Heat and Mass Transfer of Hydrodynamic Boundary Layer Flow along a Flat Plate with the Influence of Variable Temperature and Viscous Dissipation

In recent years, heat together with mass transfer of boundary layer flow gained attention because of its numerous applications in nuclear engineering as well as industries. The results from these studies are of great importance in modeling and simulation. Both thermal and concentration boundary layers develops the moment fluid at specified temperature as well as concentration flows through a surface that is at a different temperature as well as concentration. These are also flow regions over the surface where temperature and concentration variations in the direction normal to the surface are significant. Considering boundary layer flow and effects of heat and mass transfer, velocities as well as velocity gradients are usually assumed to be sufficiently small due to the neglect of effects of the energy on motion as well as and viscous energy dissipation. Hence, this study considers along a flat plate, the effects of viscous dissipation together with convection of heat and mass transfer in boundary layer flow. Gebhart [1] explained that the heat as a result of viscous dissipation is very crucial when natural convection flow is having a good size, low temperature as well as high gravity. Gebhart and Mollendof [2] considered viscous dissipative heat transfer on a natural convection flow over a semi-infinite plate

Oosthuizen and David [3] examined along flat plate effects of viscous dissipation as well as convection of heat transfer, the equation of energy is written by adding the viscous dissipation term. Analysis of mixed convection flow of a viscous dissipating fluid along a vertical flat plate was studied by Aydin et al. [4]. Several researches work on the flow of fluid as well as transfer of heat has been carried out to study effect of viscous thermal dissipation. These studies revealed that fluid flows involving greater Eckert number generates heat because of viscous thermal dissipation and controls the temperature of as well as Eckert number which cannot be zero while investigating convection heat transfer. Brinkman [5] investigated viscous dissipation on heat transfer in capillary flow. Desale and Pradhan [6] explains solved the boundary layer equations by considering of viscous dissipation influence through a flat plate in the presence of variable temperature numerically. Oyelami and Dada [7] studied viscous dissipation influence on heat transfer by natural convection in a porous channel. Tyagi [8] studied the dissipative fluid in a channel with laminar force convection. Basu et al. [9] considered the laminar heat transfer by forgone the axial conduction effect in a tube with viscous dissipation. Oyelami and Dada [10] examined heat as well as mass transfer having effects of viscous dissipation and chemical reactions using Eyring-Powell fluid model.

Alao et al. [11] analyzed heat together with mass transfer of chemically reacting fluid using spectral relaxation approach. Increase in the parameter of chemical reaction is noticed to decrease the velocity as well as concentration of the fluid. Hayat et al. [12] elucidate double stratified Powel-Eying liquid chemically reactive flow. Their study concluded that flux as a result of Fourier’s expression is due to the absence of thermal relaxation factor. Mondal et al. [13] explored MHD mixed convection mass transfer with chemically reacting fluid. Their flow equations were solved numerically and they concluded that increasing the chemical reaction retards the fluid concentration. Falodun et al. [14] considered chemical reaction effects on non-Newtonian Casson fluid in their study using spectral relaxation approach.

The study of Idowu and Falodun [15] explored steady free convection flow with chemical reaction using spectral homotopy analysis approach. In the study the chemical reaction parameter is seen to decrease the concentration distribution. Falodun and Omowaye [16] studied double-diffusive MHD convective flow with chemical reaction using spectral homotopy analysis approach. Their study concluded that increase in chemical reaction parameter increases the local heat transfer rate. Idowu and Falodun [17] considered the rate of chemical reaction in their study. It was found out in the study that increases in the rate of reaction retards the fluid concentration. The recent work of Waqas [18] was on analysis of heat transfer of ferromagnetic non-Newtonian liquid. Both homogeneous and heterogeneous chemical reactions are considered in their study. Their study concluded that rate of transportation of heat is improved due to an increased Prandtl number. Mondal et al. [19] elucidate entropy generation of varied viscosity and thermal radiation with chemical reaction. Their flow equations were solved with spectral quasi-linearization method.

Heat together with mass transfer has become an interesting area of study due to its industrial applications in science as well as engineering such as the exchanger of heat, polymer production, processing of food, petroleum reservoirs etc. Heat is spread when a hot object is in contact with any object. It occurs when an object with $0^{\circ} \mathrm{C}$ falls on object with high temperature, the temperature will be spread on the object with $0^{\circ} \mathrm{C}$ and reduces the temperature of the object at high temperature. Hence, heat has been transferred. Hayat et al. [20] examined heat generation together with absorption simultaneously with MHD flow of Maxwell nanofluid. Homotopy technique was used to solve their flow equations and it was discovered that thermal radiation enhances both temperature field as well as rate of heat transfer. Kalpana et al. [21] elucidated impact of temperature dependent viscosity together with thermal conductivity on MHD boundary layer flow of two-phase dusty fluid. Their result shows that there is an enhancement in heat transfer rate as Schmidt, Prandtl and magnetic parameter increases. Raju et al. [22] elucidates heat together with mass transfer with MHD Casson fluid flow numerically and concluded that increase in heat source parameter enhances rate of heat transfer. Reddy [23] explored the problem of heat together with mass transfer by considering MHD, thermal radiation as well as chemical reaction. Their flow equations were solved numerically and result revealed that increase in thermal radiation elevates the temperature. Khan et al. [24] considered heat together with mass transfer of Williamson nanofluid flow. It was concluded in the study that, large values of Prandtl number reduces the heat transport rate. Quwasmeh et al. [25] examined forced convection heat transfer of Casson fluid using Runge-Kutta-Fehlberg scheme. Their findings revealed that increasing Prandtl number enhances rate of heat transfer.

All the published works discussed above does not consider heat together with mass transfer of hydrodynamic boundary layer flow along a flat plate. Motivated by this assertion and to extend the problem of Desale and Pradhan [6] to heat together with mass transfer. Hence, this study is necessary because of its various applications in engineering such as in higher-polymer additives in enhancing petroleum pipelines. To the best of our knowledge, no study in literature has considered a problem of this type. The major aim of this paper is to investigate the effects of all flow parameters on local skin friction coefficient, local Nusselt as well as local Sherwood number. We considered the velocity to be high and its effect on the heat transfer has several specialized applications like warmth exchange at high speeds around gas turbine edges or a rocket motors.

This section explains the method of approach called spectral homotopy analysis method (SHAM) used in this paper. SHAM is a method of approach that combines the procedure of Chebyshev spedospectral method with the homotopy analysis method (HAM). SHAM helps in decomposing the system of nonlinear Eqns. (9)-(11) into linear equations and Chebyshev spectral collocation method is hereby used to solve the resulting equations. The physical region is first transformed at $[0, \infty)$ to $[-1,1]$using domain truncation technique [26]. Hence, the problem solution is obtained in the interval $\left[0, \eta_{\infty}\right]$ and not $[0, \infty)$ after the following algebraic mapping is applied

$\zeta=\frac{2 \eta}{L}-1, \xi \in[-1,1]$    (13)

To write the boundary conditions in homogeneous form, the following transformation is applied on the transformed equations

$f(\eta )=f(\xi )+{{f}_{0}}(\eta ),\ \theta (\eta )=\theta (\xi )+{{\theta }_{0}}(\eta ),\ \varphi (\eta )=\varphi (\xi )+{{\varphi }_{0}}(\eta )$      (14)

Substituting the above Eq. (14) into (9)-(11) to obtain

$\frac{{{d}^{3}}f}{d{{\xi }^{3}}}+\frac{1}{2}f\frac{{{d}^{2}}f}{d{{\xi }^{2}}}+{{a}_{1}}f+{{a}_{2}}\frac{{{d}^{2}}f}{d{{\xi }^{2}}}={{H}_{1}}(\eta )$     (15)

$\begin{align}  & \frac{{{d}^{2}}\theta }{d{{\xi }^{2}}}+\frac{1}{2}f\frac{d\theta }{d\xi }+{{b}_{1}}f+{{b}_{2}}\frac{d\theta }{d\xi }+EcPr\frac{{{d}^{2}}f}{d{{\xi }^{2}}}\frac{{{d}^{2}}f}{d{{\xi }^{2}}} \\ & +{{b}_{3}}\frac{{{d}^{2}}f}{d{{\xi }^{2}}}+{{b}_{4}}\frac{df}{d\xi }+{{b}_{5}}\theta ={{H}_{2}}(\eta ) \\\end{align}$      (16)

$\begin{align}  & \frac{{{d}^{2}}\varphi }{d{{\xi }^{2}}}+\frac{1}{2}f\frac{d\varphi }{d\xi }+{{c}_{1}}f+{{c}_{2}}\frac{d\varphi }{d\xi }+Sc\lambda \varphi \frac{df}{d\xi } \\ & +{{c}_{3}}\frac{df}{d\xi }+{{c}_{4}}\varphi -Sc\lambda \varphi ={{H}_{3}}(\eta ) \\\end{align}$     (17)

where,

${{a}_{1}}=\frac{1}{2}\frac{{{d}^{2}}{{f}_{0}}}{d{{\eta }^{2}}},\ {{a}_{2}}=\frac{1}{2}{{f}_{0}}\ ,{{H}_{1}}(\eta )=-\frac{{{d}^{3}}{{f}_{0}}}{d{{\eta }^{3}}}-\frac{1}{2}{{f}_{0}}\frac{{{d}^{2}}{{f}_{0}}}{d{{\eta }^{2}}}$

${{b}_{1}}=\frac{d{{\theta }_{0}}}{d\eta },\ {{b}_{2}}=\frac{1}{2}{{f}_{0}},\ {{b}_{3}}=2EcPr\frac{{{d}^{2}}{{f}_{0}}}{d{{\eta }^{2}}},\ {{b}_{4}}=-nPr{{\theta }_{0}},\ {{b}_{5}}=-nPr\frac{d{{f}_{0}}}{d\eta }$

${{H}_{2}}=-\frac{{{d}^{2}}{{\theta }_{0}}}{d{{\eta }^{2}}}-\frac{1}{2}{{f}_{0}}\frac{d{{\theta }_{0}}}{d\eta }-EcPr\frac{{{d}^{2}}{{f}_{0}}}{d{{\eta }^{2}}}\frac{{{d}^{2}}{{f}_{0}}}{d{{\eta }^{2}}}+nPr{{\theta }_{0}}\frac{d{{f}_{0}}}{d\eta }$

${{c}_{1}}=\frac{1}{2}\frac{d\varphi }{d\eta },\ {{c}_{2}}=\frac{1}{2}{{f}_{0}},\ {{c}_{3}}=Sc\lambda {{\varphi }_{0}},\ {{c}_{4}}=Sc\lambda \frac{d{{f}_{0}}}{d\eta }$

${{H}_{3}}=-\frac{{{d}^{2}}{{\varphi }_{0}}}{d{{\eta }^{2}}}-\frac{1}{2}{{f}_{0}}{{\varphi }_{0}}-Sc\lambda \varphi \frac{d{{f}_{0}}}{d\eta }+Sc\lambda {{\varphi }_{0}}$

Eqns. (15)-(17) are nonlinear equations. In the application of SHAM, the nonlinear equations are decomposed into linear and nonlinear parts. It should be noted that the derivatives of $f, \theta$ and $\phi$ are with respect to $\xi$ defined by:

$\frac{d}{d\xi }=\frac{2}{L}\frac{d}{d\eta }$      (18)

An initial guess is taken with reference to the conditions in (12) as the following functions

${{f}_{0}}(\eta )={{e}^{-\eta }}-1,\ {{\theta }_{0}}(\eta )={{\varphi }_{(}}\eta )={{e}^{-\eta }}$      (19)

The non-homogeneous linear part of Eqns. (15)-(17) are decomposed from the nonlinear part as the following equations

$\frac{{{d}^{3}}f}{d{{\xi }^{3}}}+{{a}_{1}}f+{{a}_{2}}\frac{{{d}^{2}}f}{d{{\xi }^{2}}}={{H}_{1}}(\eta )$     (20)

$\begin{align}  & \frac{{{d}^{2}}\theta }{d{{\xi }^{2}}}+{{b}_{1}}f+{{b}_{2}}\frac{d\theta }{d\xi }+{{b}_{3}}\frac{{{d}^{2}}f}{d{{\xi }^{2}}} \\ & +{{b}_{4}}\frac{df}{d\xi }+{{b}_{5}}\theta ={{H}_{2}}(\eta ) \\\end{align}$      （21）

$\begin{align}  & \frac{{{d}^{2}}\varphi }{d{{\xi }^{2}}}+{{c}_{1}}f+{{c}_{2}}\frac{d\varphi }{d\xi }+{{c}_{3}}\frac{df}{d\xi } \\ & +{{c}_{4}}\varphi -Sc\lambda \varphi ={{H}_{3}}(\eta ) \\\end{align}$      (22)

Subject to:

${{f}_{l}}(-1)=f{{'}_{l}}(1)=f{{'}_{l}}(1)=0,{{\theta }_{l}}(-1)={{\theta }_{l}}(1)=0,{{\varphi }_{l}}(-1)={{\varphi }_{l}}(1)=0$      (23)

The boundary conditions (23) are chosen to be all equals to zero with reference to the transformed domain [-1, 1], to be able to implement the linear part SHAM solution. The Chebyshev pseudospectral method is applied on Eqns. (20)-(22). The Chebyshev nodes in the transformed domain [-1, 1] are defined by the Gauss-Lobatto collocation points [27] given by

${{\xi }_{J}}=cos\left( \frac{\pi j}{N} \right)$     (24)

provided $j=0,1,...N$ and $N+1$ is the number of collocation points. The unknown functions ${{f}_{l}}(\xi ),{{\theta }_{l}}(\xi )$ and ${{\varphi }_{l}}(\xi )$ are approximated by the use of Lagrange form of interpolating polynomial which interpolates the unknown functions ${{f}_{l}}(\xi ),{{\theta }_{l}}(\xi )$ and ${{\varphi }_{l}}(\xi )$ at the Gauss-Lobatto collocation points as defined in Eq. (24).

This paper elucidates the problem of heat together with mass transfer flow of boundary layer through a flat plate. The flow of fluid particles within the flat plate is assumed to be laminar. In this case, the fluid flows on its own streamline path with no mass transfer between the adjacent fluid layers. The shear is because of molecular action. The laminar flow conditions are assumed in this study because of low Reynolds number. The flow equations were successfully solved numerically using spectral homotopy analysis method. The results in this study using SHAM is in good agreement with methods in literature. Graphical results for velocity, temperature, concentration, local skin friction coefficient, local Nusselt as well as local Sherwood number by varying pertinent flow parameters like Prandtl number, Schmidt number, Eckert number as well as chemical reaction parameter are presented. The main findings of this paper are:

1. Increase in Pr and Ec brings increase to both local skin friction, local Nusselt as well as Sherwood number;
2. Increase in Pr brings irregular behaviour to the local skin friction coefficient, local Nusselt as well as Sherwood number;
3. The velocity as well as concentration profiles drastically decreases with increase in Sc. A reverse case is observed on the temperature distribution as increase in Sc enhances the thermal boundary layer thickness; and
4. As the chemical reaction parameter increases, an increase in velocity and concentration profiles is observed.

The SHAM is more efficient because of its flexibility in the choice of linear operators in comparison to various methods in literature reviewed in this study. Hence, the choice of SHAM is because of its elegance and accuracy. The problem considered in this paper will be helpful in oil-pipeline friction reduction, surfactant applications to large-scale heating together with cooling systems. It will also be useful in high-polymer additives to enhance flow in petroleum pipe-lines which is very useful for commercial purposes.