the buoyancy terms. Under those assumptions and the Boussinesq approximations, the governing two-dimensional

> 0, *u v x y* ¶ ¶ + =

*uuu u N Bu u u v g TT g CC t xy y y K*

r

*TTT T u Q u v T T t xy y c y c*

*CCC C u v D CC*

++= - -

¶ ¶ ¶ ¶ +¶ æ öæ ö ++= + ± - ç ÷ç ÷ ¶¶¶ ¶ ¶ ç ÷

m k

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¶¶¶ ¶ ¶ æ ö

b

2 <sup>2</sup> 2 , *NNN N u u v <sup>N</sup> t x y jy j y* g

<sup>2</sup> <sup>2</sup>

where *u* and *v* are the velocity components along the *x* and*y* axes, respectively, *T* is the fluid temperature, *μ* is the component of the microrotation vector normal to the *x y* plane, *γ* is the spin gradient viscosity, *α*0 is the thermal conductivity, *Cp* is the heat capacity at constant

and concentration expansion, respectively, *β<sup>0</sup>* is the transverse magnetic field, *C* is the concen‐ tration of the solutes, *T∞* and *C∞* denote the temperature and concentration far away from the plate, respectively, and *j* is the microinertia density or microinertia per unit mass. The

 k

 r

a

*t xy y*

¶¶¶ ¶

( ) ( ) 2 2

 k

 r

<sup>0</sup> <sup>2</sup> , *p p*

> ( ) <sup>2</sup> <sup>2</sup> . *<sup>c</sup>*

*u U xt v V N T T xt C C xt y* = = == = = *ww w w* ( , , , 0, , , , at 0, ) ( ) ( ) (8)

*u T TC C y* 0, , as . = ® ® ®¥ ¥ ¥ (9)

k¥

+ + = + + -+ - - - ç ÷ ¶¶¶ ¶ ¶ è ø (4)

<sup>2</sup> \* , *t c*

 b

++= - + ç ÷ ¶¶¶ ¶ ¶ è ø (5)

 r¥

¶¶¶ ¶ (7)

¶ ¶ (3)

¥ ¥

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and *βc* are the coefficients of thermal expression

0

r

s

*p*

 n

 r

boundary layer equations are given as:

292 Numerical Simulation - From Brain Imaging to Turbulent Flows

mk

¶ ¶ ¶ +¶ ¶ æ ö

r

pressure, *g* is the acceleration due to gravity, *β<sup>t</sup>*

appropriate boundary conditions for the current model are:

In this section, we transform the partial differential equations into ordinary differential equations. Similarity techniques reduce the number of parameters, as well as improve insight into the comparative size of various terms present in the equations.

#### **3.1. Transformation of the governing equations**

In order to transform the governing Eqs. (3)–(7) into a set of ordinary differential equations, we introduce the following transformation variables [40]:

$$\begin{aligned} \eta &= \sqrt{\frac{a}{\nu \left(1 - \alpha t\right)}} \eta, \quad \nu = \sqrt{\frac{a \nu}{\left(1 - \alpha t\right)}} \ge f\left(\eta\right), \quad N = \sqrt{\frac{a^3}{\nu \left(1 - \alpha t\right)^3}} \ge h(\eta), \\\ T &= T\_\alpha + \frac{bx}{\left(1 - \alpha t\right)^2} \theta\left(\eta\right), \quad C = C\_\alpha + \frac{c\chi}{\left(1 - \alpha t\right)^2} \theta\left(\eta\right), \end{aligned} \tag{10}$$

where is the physical stream function which automatically satisfies the continuity equation. Upon substituting similarity variables into Eqs. (3)–(7), we obtain the following system of ordinary equations:

$$\left(\mathrm{l} + \Delta\right)f'' + ff'' - f'^2 - \frac{A}{2}\left(2f + \eta f'\right) + \Delta h' + \lambda\_1 \theta + \lambda\_2 \phi - \left(M\_1 + \frac{1}{K\_{\rho}}\right)f' = 0,\tag{11}$$

$$
\lambda\_\gamma h'' + f'h' - f'h - \frac{A}{2} \left( \Im h + \eta h' \right) - \Delta B \left( \Im h + f'' \right) = 0,\tag{12}
$$

$$\frac{1}{2\text{Pr}}\theta'' + f\,\theta' - f'\theta - \frac{A}{2}(4\theta + \eta\,\theta') + Ec(1+\Lambda)\,f''^2 \pm Q\_z\theta + 0,\tag{13}$$

$$\frac{1}{2\text{ Sc}}\phi'' + f\phi' - f'\phi - K\phi - \frac{A}{2}(4\phi + \eta\phi') = 0.\tag{14}$$

*Boundary conditions*

The corresponding boundary conditions become:

$$f'(0) = 1, \ f(0) = f\_w,\\ h(0) = 0, \ \theta(0) = 1, \ \phi(0) = 1,\tag{15}$$

$$f'(\infty) = 0, \ h(\infty) = 0, \ \theta(\infty) = 0, \ \phi(\infty) = 0. \tag{16}$$

with

$$A = \frac{\alpha}{a}, \ \Lambda = \frac{\kappa}{\mu}, \ \lambda\_1 = \frac{\text{g}\,\beta b}{a^2} = \frac{Gr\_x}{\text{Re}\_x^2}, \ Gr\_x = \frac{\text{g}\,\beta \left(T\_w - T\_o\right)\mathbf{x}^\dagger}{\nu^2}, \ \text{Re}\_x = \frac{U\_w \chi}{\nu}, \alpha\_0 = \frac{k}{\rho c\_p},$$

$$\lambda\_2 = \frac{\text{g}\,\beta\_c}{a^2} = \frac{Gc\_v}{\text{Re}\_x^2}, \ Gc\_x = \frac{\text{g}\,\beta\_c \left(C\_w - C\_o\right)\mathbf{x}^\dagger}{\nu^2}, \lambda\_3 = \frac{\chi}{\mu\_j}, \ B = \frac{\nu\left(1 - \alpha t\right)}{ja} = \frac{\nu\chi}{jU\_w}, \ \text{Pr} = \frac{\nu}{\alpha},\tag{17}$$

$$Ec = \frac{U\_w}{c\_p\left(T\_w - T\_o\right)}, \ M = \frac{\sigma B\_0^{-2}\left(1 - \alpha t\right)}{\rho a}, \ \frac{1}{K\_p} = \frac{\nu\left(1 - \alpha t\right)}{\rho a K\_p}, \ Q\_x = \frac{Qx}{\rho c\_p U\_w}, \ K = \frac{K\_c\left(1 - \alpha t\right)}{a}.$$

#### **3.2. Quantities of engineering interest**

The quantities of engineering interest in the present study are the skin-friction coefficient *Cfx*, the local wall couple stress *Mwx*, the local Nusselt number *Nux* and the local Sherwood number *Shx*. The quantities are, respectively, defined by:

$$C\_{\beta \epsilon} = \frac{2}{\rho U\_w^2} \left[ (\mu + \kappa) \left( \frac{\partial u}{\partial \mathbf{y}} \right)\_{\mathbf{y} = 0} + \kappa N \left|\_{\mathbf{y} = 0} \right. \right] = 2(\mathbf{l} + \Delta) \text{Re}\_x^{-1/2} f''(\mathbf{0}), \tag{18}$$

$$M\_{\rm ux} = \frac{\gamma a}{\nu} \left(\frac{\partial N}{\partial \mathbf{y}}\right)\_{\mathbf{y}=\mathbf{0}} = \text{Re}\_{\mathbf{x}}^{-1} h'(\mathbf{0}),\tag{19}$$

$$Nu\_x = \frac{-\chi}{T\_w - T\_o} \left(\frac{\partial T}{\partial \mathbf{y}}\right)\_{\mathbf{y}=\mathbf{0}} = -\text{Re}\_x^{1/2} \theta'(\mathbf{0}),\tag{20}$$

$$\text{LSh}\_x = \frac{-\chi}{C\_w - C\_o} \left( \frac{\partial C}{\partial \mathbf{y}} \right)\_{\mathbf{y} = 0} = -\text{Re}\_x^{1/2} \phi'(\mathbf{0}). \tag{21}$$
