**4. Example 2: homotopy perturbation method to analysis of squeezing flow and heat transfer of Casson nanofluid between two parallel plates embedded in a porous medium under the influences of slip, Lorentz force, viscous dissipation and thermal radiation**

Consider a Casson nanofluid flowing between two parallel plates placed at timevariant distance and under the influence of magnetic field as shown in the **Figure 2**. It is assumed that the flow of the nanofluid is laminar, stable, incompressible, isothermal, non-reacting chemically, the nanoparticles and base fluid are in thermal equilibrium and the physical properties are constant. The fluid conducts electrical energy as it flows unsteadily under magnetic force field. The fluid structure is everywhere in thermodynamic equilibrium and the plate is maintained at constant temperature.

Following the assumptions, the governing equations for the flow are given as

$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial \mathbf{y}} = \mathbf{0} \tag{27}$$

$$\rho\_{\eta f} \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p}{\partial x} + \mu\_{\eta f} \left( 1 + \frac{1}{\beta} \right) \left( 2 \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial x \partial y} + \frac{\partial^2 u}{\partial y^2} \right) - \sigma B\_o^2 u - \frac{\mu\_{\eta f} u}{K\_p} \tag{28}$$

$$\rho\_{\rm nf} \left( \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = -\frac{\partial p}{\partial y} + \mu\_{\rm nf} \left( \mathbf{1} + \frac{\mathbf{1}}{\beta} \right) \left( 2 \frac{\partial v}{\partial x^2} + \frac{\partial^2 v}{\partial x \partial y} + \frac{\partial^2 v}{\partial y^2} \right) - \frac{\mu\_{\rm nf} v}{K\_p} \tag{29}$$

$$\begin{split} \frac{\partial T}{\partial t} + u \frac{\partial T}{\partial \mathbf{x}} + u \frac{\partial T}{\partial \mathbf{y}} &= \frac{k\_{\text{nf}}}{\left(\rho \mathbf{C}\_{\text{p}}\right)\_{\text{nf}}} \left( \frac{\partial^{2} T}{\partial \mathbf{x}^{2}} + \frac{\partial^{2} T}{\partial \mathbf{y}^{2}} \right) \\ &+ \frac{\mu\_{\text{nf}}}{\left(\rho \mathbf{C}\_{\text{p}}\right)\_{\text{nf}}} \left( 1 + \frac{1}{\beta} \right) \begin{pmatrix} 2 \left( \frac{\partial^{2} u}{\partial \mathbf{x}^{2}} \right)^{2} + \left( \frac{\partial^{2} u}{\partial \mathbf{y}^{2}} + \frac{\partial^{2} v}{\partial \mathbf{x}^{2}} \right)^{2} \\ + 2 \left( \frac{\partial^{2} v}{\partial \mathbf{y}^{2}} \right)^{2} \end{pmatrix} - \frac{1}{\left( \rho \mathbf{C}\_{\text{p}} \right)\_{\text{nf}}} \frac{\partial q\_{r}}{\partial \mathbf{x}} \end{split} \tag{30}$$

where

$$
\rho\_{\eta f} = \rho\_f(\mathbf{1} - \phi) + \rho\_s \phi \tag{31}
$$

$$
\mu\_{nf} = \frac{\mu\_f}{\left(1 - \phi\right)^{2.5}}\tag{32}
$$

#### **Figure 2.**

*Model diagram of MHD squeezing flow of nanofluid between two parallel plates embedded in a porous medium.*

and the magnetic field parameter is given as

$$B(t) = \frac{B\_0}{\sqrt{1 - at}}\tag{33}$$

$$\sigma\_{\eta f} = \sigma\_f \left[ 1 + \frac{3\left\{\frac{\sigma\_\iota}{\sigma\_f} - 1\right\} \phi}{\left\{\frac{\sigma\_\iota}{\sigma\_f} + 2\right\} \phi - \left\{\frac{\sigma\_\iota}{\sigma\_f} - 1\right\} \phi} \right],\tag{34}$$

$$k\_{\eta f} = k\_f \left[ \frac{k\_s + (m - 1)k\_f - (m - 1) \left(k\_f - k\_s\right) \phi}{k\_s + (m - 1)k\_f + \left(k\_f - k\_s\right) \phi} \right],\tag{35}$$

The Casson fluid parameter, *β* ¼ *μ<sup>B</sup>* ffiffiffiffiffiffiffiffiffiffiffiffi 2*π=Py* p and *k* is the permeability constant. The radiation term is given as

$$\frac{\partial q\_r}{\partial \mathbf{y}} = -\frac{4\sigma}{3K} \frac{\partial T^4}{\partial \mathbf{y}} \cong -\frac{16\sigma T\_s^3}{3K} \frac{\partial^2 T}{\partial \mathbf{y}^2} \text{ (using Rosseland's approximation)}\tag{36}$$

The appropriate boundary conditions are given as

$$u = 0, \quad v = v\_w = \frac{dh}{dt}, \quad T = T\_H \quad \text{at } \ y = h(t) = H\sqrt{1 - at}, \tag{37}$$

$$\frac{\partial u}{\partial \mathbf{y}} = \mathbf{0}, \quad \frac{\partial T}{\partial \mathbf{y}} = \mathbf{0}, \quad v = \mathbf{0}, \quad \text{at} \quad \mathbf{y} = \mathbf{0}, \tag{38}$$

On introducing the following dimensionless and similarity variables

$$u = \frac{aH}{2\sqrt{1-a}}f'(\eta, t), \quad v = -\frac{aH}{2\sqrt{1-a}}f(\eta, t), \quad \eta = \frac{y}{H\sqrt{1-a}}, \quad \theta = \frac{T-T\_0}{T\_H - T\_0}, \quad \mathcal{L} = \frac{1}{C\_p} \left(\frac{aH}{2(1-a)}\right)^2,\tag{12.1}$$

$$Re = -\text{SA}(1-\phi)^{2.5} = \frac{\rho\_{\text{f}}HV\_w}{\mu\_{\text{f}}}, \quad S = \frac{aH^2}{2\nu\_f}, \quad Da = \frac{K\_p}{H^2}, \quad A\_1 = (1-\phi) + \phi \frac{\rho\_s}{\rho\_f}, \quad \text{Pr} = \frac{\mu C\_p}{k}, \quad \delta = \frac{H}{x}, \quad \text{Pr} = \frac{\rho\_{\text{f}}HV\_w}{\mu\_{\text{f}}}$$

$$B\_1 = \begin{bmatrix} \left(\sigma\_i + (m-1)\sigma\_f\right) + (m-1)(\sigma\_i - \sigma\_f)\phi\\ \left(\sigma\_i + (m-1)\sigma\_f\right) - (m-1)(\sigma\_i - \sigma\_f)\phi \end{bmatrix}, \quad \Lambda\_2 = (1-\phi) + \phi \frac{\left(\rho C\_p\right)\_i}{\left(\rho C\_p\right)\_f}, \quad A\_3 = \frac{k\_{\text{sf}}}{k\_f}, \quad R = \frac{4\sigma T\_w^3}{3kK} \tag{39}$$

One arrives at the dimensionless equations

$$\begin{aligned} \left(\mathbf{1} + \frac{\mathbf{1}}{\beta}\right) f^{\circ\prime} - \text{SA}\_1 (\mathbf{1} - \boldsymbol{\phi})^{2.5} \left(\boldsymbol{\eta} f^{\prime} + \mathbf{3} f^{\prime} + \boldsymbol{f} f^{\prime} - \boldsymbol{f}^{\prime} f^{\prime}\right) - \text{Ha}^2 f^{\prime\prime} - \frac{1}{\text{Da}} f^{\prime\prime} &= \mathbf{0} \\\\ \left(\mathbf{1} + \frac{\mathbf{1}}{\mathbf{1} + \mathbf{B}}\right) \boldsymbol{\theta} f^{\prime} + \text{PsS} \left(\frac{\mathbf{A}\_2}{\beta}\right) \left(\boldsymbol{\theta}^{\prime} \boldsymbol{f} - \boldsymbol{\eta} \boldsymbol{\theta}^{\prime}\right) + \frac{\text{Pr} \text{E} \text{c}}{\left(\left(\boldsymbol{f}^{\prime\prime}\right)^2 + \mathbf{A} \boldsymbol{\xi}^{2} \left(\boldsymbol{f}^{\prime}\right)^2\right) - \text{O}} \end{aligned} \tag{40}$$

$$\left(\left(1+\frac{4}{3}R\right)\theta'' + \text{PrS}\left(\frac{A\_2}{A\_3}\right)\left(\theta'f - \eta\theta'\right) + \frac{\text{PrEc}}{A\_3(1-\phi)^{2.5}}\left(\left(f''\right)^2 + 4\delta^2\left(f'\right)^2\right) = 0\tag{41}$$

with the boundary conditions as follows

$$f = \mathbf{0}, \quad f'' = \mathbf{0}, \quad \theta' = \mathbf{0}, \quad \text{when} \quad \eta = \mathbf{0}, \tag{42}$$

*f* ¼ 1, *f* <sup>0</sup> ¼ 0, *θ* ¼ 1, *when η* ¼ 1, (43)

where *m* in the above Hamilton Crosser's model in Eq. (35).

*Perturbation Methods to Analysis of Thermal, Fluid Flow and Dynamics Behaviors of… DOI: http://dx.doi.org/10.5772/intechopen.96059*
