Non-linear thermal convection in a Casson fluid flow over a horizontal plate with convective boundary condition

==Abstract==

Casson fluid flow has many practical applications such as food processing, metallurgy, drilling operations and bio-engineering operations. In this paper, we study Casson fluid flow through a plate with a convective boundary condition at the surface and quantify the effects of suction/injection, velocity ratio, and Soret and Dufour effects. Firstly we used a similarity transformation to change the governing equations to ordinary differential equations which were then solved numerically. The effect of the rheological parameters on the velocity, temperature, and concentration with skin friction, and heat and mass transfer are shown graphically and discussed briefly. It is observed that the velocity of the fluid at the surface decreases with increase of the velocity ratio while the nature of the flow is in opposite characteristics. The local Nusselt number decreases with increase in the velocity ratio. Skin friction at the surface is enhanced by buoyancy ratio and Casson number. Due to injection of the fluid in the system, the mass transfer rate at the surface increases while it decreases with the velocity ratio parameter.

==Keywords==

Casson fluid; Non-linear thermal convection; Soret and Dufour effect; Viscous dissipation; Suction

==1. Introduction==

Heat and mass transfer characteristics and convection in a non-Newtonian fluid have practical applications in many engineering and medical sciences [[#b0005|[1]]]. Casson fluid is classified as a most popular non-Newtonian fluid due to its rheological characteristics [[#b0010|[2]]], [[#b0015|[3]]], [[#b0020|[4]]], [[#b0025|[5]]] and [[#b0030|[6]]]. Kameswaran et al. [[#b0035|[7]]] have discussed the Casson fluid flow over a stretching or shrinking sheet. Mahanta and Shaw [[#b0040|[8]]] have studied a three dimensional Casson fluid flow past a porous linearly stretching sheet in the presence of convective boundary condition. The influence of the viscous dissipation on the MHD Casson fluid flow over a non-Darcy porous medium has been discussed by Makanda et al. [[#b0045|[9]]]. Casson fluid flow and heat transfer past an exponentially porous stretching surface have been discussed by Pramanik [[#b0050|[10]]]. Convective boundary condition effect on the stagnation point flow of nanofluid has been discussed by Akbar et al. [[#b0055|[11]]].

The processes of suction and injection find application in many industries such as in the design of thrust bearing and radial diffusers, and thermal oil recovery. Hartnett [[#b0060|[12]]] analyzed the significance of suction or injection on boundary layer flow and noted that suction or injection through the surface can significantly modify the flow field. The effect of the suction or injection on the boundary layer flow and heat transfer has been studied by Pantokratoras [[#b0065|[13]]]. The influence of the thermophoretic and non-linear convection has been studied by Kameswaran et al. [[#b0070|[14]]]. The non-linear convection in nano-fluid flow on stretching sheet has been studied by Shaw et al. [[#b0075|[15]]].

When heat and mass transfer occurs simultaneously in a moving fluid, the relation between the fluxes and the driving potential becomes complex [[#b0080|[16]]], [[#b0085|[17]]] and [[#b0090|[18]]]. The Soret effect (or thermal diffusion) is mainly the occurrence of a diffusion flux due to a temperature gradient, whereas the Dufour effect is due to a heat flux as a result of chemical potential gradients. Such effects are significant in the fields of geosciences and chemical engineering. Hayat et al. [[#b0095|[19]]] studied the magnetohydrodynamics flow of a Casson fluid over a stretching surface with Soret and Dufour effects. Convective boundary conditions are applicable in industries with high temperatures such as in gas turbines, nuclear plants, and thermal energy storage. Non-Newtonian fluids are used in place of Newtonian fluids in these industries [[#b0100|[20]]], [[#b0105|[21]]], [[#b0110|[22]]], [[#b0115|[23]]], [[#b0120|[24]]], [[#b0125|[25]]], [[#b0130|[26]]] and [[#b0135|[27]]].

Heat transfer at the surface of the microvessels is followed by convective heat because of the connective tissue which covers the outer wall of the microvessel. Diffusion and heat flux play a vital role during atherosclerosis, hyperthermia and other diseases. The aim of the present problem is to study the influence of suction or injection due to the permeable surface, Soret and Dufour effects due to mass and heat flux on Casson fluid flow through a plate with a convective boundary condition at the surface. We transformed the governing equations to a system of ordinary differential equation using similarity variables. The transformed equations were solved numerically for different values of the velocity ratio, suction/injection parameter, and the Soret and Dufour parameters. This paper is very useful in medical science to know the flow nature of the blood in the vessel.

==2. Mathematical formulation==

We consider steady, incompressible Casson fluid flow over a heated flat plate. It is assumed that the free stream moves with a constant velocity <math display="inline">U_{\infty }</math> over the top of the surface and the plate moving opposite or same direction of the free stream with velocity <math display="inline">U_w</math>.

By assuming homogeneity and thermal equilibrium, the governing equations are written as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=</math><math>0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

<span id='e0010'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=</math><math>\nu \left(1+\frac{1}{\beta }\right)\frac{{\partial }^2u}{\partial y^2}+</math><math>g{\beta }_c(C-C_{\infty })+g[{\beta }_0(T-T_{\infty })+</math><math>{\beta }_1{\left(T-T_{\infty }\right)}^2]\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}

<span id='e0015'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=</math><math>{\alpha }_m\frac{{\partial }^2T}{\partial y^2}+\frac{D_mK_T}{C_sC_p}\frac{{\partial }^2C}{\partial y^2}+</math><math>\frac{\mu }{\rho C_p}\left(1+\frac{1}{\beta }\right){\left(\frac{\partial u}{\partial y}\right)}^2\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

<span id='e0020'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=</math><math>D_m\frac{{\partial }^2C}{\partial y^2}+\frac{D_mK_T}{T_m}\frac{{\partial }^2T}{\partial y^2}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

The corresponding boundary conditions at the surface and far away from the surface are written as follows:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{lc}
u=U_w\mbox{,}v=V_w(x)\mbox{,}-k\frac{\partial T}{\partial y}=h_f(T_f-T)\mbox{,}C=C_w & \mbox{at}\quad y=0\mbox{,}\\
u\rightarrow U_{\infty }\mbox{,}T\rightarrow T_{\infty }\mbox{,}C\rightarrow C_{\infty } & \mbox{as}\quad y\rightarrow \infty \mbox{,}
\end{array}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}

where the velocities of the fluid <math display="inline">u</math> and <math display="inline">v</math> are along the <math display="inline">x</math> and <math display="inline">y</math>-axis, respectively, <math display="inline">\nu </math> is the apparent viscosity of the Casson fluid, <math display="inline">g</math> is the acceleration due to gravity, <math display="inline">{\beta }_0</math> and <math display="inline">{\beta }_1</math> are the linear and non-linear volumetric thermal expansion coefficient, <math display="inline">{\beta }_c</math> is the volumetric solute expansion coefficient, <math display="inline">T</math> is the temperature of the fluid, <math display="inline">T_{\infty }</math> is uniform ambient temperature, <math display="inline">C</math> is the concentration of the fluid, <math display="inline">{\alpha }_m</math> is the thermal diffusivity, <math display="inline">D_m</math> is the effective solutal diffusivity of the medium, <math display="inline">K_T</math> is the thermal diffusion ratio, <math display="inline">C_s</math> is the concentration susceptibility, and <math display="inline">C_p</math> is the specific heat capacity. Convective boundary condition is introduced at the surface of the plate. It is assumed that the bottom surface of the plate is heated by convection from a hot fluid of temperature <math display="inline">T_f</math> which is related with the heat transfer coefficient <math display="inline">h_f</math> where <math display="inline">k</math> and <math display="inline">T_w</math> are the thermal conductivity and the uniform temperature at the wall, respectively. It is noted that <math display="inline">T_f>T_w>T_{\infty }</math>. Following the work of Ishak [[#b0140|[28]]], we define the heat transfer coefficient as <math display="inline">h_f={cx}^{-1/2}</math> with constant ''c  '' and corresponding Biot number written as <math display="inline">\mbox{Bi}=\frac{c}{k}\sqrt{U/{\alpha }_m}</math>. The variable plate surface permeability function is given as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>V_w(x)=-\frac{f_w}{2}\sqrt{\frac{U{\nu }_f}{x}}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

where <math display="inline">U=U_w+U_{\infty }</math>. <math display="inline">f_w</math> is a constant with <math display="inline">f_w>0</math> representing the transpiration (suction) rate at the plate surface, <math display="inline">f_w<0</math> corresponds to injection and <math display="inline">f_w=0</math> for an impermeable surface. We introduce the following similarity transformation to convert the partial governing differential equation to system of ordinary differential equation, written as  [[#b0145|[29]]] and [[#b0150|[30]]]

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\eta =y\sqrt{\frac{U}{\nu x}}\mbox{,}\psi =\sqrt{\nu xU}f(\eta )\mbox{,}\theta (\eta )=</math><math>\frac{T-T_{\infty }}{T_w-T_{\infty }}\mbox{,}\phi (\eta )=</math><math>\frac{C-C_{\infty }}{C_w-C_{\infty }}\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}

where <math display="inline">\psi </math> is the stream function which is defined as usual way <math display="inline">u=\partial \psi /\partial y</math> and <math display="inline">v=-\partial \psi /\partial x</math>, which by default satisfied the continuity equation. So the remaining governing  [[#e0010|(2)]], [[#e0015|(3)]] and [[#e0020|(4)]] are written as

<span id='e0040'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\left(1+\frac{1}{\beta }\right)f^{{'''}}+\frac{1}{2}{ff}^{{''}}+</math><math>\lambda (\theta +\alpha {\theta }^2+N\phi )=0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}

<span id='e0045'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{\theta }^{{''}}+D_f\mbox{Pr}{\phi }^{{''}}+\frac{1}{2}Prf{\theta }^{{'}}+</math><math>\mbox{Br}\left(1+\frac{1}{\beta }\right)f^{{''}2}=</math><math>0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}

<span id='e0050'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{\phi }^{{''}}+\mbox{SrSc}{\theta }^{{''}}+\frac{1}{2}\mbox{Sc}\quad f\quad {\phi }^{{'}}=</math><math>0\mbox{,}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

and the corresponding boundary conditions are written as

<span id='e0055'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{ll}
f=f_w\mbox{,}f^{{'}}=1-r\mbox{,}{\theta }^{{'}}=-\mbox{Bi}(1-\theta (0))\mbox{,}\phi =1\mbox{,} & \mbox{at}\quad y=0\mbox{,}\\
f^{{'}}\rightarrow r\mbox{,}\theta \rightarrow 0\mbox{,}\phi \rightarrow 0\mbox{,} & \mbox{as}\quad y\rightarrow \infty \mbox{.}
\end{array}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}

The parameters used in the above equations are written as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}
\lambda =\frac{{\mbox{Gr}}_x}{{\mbox{Re}}_x^2}\mbox{,}\quad {\mbox{Gr}}_x=\frac{g{\beta }_0(T_w-T_{\infty })x^3}{{\nu }^2}\mbox{,}\quad {\mbox{Re}}_x=\frac{Ux}{\nu }\mbox{,}\quad \alpha =\frac{{\beta }_1(T_w-T_{\infty })}{{\beta }_0}\mbox{,}\quad N=\frac{{\beta }_c(C_w-C_{\infty })}{{\beta }_0(T_w-T_{\infty })}\mbox{,}\quad \mbox{Bi}=\frac{c}{k}\sqrt{\frac{\nu }{U}}\mbox{,}\\
Pr=\frac{\nu }{\epsilon {\alpha }_m}\mbox{,}\quad D_f=\frac{\epsilon D_mK_T(C_w-C_{\infty })}{\nu C_sC_p(T_w-T_{\infty })}\mbox{,}\quad \mbox{Sc}=\frac{\nu }{\epsilon D_m}\mbox{,}\quad \mbox{Br}=\frac{\epsilon U^2}{C_p(T_w-T_{\infty })}\mbox{,}\quad \mbox{Sr}=\frac{\epsilon D_mK_T(T_w-T_{\infty })}{\nu T_m(C_w-C_{\infty })}\mbox{,}r=\frac{U_{\infty }}{U_w}\mbox{,}
\end{array}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}

where <math display="inline">\lambda </math> is the mixed convection parameter, <math display="inline">{\mbox{Gr}}_x</math> is the Grashof number, <math display="inline">{\mbox{Re}}_x</math> is the Reynolds number, <math display="inline">\alpha </math> is the nonlinear convection parameter, <math display="inline">N</math> is the buoyancy force, Bi is the Biot number, Pr is the Prandtl number, <math display="inline">D_f</math> is the Dufour effect, Br is the Brinkman number, Sc is the Schmidt number, Sr is the Soret number, and ''r'' is the velocity ratio.

It is interesting to note that the assisting flow and opposing flow depend on sign of the mixed convection parameter ''λ''. For assisting flow (when ''λ'' > 0 i.e., ''T<sub>w</sub>'' > ''T''<sub>∞</sub>), the buoyancy force acts directly in the direction of the main stream flow which accelerates the fluid thus acting as a favorable pressure gradient. For the opposing flow (when ''λ'' < 0 i.e., ''T<sub>w</sub>'' < ''T''<sub>∞</sub>), the buoyancy force opposes the fluid flow, impeding the fluid flow and thus behaving as an adverse pressure gradient. This directly affects the Brinkman number as it depends on the temperature difference. The Brinkman number Br > 0 for the case of assisting flow while Br < 0 for the case of opposing flow.

===2.1. Skin friction, heat and mass transfer coefficients===

The quantities of practical interest in this study are the skin friction coefficient, heat transfer rate and mass transfer rate, which is respectively defined as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>C_f=\frac{{\tau }_w}{\rho U_{\infty }^2}\mbox{,}\quad {\mbox{Nu}}_x=</math><math>\frac{{xq}_w}{k(T_f-T_{\infty })}\mbox{,}\quad {\mbox{Sh}}_x=</math><math>\frac{x\quad q_m}{D_s(C_w-C_{\infty })}\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

The shear stress at the surface of the wall <math display="inline">{\tau }_w</math> is defined as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{\tau }_w=\left({\mu }_B+\frac{{\tau }_y}{\sqrt{2{\pi }_c}}\right){\frac{\partial u}{\partial y}}_{y=0}=</math><math>\left({\mu }_B+\frac{{\tau }_y}{\sqrt{2{\pi }_c}}\right){\left(\frac{U^3}{\nu x}\right)}^{1/2}f^{{''}}(0)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

The surface flux at the wall is defined as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>q_w=-k{\frac{\partial T}{\partial y}}_{y=0}=-k(T_w-</math><math>T_{\infty }){\left(\frac{U}{\nu x}\right)}^{1/2}{\theta }^{{'}}(0)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|}

Using the above equation, the local Nusselt number <math display="inline">{\mbox{Nu}}_x</math> is written as

<span id='e0080'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{\mbox{Re}}_x^{1/2}{\mbox{Nu}}_x=-{\theta }^{{'}}(0)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
|}

The mass flux at the surface is written as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>q_m=-D_s{\frac{\partial C}{\partial y}}_{y=0}=-D_s(C_w-</math><math>C_{\infty }){\left(\frac{U}{\nu x}\right)}^{1/2}{\phi }^{{'}}(0)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
|}

Using the above equation, the local Sherwood number <math display="inline">{\mbox{Sh}}_x</math> is written as

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>{\mbox{Re}}_x^{-1/2}{\mbox{Sh}}_x=-{\phi }^{{'}}(0)\mbox{.}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
|}

==3. Results and discussions==

In the present study, we solved the system of differential [[#e0040|(8)]], [[#e0045|(9)]] and [[#e0050|(10)]] with boundary condition Eq. [[#e0055|(11)]] to quantify the effects of various physical parameters. We used similarity variables to transform the governing partial differential equations into ordinary differential equations and then solved the later using the ‘bvp4c’ Matlab package. The effect of the different parameters e.g., suction/injection parameter, Casson parameter, Biot number, Dufour and Soret effect, velocity ratio parameter, Buoyancy ratio, Schmidt Number on the velocity, temperature and concentration on the fluid skin friction, local Nusselt number and local Sherwood number is shown by graphs. In general, the values of the parameters are as follows: <math display="inline">\lambda =0.1\mbox{,}\quad \beta =1\mbox{,}\quad \alpha =</math><math>0.1\mbox{,}\quad N=0.1\mbox{,}\quad D_f=0.1\mbox{,}\quad \mbox{Br}=</math><math>0.1\mbox{,}\quad \mbox{Sr}=0.1\mbox{,}\quad \mbox{Sc}=</math><math>5\mbox{,}\quad f_w=0.1\mbox{, Bi}=10\mbox{,}\quad r=</math><math>0.5</math>.

The influence of the different velocity ratio parameter on the velocity profile is depicted in [[#f0005|Fig. 1]]. We observed that the velocity ratio plays a vital role in the behavior of the fluid. The structure of the velocity profiles changes dramatically with the value of velocity ratio. From the boundary condition, it is clear that when ''r'' < 0, the velocity at the wall is greater than the velocity away from the wall. The velocity at the wall increases with a decrease in the value of the velocity ratio parameter ''r''. Similarly, for the case ''r'' > 0.5, the velocity at the wall is lower than the velocity near the wall and concurrently with an increase in ''r'', the velocity at the wall decreases. The velocity at the wall is unaltered when ''r'' = 0.5 and it is parallel to the ''η''-axis as the velocity. To satisfy the boundary condition, the momentum boundary layer converges at ''r'' and the boundary layer thickness increases with the velocity ratio parameter and it is observed that the fluid velocity for both assisting and opposing flow is shown in  [[#f0010|Figure 2]] and [[#f0015|Figure 3]] for different ''r''. For ''r'' = 0.5, the momentum boundary layer thickness increases for the case of assisting flow while it decreases for the case of opposing flow. We noted that the velocity is maximum near the wall, although the velocity at the wall is unaltered in line with the boundary condition. It is observed that the boundary layer thickness decreases with decrease in mixed convection parameter ''λ''. For the assisting flow, the favorable pressure gradient pushes the fluid away from the surface which increases the boundary layer thickness. However for opposing flow, the adverse pressure gradients pull the fluid toward the surface which decreases the boundary layer thickness for ''r'' > 0.5. An opposite phenomenon is observed for the case of ''r'' < 0.5.

<span id='f0005'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr1.jpg|center|366px|Velocity profile for different velocity ratio.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 1.

Velocity profile for different velocity ratio.

</span>
|}

<span id='f0010'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr2.jpg|center|365px|Velocity profile for different mixed convection parameter.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 2.

Velocity profile for different mixed convection parameter.

</span>
|}

<span id='f0015'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr3.jpg|center|366px|Velocity profile for different mixed convection parameter for r=0.5.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 3.

Velocity profile for different mixed convection parameter for ''r'' = 0.5.

</span>
|}

The convective boundary condition is related with the thermal condition on the surface. It is transformed to normal boundary condition when Bi → ''∞'' and so it has no significant effects on the governing system for higher value of Biot number. The Biot number influences the velocity due to convection which is clearly shown in  [[#f0020|Figure 4]] and [[#f0025|Figure 5]]. The momentum boundary layer thickness decreases with increase in the Biot number when ''r'' = 0.5. A similar phenomena was observed for ''r'' = 0.8 while the opposite is true when ''r'' = 0.2. Basically, with the increase of Biot number, the thermal boundary layer thickness decreases and as a result, the heat transfer rate increases. For ''r'' = 0.5 and 0.8, the fluid velocity at the wall is equal or less than the velocity away from the surface (i.e., at infinity), which helps to enhance the boundary layer thickness. For ''r'' < 0.5, it observed that the velocity at the wall is higher than the velocity away from the wall. Opposite is true for the case of ''r'' > 0.5. When ''r'' = 0.5, the velocity of the fluid at the surface and velocity of the fluid far away from the surface (i.e., at infinity) is same and so the velocity is unaltered for this case. The Casson fluid is subject to a yield stress at the surface which enhanced the shear stress at the wall and as a result the velocity is smaller for the case of Casson fluid than for Newtonian fluid. The Casson parameter is inversely proportional to the yield stress. So with an increase in the Casson parameter, the momentum boundary layer thickness decreases (see [[#f0030|Fig. 6]]).

<span id='f0020'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr4.jpg|center|366px|Velocity profile for different Biot number for r=0.5.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 4.

Velocity profile for different Biot number for ''r'' = 0.5.

</span>
|}

<span id='f0025'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr5.jpg|center|365px|Velocity profile for different Biot number.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 5.

Velocity profile for different Biot number.

</span>
|}

<span id='f0030'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr6.jpg|center|366px|Velocity profile for different Casson parameter.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 6.

Velocity profile for different Casson parameter.

</span>
|}

Suction/injection, Soret and Dufour number, Buoyancy ratio and Schmidt number play a vital role in the behavior of the velocity profile as shown in [[#f0035|Figure 7]], [[#f0040|Figure 8]], [[#f0045|Figure 9]], [[#f0050|Figure 10]], [[#f0055|Figure 11]] and [[#f0060|Figure 12]]. A similar result was observed by Mustafa et al. [[#b0155|[31]]]. The fluid velocity decreases during suction and increases during injection (see [[#f0035|Fig. 7]]). This can be attributed to an increase in the momentum boundary layer thickness during suction and decrease during injection as shown in earlier work by Motsumi and Makinde [[#b0160|[32]]]. The Soret number enhances mass diffusion near the surface and as a result the boundary layer thickness reduces (see [[#f0040|Fig. 8]]). The nonlinear convection parameter increases the momentum boundary layer thickness (see [[#f0045|Fig. 9]]). This is because the nonlinear part of the thermal convection increases the temperature gradient at the wall which forces the fluid to move away from the surface. The momentum boundary layer thickness increases for the case of increasing buoyancy and it appears much larger for higher values of buoyancy parameter (see [[#f0050|Fig. 10]]). It is observed that the momentum boundary layer thickness decreases by introducing Dufour effect and it decreases with increase of Dufour number (see [[#f0055|Fig. 11]]). In the case of opposing buoyancy, the boundary layer thickness decreases. The momentum boundary layer thickness decreases with an increase in the Schmidt number (see [[#f0060|Fig. 12]]).

<span id='f0035'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr7.jpg|center|365px|Velocity profile for different suction/injection parameter.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 7.

Velocity profile for different suction/injection parameter.

</span>
|}

<span id='f0040'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr8.jpg|center|366px|Velocity profile for different buoyancy parameter.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 8.

Velocity profile for different buoyancy parameter.

</span>
|}

<span id='f0045'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr9.jpg|center|367px|Velocity profile for different the Dufour number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 9.

Velocity profile for different the Dufour number.

</span>
|}

<span id='f0050'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr10.jpg|center|365px|Velocity profile for different Soret number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 10.

Velocity profile for different Soret number.

</span>
|}

<span id='f0055'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr11.jpg|center|367px|Velocity profile for different nonlinear convection parameter.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 11.

Velocity profile for different nonlinear convection parameter.

</span>
|}

<span id='f0060'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr12.jpg|center|371px|Velocity profile for different Schmidt number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 12.

Velocity profile for different Schmidt number.

</span>
|}

The fluid temperature is highly influenced by the Biot number, Dufour effect, suction/injection, velocity ratio and Schmidt number as shown in [[#f0065|Figure 13]], [[#f0070|Figure 14]], [[#f0075|Figure 15]], [[#f0080|Figure 16]], [[#f0085|Figure 17]] and [[#f0090|Figure 18]]. The Biot number is related to the convective boundary condition and it increases the temperature near the surface. As a result the thermal boundary layer increases with an increase in the Biot number (see [[#f0065|Fig. 13]]). The Dufour effect appears mainly due to a mass concentration gradient which is an energy flux and increases the thermal boundary layer thickness (see [[#f0070|Fig. 14]]). An opposite phenomena is observed for the case of Soret parameter (see [[#f0075|Fig. 15]]). Due to suction, the fluid is removed from the system and as a result the thermal boundary layer thickness increases as shown in [[#f0075|Fig. 15]]. For injection, the fluid injects in the system and helps to reduce the thermal boundary layer which is clearly shown in [[#f0080|Fig. 16]]. With an increase in the velocity ratio, the thickness of the thermal boundary layer increases as the fluid velocity at the wall decreases with respect to the velocity at the wall (see [[#f0085|Fig. 17]]). The viscous diffusion rate increases with increase of the Schmidt number that restricts the fluid to flow further and reduces the thickness of thermal boundary layer (see [[#f0090|Fig. 18]]).

<span id='f0065'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr13.jpg|center|366px|Temperature profile for different Biot number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 13.

Temperature profile for different Biot number.

</span>
|}

<span id='f0070'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr14.jpg|center|365px|Temperature profile for different Dufour number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 14.

Temperature profile for different Dufour number.

</span>
|}

<span id='f0075'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr15.jpg|center|365px|Temperature profile for different Soret number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 15.

Temperature profile for different Soret number.

</span>
|}

<span id='f0080'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr16.jpg|center|364px|Temperature profile for different velocity ratio.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 16.

Temperature profile for different velocity ratio.

</span>
|}

<span id='f0085'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr17.jpg|center|365px|Temperature profile for different suction/injection parameter.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 17.

Temperature profile for different suction/injection parameter.

</span>
|}

<span id='f0090'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr18.jpg|center|365px|Temperature profile for different Schmidt number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 18.

Temperature profile for different Schmidt number.

</span>
|}

The fluid concentration depends highly on the suction/injection, velocity ratio, Soret and Schmidt number as shown in [[#f0095|Figure 19]], [[#f0100|Figure 20]], [[#f0105|Figure 21]] and [[#f0110|Figure 22]]. Fluid injection increases the amount of fluid available and so reduces the concentration boundary layer thickness. The opposite phenomena is true for the suction parameter as shown in [[#f0095|Fig. 19]]. For velocity ratio ''r'' > 0.5, the velocity at the wall is less than the velocity at large distances away and so the fluid moves away from the surface. As a result the concentration layer thickness increases with ''r''. For ''r'' > 0.5, the fluid velocity at the surface is more than the velocity at further from the plate which helps to reduce the concentration boundary layer thickness as depicted in [[#f0100|Fig. 20]]. The Soret and Schmidt numbers appear directly in the concentration equation. The concentration boundary thickness decreases with increase in Schmidt number as the Schmidt number reduces the mass diffusion rate at the surface (see [[#f0105|Fig. 21]]). The particles were ejected from the surface due to the Soret effect and as a result the concentration boundary layer thickness increases which is visible in [[#f0110|Fig. 22]].

<span id='f0095'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr19.jpg|center|363px|Concentration profile for different suction/injection parameter.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 19.

Concentration profile for different suction/injection parameter.

</span>
|}

<span id='f0100'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr20.jpg|center|364px|Concentration profile for different Schmidt number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 20.

Concentration profile for different Schmidt number.

</span>
|}

<span id='f0105'></span>

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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr21.jpg|center|364px|Skin Friction vs. Brinkman number for different Casson parameter.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 21.

Skin Friction vs. Brinkman number for different Casson parameter.

</span>
|}

<span id='f0110'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr22.jpg|center|365px|Concentration profile for different velocity ratio.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 22.

Concentration profile for different velocity ratio.

</span>
|}

From Eq. [[#e0080|(16)]], it is evident that the skin friction is proportional to the Casson parameter ''β''. So with an increase in ''β'', the skin friction at the wall increases as clearly show in  [[#f0115|Fig. 23]]. The Brinkman number has no effect on the skin friction. The skin friction increases with the opposing flow. This is because the rising flow increases the shear stress at the wall which increases the skin friction. For totally opposing flow, the skin friction at the wall increases with the Biot number while a different phenomenon was observed for higher buoyancy parameter (see [[#f0120|Fig. 24]]). The skin friction at the wall decreases with the injection parameter for ''r'' < 0.5, while it increases when ''r'' > 0.5. This is because for ''r'' < 0.5, the velocity at the wall is less than the velocity at infinity which pushes the fluid away from the surface. It is also noted that the skin friction value is negative until ''r'' = 0.5 and then becomes positive as shown in [[#f0125|Fig. 25]]. The skin friction is negative for negative values of the mixed convection parameter and is positive for positive ''λ''. It is evident from  [[#f0130|Fig. 26]] that the Soret number reduces the skin friction at the wall for positive mixed convection parameter. The skin friction value is 0.33205734 for ''λ'' = ''D<sub>f</sub>'' = Br = Sr = Sc = 0 and ''β'' → ∞ which may compare with the result of Motsumi and Makinde [[#b0160|[32]]] which was 0.3321 for their cases. It shows a good agreement with our present methodology and outcomes of the problem.

<span id='f0115'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr23.jpg|center|364px|Concentration profile for different Soret number.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 23.

Concentration profile for different Soret number.

</span>
|}

<span id='f0120'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr24.jpg|center|364px|Skin Friction vs. buoyancy ratio for different Biot number.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 24.

Skin Friction vs. buoyancy ratio for different Biot number.

</span>
|}

<span id='f0125'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr25.jpg|center|362px|Skin Friction vs. velocity ratio for different injection parameter.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 25.

Skin Friction vs. velocity ratio for different injection parameter.

</span>
|}

<span id='f0130'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr26.jpg|center|365px|Skin Friction vs. mixed convection parameter for different of Soret number.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 26.

Skin Friction vs. mixed convection parameter for different of Soret number.

</span>
|}

The local Nusselt number is an increasing function of the injection parameter. This is because, fluid injection increases the heat transfer rate at the wall. It nonlinearly decreases with an increase in the velocity ratio. Lower velocity ratio values increase the velocity at the wall which enhances the heat transfer rate near the wall. The mixed convection parameter increases the local Nusselt number at the surface. The Soret number increases the temperature gradient at the surface as shown in [[#f0135|Figure 27]] and [[#f0140|Figure 28]].

<span id='f0135'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
|-
|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr27.jpg|center|365px|Local Nusselt number vs. velocity ratio for different injection parameter.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 27.

Local Nusselt number vs. velocity ratio for different injection parameter.

</span>
|}

<span id='f0140'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr28.jpg|center|365px|Local Nusselt number vs. velocity ratio for different Soret number.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 28.

Local Nusselt number vs. velocity ratio for different Soret number.

</span>
|}

The effect of the velocity ratio, injection parameter, mixed convection parameter and Soret number on local Sherwood number is shown in [[#f0145|Figure 29]] and [[#f0150|Figure 30]]. We observe that the local Sherwood number is a decreasing function of the velocity ratio. Injection reduces the mass transfer rate at the wall which increases the local Sherwood number. The local Sherwood number increases due to the Soret effect.

<span id='f0145'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr29.jpg|center|364px|Local Sherwood number vs. velocity ratio for different injection parameter.]]


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| <span style="text-align: center; font-size: 75%;">

Figure 29.

Local Sherwood number vs. velocity ratio for different injection parameter.

</span>
|}

<span id='f0150'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; max-width: 100%;" 
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|


[[Image:draft_Content_616345013-1-s2.0-S111001681630076X-gr30.jpg|center|365px|Local Sherwood number vs. mixed convection parameter for different Soret number.]]


|-
| <span style="text-align: center; font-size: 75%;">

Figure 30.

Local Sherwood number vs. mixed convection parameter for different Soret number.

</span>
|}

==4. Conclusion==

In the present problem we investigated Casson fluid flow through a permeable plate and the effect of the velocity ratio, suction or injection, Sore and Dufour number. We have considered a convective boundary condition at the surface. The nonlinear governing equations were written as a system of ordinary differential equations by using similarity transformations and then solved numerically. The velocity of the fluid decreased for velocity ratio parameter less than 0.5 and increased with ''r'' for velocity ratio parameters greater than 0.5. The momentum boundary layer thickness decreased with an increase in the Casson parameter. The Brinkmann number enhanced the boundary layer thickness for velocity ratio parameter equal to and greater than 0.5 but the opposite is true for velocity ratio parameter less than 0.5. The results of this work may find application in different industrial and medical applications where a Casson fluid is an important medium for heat and mass transfer.

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