**2. Problem Formulation**

Consider an electrically conducting and time-dependent thin film flow of Reiner-Philippoff fluid over spreading sheet. The elastic sheet start moving from fixed slit. The coordinates *oxyz* are adjusted in such way that *ox* and the plate are equal, and *oy* is along the sheet. The origin is at rest in the direction of the *x*-axis, due to the two equal and opposite forces of the stretching sheet flow. We take the *<sup>x</sup>*-axis in the direction spreading sheet and has the stress velocity *Uw*(*x*, *<sup>t</sup>*) = *<sup>γ</sup>x*(<sup>1</sup> − *<sup>ζ</sup>t*)−1, where *γ* and *ζ* represent any fix numbers, which are vertical to *y*-axis as shown in Figure 1.

**Figure 1.** Geometry of the physical model.

The wall temperature [72,73] of the liquid is

$$T\_w(\mathbf{x}, t) = \left(\frac{\gamma \mathbf{x}^2}{2(1 - \zeta t)^{1.5} \nu\_f}\right) T\_r + T\_0 \tag{1}$$

and the capacity of the nanoparticles is given by

$$\mathbb{C}\_{\mathbf{w}}(\mathbf{x},t) = \left(\frac{\gamma \mathbf{x}^2}{2(1-\zeta t)^{1.5}\nu\_f}\right)\mathbb{C}\_r + \mathbb{C}\_0\tag{2}$$

where *ν<sup>f</sup>* denotes the fluid kinematic viscosity, *T*<sup>0</sup> and *C*<sup>0</sup> denote the temperature of the slit and volume friction of the nanoparticles, while *Tr* and *Cr* represent the reference temperature and reference volume of the nanoparticles respectively. Assume that the effects of body forces are negligible in the field of flow. In light of the previous assumptions, the equation of continuity, the basic boundary governing equations, concentration, and heat transferring equations take the following forms.

The stress deformation behavior is well explained by Reiner-Philippoff in [74], and is considered one of the classical descriptions and is given by an implicit functional way:

$$\mathbf{r}\_{l\bar{j}} = \begin{bmatrix} \mu\_0 + \frac{\mu\_{\sigma o} - \mu\_0}{\frac{1}{2\tau\_0^2} (\sum\_{p=1}^3 \sum\_{q=1}^3 \tau\_{pq} \tau\_{qp})} \end{bmatrix} \mathbf{e}\_{l\bar{j}} \tag{3}$$

where the parameters *μ*0, *μ*∞, and *τ*<sup>0</sup> are greater than zero. This model gives interesting results due to its behavior, for large or small values of the model nearly agree with Newtonian fluids. Besides this, when the values of *τ*<sup>0</sup> are in between the extremes the model agrees with non-Newtonian fluids.

The Momentum equations for two dimensional flow together with the continuity equation takes the form:

$$\frac{\partial \vec{u}}{\partial x} + \frac{\partial \vec{v}}{\partial y} = 0 \tag{4}$$

$$
\rho \left( \frac{\partial \mathbf{d}}{\partial t} + \mathbf{i} \frac{\partial \mathbf{d}}{\partial \mathbf{x}} + \mathbf{i} \frac{\partial \mathbf{d}}{\partial \mathbf{y}} \right) + \frac{\partial p}{\partial \mathbf{x}} = \frac{\partial \tau\_{xx}}{\partial \mathbf{x}} + \frac{\partial \tau\_{xy}}{\partial y} \tag{5}
$$

$$
\rho \left( \frac{\partial \vartheta}{\partial t} + \vec{u} \frac{\partial \vartheta}{\partial x} + \vec{v} \frac{\partial \vartheta}{\partial y} \right) + \frac{\partial p}{\partial y} = \frac{\partial \tau\_{xy}}{\partial x} + \frac{\partial \tau\_{yy}}{\partial y} \tag{6}
$$

The components of stress, presented above, are difficult to present in a closed single explicit format. For this purpose, we assume a small *τ*<sup>0</sup> such that its higher powers greater than three vanishes. The stress components take the form:

$$
\pi\_{xy} = \pi\_{xx}^0 + \pi\_0^2 \pi\_{xx}^{'} \tag{7}
$$

with the constitutive relations defined by

$$\tau^{0}\_{\rm xx} = \mu\_0 \left( 2 \frac{\partial \mathfrak{a}}{\partial \mathbf{x}} \right), \tau^{0}\_{\mathbf{y}\mathbf{y}} = \mu\_0 \left( 2 \frac{\partial \mathfrak{v}}{\partial \mathbf{y}} \right), \tau^{0}\_{\mathbf{xy}} = \mu\_0 \left( \frac{\partial \mathfrak{a}}{\partial \mathbf{y}} + \frac{\partial \mathfrak{v}}{\partial \mathbf{x}} \right) \tag{8}$$

where *τ*<sup>0</sup> *xx* denotes the Newtonian stress component of the fluid with *μ*0, the coefficient of viscosity and *τ xx* is the residual contribution terms of the Reiner-Philippoff fluid, and prime should not be considered a derivative.

Using Equation (7) in Equation (3), we get

$$\tau\_{ij} = \left(\mu\_0 + \frac{\mu\_{\infty} - \mu\_0}{1 + \frac{1}{2\tau\_0^2} (\tau\_{xx}^{02} + 2\tau\_{xy}^{02} + \tau\_{yy}^{02} + \dots)}\right) \mathcal{E}\_{ij} \tag{9}$$

Neglecting fourth and higher order terms, we get

$$\tau\_{ij} = \left[\mu\_0 + \frac{2(\mu\_{\infty} - \mu\_0)}{\tau\_{xx}^{02} + 2\tau\_{xy}^{02} + \tau\_{yy}^{02}}\right] \varepsilon\_{ij} \tag{10}$$

Hence, *τij* is explicitly related to the gradient of the velocity by the above relation.

Let us consider *τ*<sup>0</sup> as large, such that its exponent larger than three is negligible in the relation expressed below.

$$
\tau\_{\rm xx} = \tau\_{\rm xx}^{\infty} + \frac{\tau\_{\rm xx}^{\prime\prime}}{\tau\_0^2} \tag{11}
$$

with the constitutive relations:

$$\tau\_{xx}^{\infty} = \mu\_{\infty} \left( 2 \frac{\partial \mathbf{d}}{\partial x} \right), \tau\_{yy}^{\infty} = \mu\_{\infty} \left( 2 \frac{\partial \breve{\mathbf{u}}}{\partial y} \right), \tau\_{xy}^{\infty} = \mu\_{\infty} \left( \frac{\partial \breve{\mathbf{u}}}{\partial y} + \frac{\partial \breve{\mathbf{u}}}{\partial x} \right) \tag{12}$$

where *τ*<sup>∞</sup> *xx* denotes the Newtonian fluid stress component with the viscosity coefficient *μ*<sup>∞</sup> and *τ xx*, the residual contribution terms of the Reiner-Philippoff fluid, while the prime should not be considered a derivative sign.

Using Equation (11) in Equation (1), and neglecting higher exponent terms of <sup>1</sup> *τ*0 , we get

$$\pi\_{ij} = \left[\mu\_{\infty} + \frac{(\mu\_0 - \mu\_{\infty})(\tau\_{xx}^{\infty2} + 2\tau\_{xy}^{\infty2} + \tau\_{yy}^{\infty2})}{2\tau\_0^2}\right] \varepsilon\_{ij} \tag{13}$$

Using Equations (10) and (13) in Equations (5) and (6), we get

$$\rho \left( \frac{\partial \mathcal{d}}{\partial t} + \mathcal{U} \frac{\partial \mathcal{d}}{\partial x} + \mathcal{U} \frac{\partial \mathcal{d}}{\partial y} \right) = -\frac{\partial p}{\partial x} + \frac{\partial}{\partial x} \left[ 2\lambda \frac{\partial \mathcal{d}}{\partial x} \right] + \frac{\partial}{\partial y} \left[ \lambda \left( \frac{\partial \mathcal{d}}{\partial y} + \frac{\partial \mathcal{d}}{\partial x} \right) \right] \tag{14}$$

$$\rho \left( \frac{\partial \vartheta}{\partial t} + \mathrm{il} \frac{\partial \vartheta}{\partial x} + \vartheta \frac{\partial \vartheta}{\partial y} \right) = -\frac{\partial p}{\partial y} + \frac{\partial}{\partial y} \left[ 2\lambda \frac{\partial \vartheta}{\partial x} \right] + \frac{\partial}{\partial x} \left[ \lambda \left( \frac{\partial \vartheta}{\partial y} + \frac{\partial \vartheta}{\partial x} \right) \right] \tag{15}$$

where *λ* represents the terms defined in the square brackets in Equations (10) and (13).

In the absence of pressure gradient, the boundary-layer equations are simplified to

$$
\left(\frac{\partial \vec{a}}{\partial t} + \vec{n}\frac{\partial \vec{a}}{\partial x} + \vec{v}\frac{\partial \vec{a}}{\partial y}\right) = \frac{1}{\rho} \left[\frac{\partial}{\partial y}\lambda\frac{\partial \vec{a}}{\partial y}\right] \tag{16}
$$

where

$$\frac{\partial \overline{u}}{\partial y} = \frac{\tau\_{ij}}{\mu\_{\infty} + \frac{\left(\mu\_0 - \mu\_{\infty}\right)}{1 + \left(\frac{\tau\_{ij}}{\tau\_{\infty}}\right)^2}}\tag{17}$$

$$\frac{\partial T}{\partial t} + \vec{u}\frac{\partial T}{\partial x} + \vec{v}\frac{\partial T}{\partial y} - \frac{K}{\rho c\_p}\frac{\partial}{\partial y}\left[\frac{\partial T}{\partial y}\right] = \tau \left[D\_B \left(\frac{\partial \mathbb{C}}{\partial \vec{y}} \frac{\partial T}{\partial \vec{y}}\right) + \frac{D\_T}{T\_{\infty}} \left(\frac{\partial T}{\partial y}\right)^2\right] \tag{18}$$

$$\frac{1}{D\_B} \left[ \frac{\partial \mathcal{C}}{\partial t} + \vec{u} \frac{\partial \mathcal{C}}{\partial x} + \vec{v} \frac{\partial \mathcal{C}}{\partial y} \right] - \frac{\partial^2 \mathcal{C}}{\partial y^2} = \left( \frac{D\_T}{T\_{\infty} D\_B} \right) \frac{\partial^2 T}{\partial y^2} \tag{19}$$

With the constraints defined at the boundaries

$$\text{if } \overline{u} = \mathcal{U}\_{\text{w.}} \qquad \overline{v} = 0, \quad T = T\_{\text{w.}} \qquad \mathbb{C} = \mathbb{C}\_{w} \qquad \text{at} \quad y = 0,\tag{20}$$

$$\frac{\partial \overline{u}}{\partial \mathbf{x}} = \frac{\partial T}{\partial \mathbf{x}} = \frac{\partial \mathbf{C}}{\partial \mathbf{x}} = 0, \quad \overline{v} = \frac{dh}{dt} = 0, \quad \mathbf{C} > 0,\tag{21}$$

Here, *u*˜ and *v*˜ represent the state variables, denotes the velocity components along *x*-axis and *y*-axis respectively, and *ρ* represents the density of the fluid, local temperature by *T*, and the fluid capacitance by *Cp*. The ratio (*ρ<sup>c</sup>* )*<sup>p</sup>* (*ρ<sup>c</sup>* )*<sup>f</sup>* is the characteristic ratio of the base fluid to the nanoparticles heat capacitance; *DB* represents the direct Brownian diffusion constant; *DT* represents thermophoretic diffusion constant, *K* is the thermal conductivity, and *T*∞ denotes the fluid temperature far away from the slit. Introducing the succeeding similarity transformations [75,76]

$$\eta = \sqrt{\frac{\gamma}{\upsilon(1-\zeta t)}} y,\ \psi(x,y,t) = x\sqrt{\frac{\upsilon\gamma}{1-\zeta t}} f(\eta),\ \vec{u} = \frac{\partial\psi}{\partial y} = \gamma(1-\zeta t)^{-1}\acute{f}'(\eta)$$

$$\vec{v} = \frac{\partial\psi}{\partial x} = -\sqrt{\frac{\gamma\upsilon}{(1-\zeta t)}} f(\eta),\ \tau\_{ij} = \left[x\left(\gamma(1-\zeta t)^{-1}\right)^3\right]^{\frac{1}{2}} g(\eta) \tag{22}$$

$$h(t) = \left[\frac{\upsilon}{\gamma(1-\zeta t)^{-1}}\right]^{\frac{1}{2}} \left(T\_w - T\_0\right)\theta(\eta) = T - T\_{0\prime}\left(\mathbb{C}\_w - \mathbb{C}\_0\right)\phi(\eta) = \mathbb{C} - \mathbb{C}\_0$$

where prime represents the change with respect to *η*, *β* = ! *ζ <sup>υ</sup>*(1−*ζt*) *<sup>h</sup>*(*t*) represents the liquid film thickness, *ψ* denotes the stream function, and *υ* = *<sup>μ</sup> <sup>ρ</sup>* is the kinematics viscosity. From the dimensionless film thickness, we can write <sup>d</sup>*<sup>h</sup>* <sup>d</sup>*<sup>t</sup>* <sup>=</sup> <sup>−</sup>*c<sup>β</sup>* 2 ! *υ <sup>ζ</sup>*(1−*ζt*) , for detail see [77,78]. With the help of the newly introduced similarity transformations, Equations (14)–(21) are reduced to the following equations, while the continuity equation is satisfied identically.

$$\frac{\mathbf{d}\frac{\mathbf{d}}{\mathbf{d}\eta} - \frac{1}{\epsilon} \left[ S \left( \frac{\eta}{2} \frac{\mathbf{d}^2 f}{\mathbf{d}\eta^2} + \frac{\mathbf{d}f}{\mathbf{d}\eta} \right) + \left( \frac{\mathbf{d}f}{\mathbf{d}\eta} \right)^2 - f \frac{\mathbf{d}f}{\mathbf{d}\eta} \right] = 0 \tag{23}$$

$$g - \epsilon \frac{\mathbf{d}^2 f}{\mathbf{d}\eta^2} \left(\frac{\varrho^2 + \lambda \gamma}{\delta^2 + \gamma}\right) = 0 \tag{24}$$

$$\frac{1}{\mathcal{P}r} \begin{pmatrix} \frac{d^2\theta}{d\eta^2} \end{pmatrix} + f \frac{\mathbf{d}\theta}{\mathbf{d}\eta} - 2\theta \begin{pmatrix} \frac{d f}{d\eta} \end{pmatrix} - \frac{S}{2} \begin{pmatrix} 3\theta + \eta \frac{\mathbf{d}\theta}{d\eta} \end{pmatrix} + Nt \begin{pmatrix} \frac{d\theta}{d\eta} \end{pmatrix}^2 + Nb \begin{pmatrix} \frac{d\theta}{d\eta} \end{pmatrix} \begin{pmatrix} \frac{d\phi}{d\eta} \end{pmatrix} = 0 \tag{25}$$

$$\frac{\mathrm{d}^2 \phi}{\mathrm{d}\eta^2} + \mathrm{Sc} \left[ f \frac{\mathrm{d}\phi}{\mathrm{d}\eta} - 2\phi \left( \frac{\mathrm{d}f}{\mathrm{d}\eta} \right) - \frac{\mathrm{g}}{2} \left( 3\phi + \eta \frac{\mathrm{d}\phi}{\mathrm{d}\eta} \right) \right] + \frac{Nt}{N\mathrm{b}} \left( \frac{\mathrm{d}^2 \theta}{\mathrm{d}\eta^2} \right) = 0 \tag{26}$$

The boundary constraints of the problem are:

$$f'(0) = 1, \quad f(0) = 0, \quad \theta(0) = \phi(0) = 1, \quad \text{g}(0) = 0, \quad \text{g}'(0) = 1 \tag{27}$$

$$f(\beta) = \frac{S\beta}{2}, \quad f''(\beta) = 0, \quad \theta'(\beta) = \phi'(\beta) = 0, \quad \text{g}''(\beta) = 0, \quad \text{g}(\beta) = \frac{S\beta}{2} \tag{28}$$

The generalized physical constraints obtained are defined as: *S* = *<sup>γ</sup>* is the non-dimensional measure of unsteadiness, = !*<sup>x</sup> <sup>υ</sup>* , *<sup>λ</sup>* <sup>=</sup> *<sup>μ</sup>*<sup>0</sup> *<sup>μ</sup>*<sup>∞</sup> and *<sup>γ</sup>* <sup>=</sup> *<sup>τ</sup>*<sup>2</sup> *s* -1−*κt cx* 1 3 3 are parameters of Reiner-Philippoff fluid, *Pr* = *ρνcp <sup>K</sup>* is the Prandtl number, *Nt* <sup>=</sup> *<sup>τ</sup>Dw*(*Tw*−*T*∞) *<sup>ν</sup>T*<sup>∞</sup> represents thermophoresis constraint, *Nb* = *<sup>τ</sup>DB*(*Cw*−*C*∞) *<sup>ν</sup>* represents the limitation of the Brownian motion, and *Sc* <sup>=</sup> *<sup>ν</sup> DB* denotes Schmidt number. All these parameters and numbers are well defined and explained briefly in literature. *Cf x* and *Nux* represent the local skin-friction coefficient and local Nusselt number respectively, and are defined as:

$$\mathcal{C}\_{fx} = \frac{(\tau)\_{y=0}}{\frac{\rho \hat{\Omega}\_{\text{\tiny}}^2}{2}} \tag{29}$$

or 
$$\frac{\mathbb{C}\_{fx}\sqrt{\mathbb{R}e\_x}}{2} = \tau\_w \sqrt{\frac{\mathbf{x}}{\mathcal{U}\_w^3}} = \mathbf{g}(0, \mathbf{x}) \tag{30}$$

where *Rex* is known as the local Reynolds number and is defined as *Rex* <sup>=</sup> *<sup>U</sup>*˜ *wx <sup>ν</sup>*˜ and *τ<sup>w</sup>* is the value of *τ* on *η* = 0. *Nu* is the Nusselt number and is defined as *Nu* = *Qw* ˆ *k*(*Tw*−*T*0) , while *Qw* denotes the heat flux and *Qw* <sup>=</sup> <sup>−</sup><sup>ˆ</sup> *k* -*∂T ∂y η*=0 . *Sh* = *DB Jw* ˆ *<sup>k</sup>*(*Tw*−*T*0) represents the Sherwood number in which *Jw* is the mass flux, where *Jw* = −*DB* - *∂c ∂y η*=0 .

Sherwood number *Sh* and Nusselt number *Nu* take the dimensionless forms:

$$\text{Nu} = \boldsymbol{\Theta}^{\prime}(0), \quad \text{Sh} = -\boldsymbol{\Phi}^{\prime}(0) \tag{31}$$
