MHD Stagnation-Point Flow of Casson Fluid and Heat Transfer over a Stretching Sheet with Thermal Radiation

The two-dimensionalmagnetohydrodynamic (MHD) stagnation-point flowof electrically conducting non-NewtonianCasson fluid and heat transfer towards a stretching sheet have been considered.The effect of thermal radiation is also investigated. Implementing similarity transformations, the governing momentum, and energy equations are transformed to self-similar nonlinear ODEs and numerical computations are performed to solve those. The investigation reveals many important aspects of flow and heat transfer. If velocity ratio parameter (B) and magnetic parameter (M) increase, then the velocity boundary layer thickness becomes thinner. On the other hand, for Casson fluid it is found that the velocity boundary layer thickness is larger compared to that of Newtonian fluid. The magnitude of wall skin-friction coefficient reduces with Casson parameter (β). The velocity ratio parameter, Casson parameter, and magnetic parameter also have major effects on temperature distribution. The heat transfer rate is enhanced with increasing values of velocity ratio parameter. The rate of heat transfer is enhanced with increasing magnetic parameterM for B > 1 and it decreases withM for B < 1. Moreover, the presence of thermal radiation reduces temperature and thermal boundary layer thickness.


Introduction
In fluid dynamics the effects of external magnetic field on magnetohydrodynamic (MHD) flow over a stretching sheet are very important due to its applications in many engineering problems, such as glass manufacturing, geophysics, paper production, and purification of crude oil.The flow due to stretching of a flat surface was first investigated by Crane [1].Pavlov [2] studied the effect of external magnetic field on the MHD flow over a stretching sheet.Andersson [3] discussed the MHD flow of viscous fluid on a stretching sheet and Mukhopadhyay et al. [4] presented the MHD flow and heat transfer over a stretching sheet with variable fluid viscosity.On the other hand, Fang and Zhang [5] reported the exact solution of MHD flow due to a shrinking sheet with wall mass suction.Bhattacharyya and Layek [6] showed the behavior of solute distribution in MHD boundary layer flow past a stretching sheet.Furthermore, many vital properties of MHD flow over stretching sheet were explored in various articles [7][8][9][10][11][12] in the literature.Several important investigations on the flow due to stretching/shrinking sheet are available in the literature [13][14][15][16].
Chiam [17] investigated the stagnation-point flow towards a stretching sheet with the stretching velocity of the plate being equal to the straining velocity of the stagnationpoint flow and found no boundary layer structure near the sheet.Mahapatra and Gupta [18] reconsidered the stagnationpoint flow problem towards a stretching sheet taking different stretching and straining velocities and they observed two different kinds of boundary layer near the sheet depending on the ratio of the stretching and straining constants.The detailed discussion on the stagnation-point flow over stretching/shrinking sheet can be found in the works of Mahapatra and Gupta [19], Nazar et al. [20], Layek et al. [21], Nadeem et al. [22], Bhattacharyya [23][24][25], Bhattacharyya et al. [26][27][28], Bhattacharyya and Vajravelu [29], and Van Gorder et al. [30].
Many fluids used in industries show non-Newtonian behaviour, so the modern-day researchers are more interested in those industrial non-Newtonian fluids and their dynamics.A single constitutive equation is not enough to cover all properties of such non-Newtonian fluids and hence many non-Newtonian fluid models [31][32][33][34] have been proposed to clarify all physical behaviours.Casson fluid is one of the types of such non-Newtonian fluids, which behaves like an elastic solid, and for this fluid, a yield shear stress exists in the constitutive equation.Fredrickson [35] investigated the steady flow of a Casson fluid in a tube.The unsteady boundary layer flow and heat transfer of a Casson fluid over a moving flat plate with a parallel free stream were studied by Mustafa et al. [36] and they solved the problem analytically using homotopy analysis method (HAM).Bhattacharyya et al. [37,38] reported the exact solution for boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet with and without external magnetic field.The important characteristics of the flows of various non-Newtonian fluids over a stretching/shrinking sheet can be found in the articles [39][40][41][42][43][44][45][46].
Motivated by the previously mentioned investigations on flow of non-Newtonian fluids due to a stretching sheet and its vast applications in many industries, in the present paper, the steady two-dimensional MHD stagnation-point flow of electrically conducting non-Newtonian Casson fluid and heat transfer past a stretching sheet in presence of thermal radiation effect are investigated.Using similarity transformations, the governing equations are transformed.The converted self-similar ordinary differential equations are solved by shooting method.The numerical results are plotted in some figures to see the effects of physical parameters on the flow and heat transfer.

Mathematical Analysis of the Flow
Consider the steady two-dimensional incompressible flow of electrically conducting Casson fluid bounded by a stretching sheet at  = 0, with the flow being confined in  > 0. It is also assumed that the rheological equation of state for an isotropic and incompressible flow of a Casson fluid can be written as [37,47] where   is plastic dynamic viscosity of the non-Newtonian fluid,   is the yield stress of fluid,  is the product of the component of deformation rate with itself, namely,  =     ,   is the (, )th component of the deformation rate, and   is critical value of  based on non-Newtonian model.Under the previous conditions, the MHD boundary layer equations for steady stagnation-point flow can be written as where  and V are the velocity components in  and  directions, respectively,   =  is the straining velocity of the stagnation-point flow with  (>0) being the straining constant,  is the kinematic fluid viscosity,  is the fluid density,  =   √2  /  is the non-Newtonian or Casson parameter,  is the electrical conductivity of the fluid, and  0 is the strength of magnetic field applied in the  direction, with the induced magnetic field being neglected.The boundary conditions for the velocity components are where   =  is stretching velocity of the sheet with  (>0) being the stretching constant.
The stream function Ψ is introduced as For relations of ( 5), ( 2) is satisfied automatically and (3) takes the following form: Also, boundary conditions in (4) reduce to Now, the dimensionless variable for the stream function is implemented as where the similarity variable  is given by  = √/.Using relation (8) and similarity variable, (6) finally takes the following self-similar form: where primes denote differentiation with respect to ,  =  2 0 / is the magnetic parameter, and  = / is the velocity ratio parameter.
The boundary conditions reduce to

Analysis of Heat Transfer
For the temperature distribution in the flow field with thermal radiation, the governing energy equation can be written as where  is the temperature,  is the thermal conductivity,   is the specific heat, and   is the radiative heat flux.
The appropriate boundary conditions are where   is the constant temperature at the sheet and  ∞ is the free stream temperature assumed to be constant.Using the Rosseland approximation for radiation [48],   = −(4 * /3 1 ) 4 / is obtained, where  * is the Stefan-Boltzmann constant and  1 is the absorption coefficient.We presume that the temperature variation within the flow is such that  4 may be expanded in a Taylor's series.Expanding  4 about  ∞ and neglecting higher-order terms we get Next, the dimensionless temperature  is introduced as Using ( 8), ( 14), and the similarity variable, (13) reduces to where primes denote differentiation with respect to , Pr =   / is the Prandtl number, and  =  *  1 /4 3 ∞ is the thermal radiation parameter.
The boundary conditions for  are obtained from (12) as

Skin Friction Coefficient and Nusselt Number
The physical quantities of interest are the wall skin friction coefficient   and the local Nusselt number Nu  , which are defined as where   is the shear stress or skin friction along the stretching sheet and   is the heat flux from the sheet and those are defined as Thus, we get the wall skin friction coefficient   and the local Nusselt number Nu  as follows: where Re  =   / is the local Reynolds number.

Numerical Method for Solution
Equations ( 9) and ( 15) along with boundary conditions (10) and ( 16) are solved using shooting method [49][50][51] by converting them to an initial value problem.In this method, it is necessary to choose a suitable finite value of  → ∞, say  ∞ .The following system is set with the boundary conditions In order to integrate (20) with (21) as an initial value problem, the values of (0), that is,   (0) and (0), that is,   (0), are required, but no such values are given in the boundary conditions.The suitable guess values for   (0) and   (0) are chosen and then integration is carried out.Then, the calculated values for   and  at  ∞ = 15 (say) are compared with the given boundary conditions   (15) =  and (15) = 0 and the estimated values,   (0) and   (0), are adjusted to give a better approximation for the solution.We take the series of values for   (0) and   (0) and apply the fourth-order classical Runge-Kutta method with step-size Δ = 0.01.The previous procedure is repeated until we get the asymptotically converged results within a tolerance level of 10 −5 .

Results and Discussion
The abovementioned numerical scheme is carried out for various values of physical parameters, namely, the velocity ratio parameter (), the magnetic parameter (), the Casson parameter (), the Prandtl number (Pr), and the thermal radiation parameter () to obtain the effects of those parameters on dimensionless velocity and temperature distributions.
The obtained computational results are presented graphically in  in the absence of external magnetic field for different values of velocity ratio parameter (B) are compared with those obtained by Mahapatra and Gupta [19], Nazar et al. [20] in Table 1 in order to verify the validity of the numerical scheme used and those are found in excellent agreement.
The velocity boundary layer thickness () and thermal boundary layer thickness (  ) are, respectively, described by the equations  =   √/ and   =   √/.The dimensionless boundary layer thicknesses   and   are defined as the values of  (nondimensional distance from the surface) at which the difference of dimensionless velocity   () and the parameter  has been reduced to 0.001 and the dimensionless temperature () has been decayed to 0.001, respectively.The velocity and thermal boundary layer thicknesses for various parametric values are given in Table 2.The velocity boundary layer thickness decreases with increasing values  (both for  > 1 and  < 1) and also the thermal boundary layer thickness decreases with increasing .So, when the straining velocity rate increases compared to that of stretching velocity rate, then both the boundary layer thicknesses reduce.Actually, downward vorticity due to straining velocity causes the reduction of boundary layer thickness.Similar to velocity ratio parameter, the increase of Casson parameter  also makes the velocity boundary layer thickness thinner.So, the velocity boundary layer thickness for Casson fluid is larger than that of Newtonian fluid.Thus, the plasticity of the fluid causes the increment of the velocity boundary layer thickness.On the other hand, for  = 0.1 (<1), the thermal boundary layer thickness increases with increasing values of Casson parameter, but for  = 2 (>1) the thermal boundary layer thickness decreases with Casson parameter.Furthermore, due to magnetic field, the velocity boundary layer thickness reduces in all cases.But the thermal boundary layer thickness reduces (increases) for  = 2 ( = 0.1) with stronger magnetic field.Finally, for the Prandtl number and for radiation parameter, the thermal boundary layer thickness decreases, which is the same as that of Newtonian fluid case.The velocity and temperature profiles for various values of velocity ratio parameter  are plotted in Figures 1 and 2, respectively.Depending on the velocity ratio parameter, two different kinds of boundary layers are obtained as described by Mahapatra and Gupta [18] for Newtonian fluid.In the first kind, the velocity of fluid inside the boundary layer decreases from the surface towards the edge of the layer (for  < 1) and in the second kind the fluid velocity increases from the surface towards the edge (for  > 1).Those characters can be seen from velocity profiles in Figure 1.Also, it is important to note that if  = 1 ( = ), that is, the stretching velocity and the straining velocity are equal, then there is no boundary layer of Casson fluid flow near the sheet, which is similar to that of Chiam's [17] observation for Newtonian fluid.From Figure 2, it is seen that in all cases thermal boundary layer is formed and the temperature at a point decreases with .
The effects of Casson parameter  on the velocity and temperature fields are depicted in Figures 3 and 4. It is worthwhile to note that the velocity increases with the increase in values of  for  = 2 and it decreases with  for  = 0.1.Consequently, the velocity boundary layer thickness reduces for both values of .Due to the increase of Casson parameter , the yield stress   falls and consequently  In Figures 5 and 6, the velocity and temperature profiles are presented for several values of magnetic parameter .Similar to that of Casson parameter, due to the increase of magnetic parameter the dimensionless velocity at fixed  increases for  = 2 and for  = 0.1 the velocity decreases.Consequently, for both types of boundary layers, the thickness decreases.The Lorentz force induced by the dual actions of electric and magnetic fields reduces the velocity boundary layer thickness by opposing the transport phenomenon.Also, for  = 2, the temperature decreases with  and increases with  for  = 0.1.
The dimensionless temperature profiles () for several values of Prandtl Number Pr and thermal radiation parameter  are exhibited in Figures 7 and 8, respectively, for two values of .In both cases ( = 0.1 and 2), the temperature decreases with increasing values of Prandtl number and radiation parameter and the thermal boundary layer thickness becomes smaller in all cases.Actually, the rate of heat transfer is enhanced with Prandtl Number and radiation parameter and this causes the reduction of thermal boundary layer thickness.
The physical quantities, the wall skin friction coefficient   , and the local Nusselt number Nu  , which have immense engineering applications, are proportional to the values of (1 + 1/)  (0) and −  (0), respectively.The values of (1 + 1/)  (0) and −  (0) against the magnetic parameter  are plotted in Figures 9 and 10 for different values of .From the figures, it is observed that the magnitude of wall skin friction coefficient decreases with increasing values of velocity ratio parameter  when  < 0.1, whereas for  > 0.1 the magnitude of skin-friction increases with .The local Nusselt number (Figure 10) increases with ; that is, the heat transfer rate is enhanced with .Due to higher values of Casson parameter , the magnitude of (1 + 1/)  (0) decreases (Figure 11) for both values of  (for  > 1 as well as for  < 1).On the other hand, the value of −  (0), that is, the heat transfer (Figure 12), increases with  for  = 2 (>1) and decreases with  when  = 0.1 (<1).Finally, from those figures (Figures 9-12), it can be noticed that the wall skin friction coefficient always becomes larger when the external magnetic field is stronger and the rate of heat transfer is enhanced (reduced) with increasing magnetic parameter  for  > 1 ( < 1).

Conclusions
The MHD stagnation-point flow of Casson fluid and heat transfer over a stretching sheet are investigated taking into consideration the thermal radiation effect.Using similarity transformations, the governing equations are transformed to self-similar ordinary differential equations which are then solved using shooting method.From the study, the following remarks can be summarized.
(a) The velocity boundary layer thickness reduces with velocity ratio parameter and magnetic parameter.(b) The velocity boundary layer thickness for Casson fluid is larger than that of Newtonian fluid.(c)For Casson fluid, that is, for decrease of Casson parameter, the thermal boundary layer thickness decreases for  = 0.1 (<1) and, in contrast, for  = 2 (>1) the thickness increases.

( d )
Due to thermal radiation, the temperature inside the boundary layer decreases.(e) The magnitude of wall skin-friction coefficient decreases with Casson parameter .Nomenclature : Straining constant : Velocity ratio parameter : Stretchingconstant   : Wall skin friction coefficient   : Specific heat : Dimensionless stream function   : Dimensionless velocity  0 : Strength of magnetic field applied in the  direction  1 : Absorption coefficient : Magnetic parameter Nu  : Local Nusselt number Pr: Prandtl number : Avariable   : Yield stress of fluid : Avariable   : Radiative heat flux   : Heat flux from the sheet : Thermal radiation parameter Re  : Local Reynolds number : Temperature   : Constant temperature at the sheet  ∞ : Free stream temperature   : Straining velocity of the stagnation-point flow   : Stretching velocity of the sheet : Velocity component in  direction V: Velocitycomponentin direction : Distance along the sheet : Distance perpendicular to the sheet : A variable.Greek Symbols : Non-Newtonian/Casson parameter : Velocity boundary layer thickness   : Thermal boundary layer thickness : Similarity variable  ∞ : Finite value of    : Dimensionless velocity boundary layer thickness   : Dimensionless thermal boundary layer thickness : Thermal conductivity   : Plastic dynamic viscosity of the non-Newtonian fluid : Product of the component of deformation rate with itself   : Critical value of  : Kinematic fluid viscosity : Fluid density Ψ: Stream function : Electrical conductivity of the fluid  * : Stefan-Boltzmann constant   : Shear stress : Dimensionless temperature.

Table 1 :
Values of   (0) for several values of B with  = 0 and  = ∞ (Newtonian fluid case without magnetic field).

Table 2 :
Values of   and   for several values of B, , M, Pr, and R.