The Effect of Viscous Dissipation and Low Pressure Gradient on a Magnetohydrodynamics Flow over a Flat Plate

The concerned study about MHD boundary layer flow with viscous incompressible fluid flow has multitudinous applications in science; engineering era and industries such are textile industry, polymer technology, purification of crude oil, geothermal engineering, metal and polymer handling business, liquid metal and plasma flows and many more eras. Many Scientist and researchers have been received wide attention on this field. In view of these applications, Chen [1] has meticulously revealed the analytical solution of Magneto hydrodynamic flow of heat transfer with special types of viscoelastic fluids over a stretching sheet with energy dissipation, internal heat source and thermal emission. Moreover, huge attention has been drawn on fluid flow over a surface moving with either variable or constant velocity. Turkyilmazoglu [2] is the pioneer who reported the analytical solution of mixed convection with heat transmission of MHD viscoelastic fluid throughout a pervious stretching surface. Hayat et al. [3] have carried out their research on the three–dimensional fluid flow with generation/absorption of heat. The convective flow of micropolar fluid with isothermal porous surface moreover viscous dissipation and joule heating also well agreement by Rahman [4]. The MHD flow with unfreezing heat transmission of Cu-water nanofluid in connection with viscous dissipation and joule heating has observed by Hayat et al. [5]. Mahanthesh et al. [6] have introspected the effect of three-dimensional MHD flow of a nanofluid past a convectively heated stretching sheet with thermal emission and viscous dissipation. Khan et al. [7] focused the influence of the chemically reactive flow of micro polar fluid including viscous dissipation and Joule heating. Kumar [8] elucidated the influence of radiative heat conduction and viscous dissipation with hydro magnetic flow throughout a stretching surface in existence with heat flux. Sangapatnam et al. [9] have described the impact of emission and mass transfer on MHD flow past an impulsively isothermal perpendicular sheet with viscous dissipation. Kar et al. [10] have studied extensively the effect of chemical reaction and viscous dissipation on magneto hydrodynamic fluid flow including heat with mass transfer past a perpendicular porous plate.

The fluid flow throughout a horizontal plate plays a crucial role on both the theoretical and practical aspects because of inherent huge implementations in hydrometallurgical industries, chemical technology and plastic engineering. The wobbly MHD convective of nanofluids with boundary-layer flow over a extending sheet in addition with thermal emission and influence of viscous dissipation has analyzed by Khan et al. [11]. Motsumi and Makinde [12] have premeditated nanofluids flow throughout a pervious movable horizontal sheet due to showing of thermal radiation and viscous dissipation. This study has motivated by young researchers on fluid flow problems, which deals with a wide several boundary lawer problems with fluid flows over a stretching sheeting both Newtonian and non-Newtonian fluids. The study on MHD with the assorted convection heat transfer from rampant stretching sheet with due impact of viscous immoderation has been proposed by Partha et al. [13]. Vyas and Ranjan [14] have introspected on MHD flow with boundary conditions in a porous media over a stretching sheet in appearance of thermal emission with dissipation. Anjali and Ganga. [15] analysed the study on nonlinear MHD flow past a porous surface with irregularity of heat and dissipation effects. Ferdows et al. [16] have examined the boundary layer with MHD flow with heat transfer of a nanofluid throughout a permeable precarious stretching sheet with viscous dissipation. Nia et al [17] reported the effect of the steadiness of nonlinear differential equations by the strengthen of Homotopy Perturbation Technique. Venkateswarlu et al. [18] investigated the viscous dissipation effects on MHD flow over a movable surface along with constant heat source. Vajravelu et al [19] have investigated the volatile convective boundary layer viscous fluid flow with perpendicular surface and irregular fluid properties. Hunegnaw and Kishan [20] have premeditated the porous MHD heat and mass transfer flow throughout stretching sheet in porous media assuming with viscous dissipation and chemical reaction.

Although significant studies have been made on magnetic, radiative and electrically conducting flows from horizontal or vertical stretching sheets by researchers and sciencists. But the heat and mass diffusion along a perpendicular plate with viscosity of the natural convection flows has investigated by Dimian and Hadhoda [21].The natural convection of MHD of nanofluids with the impact of size, shape of nanoparticles and temperature of the base fluid studied by Reddy and Chamkha [22].The effects of chemical reaction and thermal emission in existence of viscosity of nanofluid flow throughout a permeable wedge ingrained in impregnated porous media inspected by James et al. [23]. The chemical reaction effects with heat and mass transfer through a porous medium of magnetohydrodynamics flow of nanofluids in existence with thermal emission, viscous dissipation experimented by Haile and Shankar [24]. An exponentially accelerated slanted plate immersed in a porous medium with thermal conductivity in existence of emission has inspected by Ahmmed et al. [25]. Hussain [26] is the pioneer who reported the solution for MHD nanofluid flow with heat and mass transfer across a porous surface with thermal emission, viscous dissipation as well as chemical reaction effect. Makinde and Mutuku [27] have evolved the MHD nanofluids with hydromagnetic thermal boundary layer flow throughout a heated horizontal plate with viscous dissipation and ohmic heating. The Lorentz force on convection nanofluid flow throughout a stretched sheet has meticulously scrutinized by Sheikholeslami et al. [28]. Kar et al. [29] have experimented the influence of the mass transfer effects on volatile convective MHD incompressible fluid flow along an exponentially accelerated perpendicular porous plate in the presence of heat source. Analytical solution of MHD flow along an imprudently commenced the perpendicular plate with the mass transfer effect also experimented by Swain and Senapati [30].

The preeminent purpose of the present paper is to quite develop the work of Jhankal [31] by considering heat transfer in presence of viscous dissipation. The irreversible process in which work done by the adjacent layer of fluid because of shearing forces is altered into heat is defined as viscous dissipation. It has many applications in real life world problem such are the aerodynamic heating in the thin boundary layer whereas the raises in the temperature of the skin of high speed aircraft. The reduced governing boundary layer equations are solved by Homotopy perturbation method and Runge-Kutta numerical integration with shooting technique. The comparison between the two methods proves the accuracy of solution.

3.1 Homotopy perturbation method

Basic concept of homotopy perturbation method (HPM), we consider the following nonlinear differential equation:

$A(u)-f(r)=0, r \in \Omega$     (9)

Subject to the boundary conditions

$B\left(u, \frac{\partial u}{\partial \eta}\right)=0, r \in \Gamma$    (10)

where, $\mathrm{A}$ is a general differential operator, $\mathrm{B}$ is a boundary operator, $f(r)$ is a known an analytical function and $\Gamma$ is the boundary of the domain $\Omega$.

Operator A can be divided into two parts, which are L and N, where L is linear and N is nonlinear, therefore Eq. (9) can be

$L(u)+N(u)-f(r)=0 . r \in \Omega$     (11)

By the homotopy perturbation technique,

We construct a homotopy $U(r, p): \Omega \times[0,1] \rightarrow R$, which satisfies:

$\mathrm{H}(\mathrm{U}, \mathrm{p})=(1-p)\left[L(U)-L\left(u_{0}\right)\right]+p[A(U)-f(r)]$,$p \in[0,1] r \in \Omega$     (12)

where, $p \in[0,1]$ is an embedding parameter, $u_{0}$ is an initial approximation, which satisfies the boundary condition. Obviously, from these definitions we have

$\mathrm{H}(\mathrm{U}, 0)=\mathrm{L}(\mathrm{U})-L\left(u_{0}\right)=0$, $\mathrm{H}(\mathrm{U}, 1)=\mathrm{A}(\mathrm{U})-f(r)=0$

The changing process of $p$ from zero to unity is just that of $U(r, p)$ from $u_{0}(r)$ to $u(r)$. In topology, $\mathrm{L}(\mathrm{U})-L\left(u_{0}\right)$ and $\mathrm{A}(\mathrm{U})-f(r)$ are called homotopy. According to the (HPM), we can first use the embedding parameter $p$ as a small parameter and assume that the solution of Eq. (12) can be written as a power series in $p$ :

$U=U_{0}+p U_{1}+p^{2} U_{2}+p^{3} U_{3}+\cdots$     (13)

Setting $p=1$, results in the approximate solution of Eq. (9)

$u=\lim _{p \rightarrow 1} U=U_{0}+U_{1}+U_{2}+U_{3}+\cdots$     (14)

Using Homotopy perturbation method, the homotopy forms of Eqns. (6) and (7) are given below:

$(1-p)\left(f^{\prime \prime \prime}-M^{2} f^{\prime}\right)+p\left(f^{\prime \prime \prime}+\frac{1}{2} f f^{\prime \prime}-M^{2} f^{\prime}\right)$$=-\lambda$     (15)

$(1-p)\left(\theta^{\prime \prime}\right)+p\left(\theta^{\prime \prime}+\frac{P r}{2} \theta^{\prime} f+\operatorname{PrEc} f^{\prime \prime^{2}}\right)=0$     (16)

Here we assume $f$ and $\theta$ as the following forms: -

$f=f_{0}+p f_{1}+p^{2} f_{2}+p^{3} f_{3}+p^{4} f_{4}$. $\cdots \ldots \ldots \ldots . .$

$\theta=\theta_{0}+p \theta_{1}+p^{2} \theta_{2}+p^{3} \theta_{3}+p^{4} \theta_{4} \ldots \ldots \ldots . .$     (17)

Substituting Eq. (17) into Eq. (15) and Eq. (16), we obtain independent term of $p$ as follow

$f_{0}^{\prime \prime \prime}-M^{2} f_{0}^{\prime}=-\lambda$     (18)

$\theta_{0}^{\prime \prime}=0$     (19)

The boundary conditions are

$f_{0}(0)=0, f_{0}^{\prime}(0)=0, f_{0}^{\prime}(\infty)=1, \theta_{0}(0)=1$, $\theta_{0}(\infty)=0 .$     (20)

Terms containing only $p$ are given below:

$f_{1}^{\prime \prime \prime}-M^{2} f_{1}^{\prime}+\frac{1}{2} f_{0}^{\prime \prime} f_{0}=0$     (21)

$\theta_{1}^{\prime \prime}+\frac{\operatorname{Pr}}{2} \theta_{0}^{\prime} f_{0}+\operatorname{PrEc} f_{0}^{\prime \prime^{2}}=0$     (22)

The boundary conditions are given below:

$f_{1}(0)=0, f_{1}^{\prime}(0)=0, f_{1}^{\prime}(\infty)=0, \theta_{1}(0)=0$, $\theta_{1}(\infty)=0$     (23)

Now, solving Eq. (18)-(19) and Eq. (21)-(22) with boundary conditions (20) and (23) respectively, we obtain

$f_{0}=C_{1}+C_{2} e^{M \eta}+C_{3} e^{-M \eta}+\frac{\lambda \eta}{M^{2}}$     (24)

$\begin{aligned} f_{1}=C_{4}+C_{5} e^{M \eta} &+C_{6} e^{-M \eta}+J_{3} \eta e^{M \eta}+J_{4} \eta e^{-M \eta} \\ &+J_{5} \eta^{2} e^{M \eta}+J_{6} \eta^{2} e^{-M \eta}+J_{7} e^{2 M \eta} \\ &+J_{8} e^{-2 M \eta}+J_{9} \end{aligned}$     (25)

$\theta_{0}=-\frac{\eta}{6}+1$     (26)

$\theta_{1}=C_{9}+C_{10} \eta+J_{11} e^{M \eta}+J_{12} e^{-M \eta}+J_{13} e^{2 M \eta}$$+J_{14} e^{-2 M \eta}+J_{15} \eta^{2}+J_{16} \eta^{3}$     (27)

The above assigned constant coefficients can be estimated using boundary conditions. η_∞ is assumed as 6.

3.2 Numerical integration by Runge-Kutta technique

With due implementation of numerical integration, a set of first order differential equations is developed from Eqns. (6) and (7). We generate the following substitutions

$f=y_{1}, f^{\prime}=y_{2}, f^{\prime \prime}=y_{3}, \theta=y_{4}, \theta^{\prime}=y_{5}$     (28)

The reduced equations are

$\begin{aligned} y_{3}^{\prime} &=-\frac{1}{2} y_{1} y_{3}+M^{2} y_{2}-\lambda \\ y_{5}^{\prime} &=-\frac{P_{r}}{2} y_{1} y_{5}+P_{r} E_{c} y_{3}{ }^{2} \end{aligned}$     (29)

Corresponding boundary conditions are given by

$y_{1}=0, y_{2}=0, y_{3}=?, y_{4}=1, y_{5}=?$, at $\eta=0$     (30)

$y_{2} \rightarrow 1, y_{4} \rightarrow 0$ at $\eta \rightarrow \infty$.     (31)

To initiate the integration, the result of y₃ and y₅ at η=0 are provided as guess values whereas the stepwise integration is employed with step interval 0.001 adopting shooting mechanism with code of MATLAB along with error bound 10^-3_.

• Velocity of the fluid flow increases with increasing values of pressure gradient. i.e., the pressure gradient forces enhances the velocity resulting a linear like curve. Further increasing the values of λ, we got concave curves, which is obvious.
• Magnetic parameter creates a resistive force which diminishes the velocity. Therefore magnetic force can be applied in different physiological fluid flow of in industries. Whenever a reduction of velocity is needed.
• Higher viscous dissipative heat raises the temperature significantly. So, a heating system can be formed considering viscous dissipation.
• Pressure gradient parameter is inversely proportional to the temperature.
• Prandtl number P_r diminishes the temperature in the presence of higher pressure gradient whereas P_r makes the flow domain more hot with help of larger Eckert number.
• With improve in M; there is a rise in temperature. Because higher magnetic parameter involves stronger Lorentz force which leads to a higher temperature.