Lare, A. (2015). Casson Fluid Flow with Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation. Journal of Heat and Mass Transfer Research(JHMTR), 2(2), 6378. doi: 10.22075/jhmtr.2015.346
Animasaun Isaac Lare. "Casson Fluid Flow with Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation". Journal of Heat and Mass Transfer Research(JHMTR), 2, 2, 2015, 6378. doi: 10.22075/jhmtr.2015.346
Lare, A. (2015). 'Casson Fluid Flow with Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation', Journal of Heat and Mass Transfer Research(JHMTR), 2(2), pp. 6378. doi: 10.22075/jhmtr.2015.346
Lare, A. Casson Fluid Flow with Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation. Journal of Heat and Mass Transfer Research(JHMTR), 2015; 2(2): 6378. doi: 10.22075/jhmtr.2015.346
Casson Fluid Flow with Variable Viscosity and Thermal Conductivity along Exponentially Stretching Sheet Embedded in a Thermally Stratified Medium with Exponentially Heat Generation
^{}Federal University of Technology, Akure, Ondo State, Nigeria.
Receive Date: 10 June 2014,
Revise Date: 05 December 2014,
Accept Date: 30 December 2015
Abstract
The motion of temperature dependent viscosity and thermal conductivity of steady incompressible laminar free convective (MHD) nonNewtonian Casson fluid flow over an exponentially stretching surface embedded in a thermally stratified medium are investigated. It is assumed that natural convection is induced by buoyancy and exponentially decaying internal heat generation across the space. The dimensionless temperature is constructed such that the effect of stratification can be revealed. Similarity transformations were employed to convert the governing partial differential equations to a system of nonlinear ordinary differential equations. The numerical solutions were obtained using shooting method along with the RungeKutta Gill method. The behaviour of dimensionless velocity, temperature and temperature gradient within the boundary layer has been studied using different values of all the controlling parameters. The numerical result show that increase in the magnitude of temperature dependent fluid viscosity parameter leads to an increase in velocity, decrease in temperature, decrease in temperature gradient near the wall and increase in temperature gradient far from the wall. The velocity profile increases, temperature distribution increases and temperature gradient increases near the wall only by increasing the magnitude of temperature dependent thermal conductivity parameter.
The theoretical study of two dimensional nonNewtonian incompressible fluid flows over a surface with stretching or shrinking properties has taken the significant attention in the past few years due to its wide applications in engineering fields as well as in the industry. Some applications include the production of toothpaste, shampoo, custard solution, blood treatment, glass fibre production and design of the plastic films. Crane [1] investigated boundary layer flow past a stretching sheet whose velocity is proportional to the distance from the sheet. In fluid dynamics, fluids are divided into two broad groups which are Newtonian and nonNewtonian. NonNewtonian transport phenomena arise in many fields of mechanical and chemical engineering and also in food processing. Some materials e.g. muds, condensed milk, glues, printing ink, emulsions, paints, sugar solution, shampoos and tomato paste exhibit almost all the properties of nonNewtonian fluids. One of the properties of Newtonian fluid is that coefficient of viscosity does not change with the rate of deformation of the fluid. This property can be found in the motion of water, kerosene and air. In addition, nonNewtonian fluids do not exhibit the property of Newtonian fluids where shear stress is directly proportional to shear rate. There are three broad classifications of nonNewtonian fluids. These are timedependent, timeindependent and viscoelastic fluids. The timeindependent nonNewtonian fluids are those fluids in which the shear rate at a given point is a function of the stress at that point only. Examples are Casson, Bingham, Dilatant and Pseudoplastic fluids. The governing equations of nonNewtonian fluids are highly nonlinear and much more complicated than those of Newtonian fluids. The More research is needed to investigate such fluids for understanding the flow characteristics. See Mukhopadhyay [2]. This rheological model was introduced originally by Casson [3] in his study on a flow equation for pigment oilsuspensions of printing ink. Bird et al.[4] investigated the rheology and flow of plastic fluid model which exhibits shear thinning characteristics, yield stress and high shear viscosity. Venkatesan et al. [5] stated that the blood shows Newtonian fluid’s characteristics when it flows through the larger diameter arteries at the high shear rates, but it exhibits a significant nonNewtonian behaviour when it flows through the small diameter arteries at low shear rates.
Internal energy generation can be explained as a scientific method of generating heat energy within a body by a chemical, electrical or nuclear process. The natural convection induced by the internal heat generation is a common phenomenon in nature. Examples include motion in the atmosphere where heat is generated by absorption of sunlight. see Tasaka et al. [6]. Crepeau and Clarksean [7] carried out a similarity solution for a fluid with an exponentially decaying heat generation term and a constant temperature vertical plate under the assumption that the fluid has an internal volumetric heat generation. An exponential form is used to account for the internal energy generation term. It was reported that the effect of internal heat generation is important in several applications i.e. reactor safety analyses, fire and combustion studies. In many situations, there may be appreciable temperature difference between the surface and the ambient fluid. This necessitates the consideration of temperature dependent heat sources that may exert a strong effect on the heat transfer characteristics ElAziz and Salem [8]. The study of the heat generation or absorption effects is important in view of several physical problems such as fluids under the exothermic or endothermic chemical reaction; although, the exact modeling of internal heat generation or absorption is completely difficult, some simple mathematical models can express its average behaviour for most physical situations see ElAziz and Salem [9].
The thermal stratification can be defined as the scientific term that describes the layering of bodies of water based on their temperature. This concept divides water bodies about a surface/plate into three layers known as epilimnion, metalimnion and hypolimnion. The epilimnion is the highest and warmest layer; the metalimnion is the transition layer between the upper warm regions of the fluid and the cool layer near the bottom is the hypolimnion. Recently, many researchers have reported free convection flow over a surface/plate embedded in a thermally stratified medium due to its real life application.
Moorthy and Senthilvadivu [10] have studied the effect of variable viscosity on free convective flow of nonNewtonian powerlaw fluids along a vertical surface with the thermal stratification. It was reported that as the thermal stratification parameter increases the heat transfer rate also increases for and . In the research, is the parameter characterizing the effect of temperature on viscosity. The flow, heat and nanoparticle mass transfer characteristics in the free convection from a vertical plate in a thermally linearly stratified nanofluid saturated nonDarcy porous medium under the convective boundary condition have been investigated by RamReddy et al. [11]. They assumed that the fluid flow is moderate, so the pressure drop is proportional to the linear combination of the fluid velocity and the square of the velocity. They reported that the fluid velocity and temperature increase as the thermal stratification parameter increases. A theoretical study on magnetohydrodynamic boundary layer flow with constant viscosity and the thermal conductivity towards an exponentially stretching sheet embedded in a thermally stratified medium subject to suction is investigated by Mukhopadhyay [12].It was reported that increase in stratification parameter in the absence of suction and also in the presence of suction corresponds to decrease in temperature profiles and increase in temperature gradient. The convective boundary condition invoked upon temperature profile in [11] was removed; behaviour of the fluid flow was reconsidered and explained in [13]. The resulting system of equations was solved numerically by using an implicit finite difference method. It was reported that the positive values of the thermal stratification parameter have the tendency to increase the boundary layer thickness, due to the enhancement in the temperature difference between the plate and the free stream in the presence of the nanofluids [13].
In most of the published articles, Casson fluid flow along the heated surface has been treated with constant viscosity and thermal conductivity. The heat transfer is energy in transit due to the temperature difference; whenever there is a temperature difference in a medium or between two medium of difference temperature, the heat transfer must occur significantly. The temperature difference is the driving force for heat transfer. Hence, viscosity and thermal conductivity cannot be assumed as a constant. This fact shows the occurrence of a phase of the variable viscosity. Due to the nature and behaviour of Casson fluid flow, it may not flow very well along a vertical surface. The motivation to this study is to research more on the areas that have been neglected in other works of many researchers. This will provide results on the effects of variable viscosity and thermal conductivity, stratification parameter and exponentially decaying heat generation on velocity, temperature and temperature gradient of Casson flow along exponentially stretching vertical surface embedded in a thermally stratified medium.
The aim of this theoretical study is to unravel the effect of the emerging controlling parameters on velocity, temperature and temperature gradient of Casson fluid with variable viscosity and thermal conductivity in the boundary layer over a vertical surface embedded in a stratified medium with suction and exponentially decaying space dependent internal heat generation. The governing partial differential equations are modified and converted to nonlinear ordinary differential equations by using the suitable similarity transformations. The transformed selfseminar ODE’s are solved by using the shooting method and quadratic interpolation. The effects of the embedded flow controlling parameters on the fluid velocity, temperature, temperature gradient and heat transfer rate have been demonstrated graphically and discussed. A comparative study is also presented.
2. Mathematical Formulation
A steady incompressible twodimensional laminar free convective electrically conducting viscous fluid flow along a vertical exponentially stretching sheet embedded in a thermally stratified medium in the presence of suction is considered for a theoretical study. The vertical surface is elastic. The motion of an incompressible nonNewtonian fluid is induced by the stretching property of the vertical surface, buoyancy effect generated by gradients in the temperature field and space dependent internal heat generation. This occurs in view of the elastic properties of the surface parallel to the axis through equal and opposite forces when the origin is fixed at. A variable magnetic field of constant intensity is applied in a direction transverse to the plate and the electrical conductivity of the fluid is assumed to be small so that the induced magnetic field can be neglected in comparison to applied magnetic field. The surface temperature is embedded in a thermally stratified medium of variable ambient temperature where. The following wall conditions and the free stream temperature embedded in a thermally stratified medium are stated in Mukhopadhyay [12].
Figure 1. Physical model and coordinate system
Since the fluid pressure is constant throughout the boundary layer, it is assumed that the induced magnetic field is small in comparison to the applied magnetic field; hence it is neglected. Under the above assumptions and invoking the Boussinesq approximation, the boundary layer equations governing the flow and heat transfer of a viscous incompressible fluid can be written as
where is the fluid temperature , is the kinematic viscosity, is the fluid viscosity and is the fluid density, is the thermal diffusivity with being the fluid thermal conductivity and is the heat capacity at constant pressure. The dimensionless space internal heat generation term in energy equation is formulated by using the concept introduced in Salem and ElAziz [9] where is known as a coefficient of the dimensionless spacedependent internal heat generation. From the concept of viscosity, according to Mukhopadhyay [14] and Hayat et al. [15] it is assumed that the rheological equation of an isotropic and incompressible flow of a Casson fluid can be written as
According to [12], is known as the fluid yield stress that mathematically expresses as
is known as the plastic dynamic viscosity of the nonNewtonian fluid, is the product of the deformation rate component with itself i.e. , where is the component of the deformation rate and is the critical value of based on nonNewtonian model. In a case of Casson fluid (nonNewtonian) flow, where, it’s possible to say that
Substituting (5) into (6) then simplify
The kinematics viscosity of Casson fluid is now a function depending on plastic dynamic viscosity, density and Casson parameter
The Rosseland approximation requires that the media is optically dense media and the radiation travels only a short distance before being scattered or absorbed. The another objective of this research is to study the radiation of heat within optically thick Casson fluid before the heat is scattered, radiative heat transfer is taken into account, and the Rosseland equation is used to estimate the radiative thermal conductivity in Casson fluid. The Rosseland equation is a simplified model of Radiative Transfer Equation (RTE). When the material has a great extinction coefficient, it can be treated as optically thick. is the radiative heat flux and is defined by using the Rosseland approximation Chamkha et al. [16] as
where is the StefanBoltzmann constant and is known as the absorption coefficient. By assuming that the temperature difference within the flow is such that may be expanded in a Taylor series and expanding about and neglecting higher orders. Next is to consider Taylor Series Expansion of about, Considering the Taylor’s series expansion of a function about
Likewise, expansion of about , by neglecting higher order
In this study, it is assumed that the plastic dynamic viscosity and the thermal conductivity of Casson fluid vary as a linear function of temperature. This assumption is valid since it is known that the physical properties of the fluid may change significantly with temperature. For lubricating fluids, the heat generated by the internal friction and the corresponding rise in temperature affect on the fluid viscosity and so the fluid viscosity can no longer be assumed constant. In industry, the fluids can be subjected to extreme conditions such as high temperature, pressure, high shear rates and external heating (Ambient Temperature) and each of these factors can lead to high temperature being generated within the fluid. According to Anyakoha [17], Batchelor [18] and Vajravelu et at. [19]. The following relations are now introduced for and as and respectively. Here is the stream function. These automatically satisfy continuity equation (1). Modified governing equations of (2) and (3) are:
In Physics, it is a wellknown fact that, if an object is on an elastic surface at rest, when the surface is stretched; the object also tends to move towards the direction of the pull. The surface of the plate is assumed to be highly elastic and is stretched in the vertical direction with velocities
,
.
is the reference temperature, and are constants. Equations (11) and (12) are subject to following boundary conditions
In this study, is a constant and is the reference length. It is very important to note that, the exponential velocity is valid only when. When , it is obvious that the effect of the exponential property on wall velocity may skyrocket. Also, in the third term of Equation (13); it is obvious that . By introducing the stream function and similarity variables as
Dimensionless temperature, thermal conductivity model in Salem and Fathy [20] temperature dependent viscosity model in Layek et al. [21] and respectively as
By substituting all into Equations (11) to (14) we obtain the following locally similar ordinary differential equations:
Together with the boundary conditions
Here all the primes denote the differentiation with respect to , is known as temperature dependent variable plastic dynamic viscosity parameter, is the nonNewtonian Casson parameter, is the local modified Grashof related parameter, is known as temperature dependent variable thermal conductivity parameter, is known as thermal radiation parameter, is the Prandtl number, is the space dependent internal heat source parameter and is the stratification parameter. For practical applications, the major physical quantities of interest are the local skin friction coefficient and Nusselt number. The first physical quantity of interest is the wall skin friction coefficient defined as
is known as the shear stress or the skin friction along the stretching sheet
Another physical quantity of interest is the local Nusselt number, which is defined as
is known as heat flux from the sheet
Here local Reynold number.
3. The Numerical Technique
The set of strong nonlinear coupled differential equations (17) and (18) together with the boundary conditions (19) and (20) are solved numerically by using Runge Kutta Gill method along with shooting techniques by using the prescribed parameters. The quadratic interpolation is based on local approximation of the nonlinear functions in RHS of (25)(29) by a quadratic function and the root of the quadratic function is taken as an improved approximation to the root of nonlinear functions. The procedure is applied repetitively to converge. There are two types of error involved in Runge Kutta as an approximation method of ordinary differential equations. They are round off error and truncation error. Runge Kutta Gill method is selected because it reduces (minimize) round off error and this method of the integration systems of first order does not require preceding function values to be known see Gill [22]. According to Finlayson [23], Order analysis, Consistency analysis and Stability analysis shows that Runge Kutta Gill is also of fourth order, stable and consistent. The constants are selected to reduce the amount of storage required in the solution of a large number of simultaneous first order differential equation, in addition; the Runge Kutta Gill variant method is probably used more often in machine integration due to the storage saving. The BVP cannot be solved on an infinite interval, and it would be impractical to solve it on a very large finite interval. In this study, the author imposed the infinite boundary condition at a finite point of . By following Na [24], superposition method is adopted to reduce the governing dimensionless equations, (17) and (18) together with boundary conditions (19) and (20) to system of first order nonlinear autonomous ordinary differential equations. Let
Now becomes
According to the shooting method, equation (20) is used to obtain Guess 1 and Guess 2. To integrate the corresponding IVP (25) to (29), Guess 1 and Guess 2are required but no such values exist after the nondimensionalization of the boundary conditions. The suitable guess values are chosen and then the integration is carried out. The calculated values for and are compared with that of boundary condition (20). The interpolation is employed and the better estimation for Guess 1 and Guess 2 are obtained, IVP are solved by using the Runge Kutta Gill method with . To improve the solutions, quadratic interpolation method which is superior (i. e. faster convergence rate) is adopted more than linear interpolation namely secant method Hoffman [25]. In very sensitive problems like this, the quadratic interpolation may misbehave and require bracketing techniques Hoffman [25]. Hence, the guess values were chosen wisely. The above procedure is repeated until results up to the desired degree of accuracy 0.000001 is obtained. From the numerical computation, Guess 1 is proportional to the skinfriction coefficient and Guess 2 is proportional to Nusselt Number, which are and. They are also sorted out and their values are presented in a tabular form.
4. Results and Discussion
In order to analyse the numerical results, the computation have been carried out by using the method described in the previous section for various values of the temperature dependent plastic dynamic variable viscosity parameter , nonNewtonian Casson parameter , local modified Grashof related parameter , the temperature dependent variable thermal conductivity parameter , thermal radiation parameter Prandtl number , space dependent internal heat source parameter , intensity of internal heat generation parameter on space and stratification parameter . In order to show the results, the numerical values were plotted in figures 2 to 22. For the accuracy verification of the applied numerical scheme, has been made a comparison of the present results corresponding to the values of heat transfer coefficient for the thermal radiation parameter values and Prandtl number with the available published results of Bidin and Nazar [26], Nadeem et al. [27] and Pramanik [28] when and It is very important to note that in all of the above mentioned study and in this present study . Table 1 shows the good agreement between the present study result and the results reported by [27  29]. The numerical values of and for five different values of the temperature dependent variable plastic dynamic viscosity parameter, temperature dependent thermal conductivity of the Casson fluid parameter, Stratification parameter , the nonNewtonian Casson fluid parameter , The space dependent internal heat source parameter and the magnetic field parameter are shown in Table 2 to 7.
4.1 The Velocity Profiles
Figure 2 illustrates the velocity profiles for the different values of temperature dependent plastic dynamic viscosity parameter when magnetic field is present (i.e.), the wall temperature is (since ) and in the presence of suction ().
Table 1a: Various values of for several values of Prandtl number and thermal radiation
Bidin and
Nazar [26] with
and
Bidin and
Nazar [26] with
and
Nadeem et al. [27] for PES case with E = λ1 = 0 B = ω = 0 and N = 0.5
Nadeem et al. [27] for PES case with E = λ1 = 0 B = ω =0and N
Table 1b: Various values of for several values of Prandtl number and thermal radiation
Pramanik
[28] with nd
Pramanik
[28] with and
Present
Study when
Present
Study when
Table 2: The numerical values of skin friction and Nusselt number for different values of when, , , , , , , , and

Table 3: The numerical values of skin friction and Nusselt number for different values of when, , , , , , , , and

Table 4: The numerical values of skin friction and Nusselt number for different values of when, , , , , , , , and

Table 5a: The numerical values of skin friction and Nusselt number for different values of when , , , , , , , , with stratification i. e. and Constant viscosity i.e.

Table 5b: The numerical values of skin friction and Nusselt number for different values of when , , , , , , , , without stratification i. e. and variable viscosity i.e.

Table 6a: The numerical values of skin friction and Nusselt number for different values of when, , , , , , , , and

Table 7: The numerical values of skin friction and Nusselt number for different values of when, , , , , , , , and without stratification i. e.

Figure 3. Effects of the variable plastic dynamic viscosity parameter over the Temperature profiles
Figure 4. Effects of the variable plastic dynamic viscosity parameter over the Temperature gradients
Figure 5. Effects of the variable thermal conductivity parameter over the Velocity profiles
Figure 6. Effects of the variable thermal conductivity parameter over the Temperature profiles
Figure 7. Effects of the variable thermal conductivity parameter over the Temperature gradients
Figure 8. Effects of nonNewtonian Casson parameter over Velocity profiles
Figure 9. Effects of the nonNewtonian Casson parameter over Temperature profiles
Figure 10. Effects of the the nonNewtonian Casson parameter over Temperature gradients
Figure 11. Effects of the Stratification parameter over Velocity profiles
Figure 12. Effects of the Stratification parameter over Temperature profiles
Figure 13. Effects of the Stratification parameter over Temperature gradients
Figure 14. Effects of the intensity of the internal heat generation parameter on space over Velocity profiles
Figure 15. Effects of the intensity of internal heat generation parameter on space over Temperature profiles
Figure 16. Effects of the intensity of the internal heat generation parameter on space over Temperature gradients
Figure 17. Effects of the internal heat generation parameter on space over Velocity profiles
Figure 18. Effects of the internal heat generation parameter on space over Temperature profiles
Figure 19. Effects of the internal heat generation parameter on space over Temperature gradients
Figure 20. Effects of the Magnetic field parameter with and without stratification over Velocity profiles
Figure 21. Effects of the Magnetic field parameter with and without stratification over Temperature profiles
Figure 22. Effects of the Magnetic field parameter with and without stratification over Temperature gradient
It is observed that when Casson fluid is considered as the fluid with constant plastic dynamic viscosity, the velocity is found to be very small in magnitude throughout the boundary layer compare to when considered as variable plastic dynamic viscosity. This figure demonstrates the effect of increasing i.e. to increase the resulting temperature difference which makes the intermolecular forces (bond) between the Casson fluid to become weaker and drastically decreases the strength of plastic dynamic viscosity. This effect eventually increases the transport phenomena across the momentum boundary layer. From Figure 5 it is observed that as the temperature dependent variable thermal conductivity parameter increases, the velocity profiles increases. Effect of parameter is more negligible few distances away from the wall. This effect is due to increment in temperature difference between temperature at the wall and reference temperature. Maximum velocity is found very close to the surface that is embedded in a thermal stratification (i.e.). When the problem is investigated using high value of stratification parameter , together with the same values of the remain parameters (i.e. , , , , , , , , and ); it is observed that parameter ( has no significant effect on velocity profile of Casson fluid flow.
In order to investigate the dynamic of Casson fluid flow over a surface embedded in a thermally stratified medium, two different cases were considered at a fixed value of , , , , , , and . Figure 8 depicts both cases. In the first case, Casson fluid is considered as fluid with constant plastic dynamic viscosity (i.e.) when stratification parameter (i.e.). The velocity decreases with an increase in the value of throughout the fluid domain. Increase in means increase in free stream temperature ( or decrease in surface temperature. It is obvious that both cases could not produce sufficient temperature to break down the molecules which makes up plastic dynamic viscosity; hence the velocity decreases since the viscosity of nonNewtonian Casson fluid is naturally high. In the second case, Casson fluid is treated as fluid with variable plastic dynamic viscosity (i.e.) without stratification (i.e.), the velocity increases very close to the wall () and after this interval velocity decreases as increases from nonNewtonian fluid to Newtonian fluid (i.e.). The result is obvious for this case, means the temperature of the exponentially stretching surface and more heat is injected since; hence the intermolecular forces within plastic dynamic viscosity is broken.
Next, the effects of thermal stratification parameter on velocity profiles of nonNewtonian Casson fluid when heat is injected greatly by setting, in the presence of magnetic field and uniform suction. It is found that velocity decreases. This result can be traced to the fact that, as increases, the surface temperature within thermally stratified medium ranges from epilimnion to hypolimnion. Since the wall temperature decreases, coldness is introduced and this makes the molecules and intermolecular forces of Casson fluid to become stronger. This explains the decrement in velocity with an increase in .
Figure 14 exhibits the velocity profiles for different values of intensity of internal heat generation parameter. The velocity decreases as ranges from to. This parameter is further investigated within; the velocity profiles decreases as increases. Figure 17 exhibits the velocity profiles for different values of exponentially decaying internal heat generation parameter when and. The velocity increases as ranges from to. This parameter is further investigated within; the velocity profiles increases as increases. Figure 20 represents the velocity profiles for the variation of magnetic field parameter with thermal stratification and without thermal stratification. In both cases, the velocity decreases. Application of a magnetic field to an electrically conducting Casson fluid produces a kind of draglike force called Lorentz force. This force causes reduction in the fluid velocity within boundary layer. The effect of Lorentz force on velocity profiles is highly experienced when thermal stratification is set to hypolimnion (i.e.). It is also observed that maximum velocity exist when thermal stratification is adjusted to epilimnion (i.e.)
4.2 Temperature Profiles
In Figure 3, variations of temperature field against for several values of byusing, and are shown. This figure indicates the drastic effect of the internal heat generation intensity across the space. The parabolic profiles of the temperature distribution with the pick slightly far from the wall are found to be lower when. It is observed that the temperature decreases as. The increase of temperature dependent plastic dynamic viscosity parameter leads to decrease of thermal boundary layer thickness, which results in decreasing of temperature profile. Decrease in temperature profiles across the thermal boundary layer means a decrease in the velocity of the Casson fluid. In fact, in this case, the fluid particles undergo two opposite forces which are: (i) one force increases the fluid velocity due to decrease in the fluid viscosity with increase in the values of , (ii) the second force decreases the fluid velocity due to decrease in temperature; since decreases with increasing. Very close to the vertical surface, as the temperature is high, the first force dominates and far away from the surface, the temperature is low; this implies that the second force dominates in that region.
From Figure 6 it is observed that as the temperature dependent variable thermal conductivity parameter increases, the temperature distribution increases significantly within the space. The effect of is negligible very close to the wall and also far from the wall when the value of the stratification () and also far from the wall. When the problem is investigated again by using high value of stratification parameter , together with the same values of the remain parameters i.e. , , , , , , , , and ; it is observed that has no significant effect on temperature profile of Casson fluid flow. Figure 9 shows the effects of nonNewtonian Casson fluid parameter on the temperature for fixed values of, and uniform suction. In order to investigate the dynamic of Casson fluid flow along with a vertical surface, two different cases were considered. In the first case, Casson fluid is treated as fluid with constant plastic dynamic viscosity (i.e.) and stratification parameter set to hypolimnion (). It is observed that the temperature distribution increases with an increase in the value of throughout the fluid domain. In the second case, Casson fluid is treated as fluid with variable plastic dynamic viscosity (i.e.) without stratification (i.e.) this corresponds to epilimnion layer. The temperature decreases negligibly as increases from nonNewtonian fluid to Newtonian fluid (i.e.). This result is in good agreement with a report on effects of Casson fluid parameter β, temperature dependent viscosity ξ and temperature dependent thermalconductivity parameter ε over temperature profiles in [30]. Figure 12 illustrates the effects of thermal stratification parameter on the temperature profiles of nonNewtonian Casson fluid when the heat is injected greatly by setting, in the presence of internal heat generation on dimensionless space and intensity. It is found that the temperature decreases. This result can be traced to the fact that, as increases, the wall temperature decreases, this effect dominate the temperature distribution. Figure 15 exhibits the temperature profiles for different values of intensity of exponentially decaying internal heat generation on dimensionless space. The temperature profiles decreases as ranges from to . Figure 18 exhibits the temperature profiles for different values of exponentially decaying internal heat generation parameter when. The temperature profiles increases as q1 ranges from to. Figure 21 depicts the effect of magnetic parameter with thermal stratification and without thermal stratification) on temperature gradient. In both cases, temperature distribution increases. Maximum temperature is observed very close to the wall when thermal stratification is at epilimnion (i.e. ) and maximum temperature in a parabolic profiles is observed when adjusted to hypolimnion (i.e. ).
4.3 Temperature gradient
The rate of heat transfer in a certain direction depends on the magnitude of the temperature gradient (the temperature difference per unit length or the rate of temperature changes) in that direction. The higher the temperature gradient is caused the higher the rate of heat transfers.
The effects of temperature dependent plastic dynamic viscosity parameter on the temperature gradient as Casson fluid flows over a stretchable surface embedded in thermally stratified medium with suction is indicated in figure 4. With an increase in the value of parameter , the temperature gradient of Casson fluid decreases near the wall. Within turning point of each profile exist and temperature gradient increases thereafter.
From Figure 7 it is observed that as the temperature dependent variable thermal conductivity parameter increases, temperature gradient increases significantly within. Within this interval, maximum value of the temperature gradient is obtained when (i.e. Casson fluid is treated as fluid with variable thermal conductivity) as at. When the flow is investigated again by using the high value of stratification parameter , together with the same values of the remain parameters i.e. , , , , , , , , and ; it is observed that has no significant effect on temperature gradient except within where the effect is slightly significant. When Casson fluid is treated as fluid with constant plastic dynamic viscosity (i.e.) and stratification parameter is set to hypolimnion (). It is observed that the temperature gradient increases close to the wall and decreases far away from the wall. In the second case, Casson fluid is treated as fluid with the variable plastic dynamic viscosity (i.e.) without stratification (i.e.) this corresponds to epilimnion layer. The corresponding effect on the temperature gradient and heat transfer coefficient as increases from nonNewtonian fluid to Newtonian fluid (i.e.) is presented in Figure 10. The temperature gradient increases significantly with an increase in stratification (Fig. 13). The corresponding effect of intensity of exponentially decaying internal heat generation on temperature gradient and heat transfer coefficient as increases is shown in Figure 16. The temperature gradient increases significantly with an increase in exponentially decaying internal heat generation parameter as ranges from to. (see Fig. 19). Turning point is observed within, thereafter, the temperature gradient decreases. The effect of Magnetic field parameter with and without thermal stratification on temperature gradient is shown in figure 22.
5. Conclusion
Laminar free convective MHD boundary layer flow of nonNewtonian Casson fluid flow over an exponentially stretching surface embedded in a thermally stratified medium has been studied. The numerical approach was utilized to study the effect of all the controlling parameters on the flow’s velocity and temperature profiles in the boundary layer. The results show that:
i. An increase in the variable plastic dynamic viscosity parameter of Casson fluid would increase the velocity profiles, but it would decrease the magnitude of temperature throughout the domain and temperature gradient close to the wall in the boundary layer.
ii. An increase in the variable thermal conductivity parameter of Casson fluid would increase the velocity and temperature profiles; temperature gradient also increases near the wall.
iii. Based on the results of the present study, it can be concluded that the effect of Casson fluid parameter when treated as fluid which possess constant plastic dynamic viscosity; the velocity decreases, temperature distribution increases and temperature gradient increases (near the wall). And, when treated as temperature dependent variable plastic dynamic viscosity; the velocity profile increases, temperature distribution decreases and temperature gradient decrease (near the wall).
iv. Increasing the stratification parameter results in reduction of velocity and temperature profiles.
v. It can be concluded that the effect of intensity parameter embedded in the exponentially decaying heat source decreases both velocity and temperature profiles.
vi. Variation of exponentially decaying heat source parameter show significant effect on the thickness of the boundary layer profiles (i. e. velocity, temperature and temperature gradient).
vii. The magnetic field reduces the heat transfer rate, though it causes the increment in the temperature inside the boundary layer when the stratification parameter is adjusted to epilimnion and hypolimnion.
Acknowledgements
The author expresses his profound gratitude to the anonymous Reviewer for their valuable comments and suggestions.
Nomenclature
Distance along the surface
Distance perpendicular to the surface
Velocity along direction (Streamwise velocity)
Velocity along direction (Wall normal velocity)
Variable magnetic field
Acceleration due to gravity
Ambient temperature
Reference temperature
Fluid temperature
Heat capacity at constant pressure
Radiative heat flux in direction
Coefficient of space dependent heat generation
Reference length
Reference velocity
Fluid yield stress
Prescribed surface temperature
Variable free stream temperature
Local skin friction
Local Nusselt number
Heat flux
Local modified Grashof related parameter
Magnetic field parameter
Prandtl number
Thermal radiation parameter
Space dependent internal heat source parameter
Stratification parameter
Intensity of exponentially decaying heat source
Greek Symbols
Variable Plastic dynamic viscosity parameter
Variable thermal conductivity parameter
NonNewtonian Casson parameter
Kinematic viscosity
Density
Nondimensional temperature
Shear stress
Electrical conductivity of Casson fluid
Plastic dynamic viscosity
Plastic dynamic viscosity of the ambient fluid
Thermal conductivity
Thermal conductivity of the ambient fluid
Electric conductivity
Volumetric coefficient of thermal expansion
Stream function
Product of the deformation component
Constant related to temperature dependent
Constant related to temperature dependent
Similarity variable
Dynamic Viscosity
Subscripts
Reference temperature close to the surface
Property at the wall
Property at ambient
Reference
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