Convective condition and exponentially decaying heat source on MHD Carreau Dusty fluid over a stretching sheet filled with Darcy porous.

 

Santhosh H B^1*, Mahesha², CSK Raju³

¹Lecturer, Dept. of Mathematics, GITAM University, Bangalore-562163(Karnataka), India

²Associate Professor, Dept. of Mathematics UBDT College of Engineering, Davangere 577004 (Karnataka), India

³Assistant Professor, Dept. of Mathematics, GITAM University, Bangalore-577004 (Karnataka), India

 

ABSTRACT:

This study presents the effect of magnetic field on the flow and heat transfer of an incompressible Carreau fluid over stretching sheet with uniform suspended dust particle. The relevant governing equations are first simplified under usual boundary layer assumptions and then transformed into ordinary differential equations (ODEs) by similarity transformations. The transformed ordinary nonlinear differential equations are solved numerically by R-K with Shooting Technique. The effects of certain parameters on the dimensionless velocity, temperature profile are presented graphically and also calculated the thermo physical properties of the flow friction factor, Local Nusselt and Sherwood number. The important outcome of this study is magnetic parameter enhances the temperature profile of fluid phase but decreases the velocity of both fluid and dust phase.

 

 

 

INTRODUCTION:

The boundary layer flow and heat transfer over stretching has enormous application in industry and manufacturing processes, such as polymer extrusion, drawing of copper wires, continuous stretching of plastic films, dust entrainment in a cloud during nuclear explosion, petroleum industry, purification shock waves, also such flow occur in wide range of areas of technical importance like fluidization, flow in rocket tubes, combustion, paints spraying and more recently blood flow in capillaries.

 

Mankind et al. [1] investigated about Brickman flow of nano fluid with navier slip, viscous dissipation and connective cooling. Earlier, Alfin [2] was investigated on existence of electromagnetic-hydrodynamic waves. Chamka [3] has investigated on MHD flow of uniformly stretched vertical permeable surface in presence of heat generation/absorption and chemical reaction.

 

Raju et al. [4] have investigated about heat and mass transfer in MHD non-Newtonian bio connective flow over a rotating cone/plate with cross section diffusion. Mahamoud [5] investigated the thermal radiation on MHD flow of micro polar fluid over stretching sheet surface with variable thermal conductivity. Ibrahim et al. [6] studied the magnetic stagnation point flow and Casson nanofluid past a stretching sheet with slip and convective boundary condition. Hayat et al. [7] investigated on Boundary layer flow of Carreau fluid over a convectively heated stretching sheet.  Raju et al. [8] examined the Falkner- Skan flow of a magnetic- Carreau fluid past a wedge in the presence of cross diffusion. Ishak et al.  [9] Investigated Heat transfer over a stretching surface with variable heat flux in micro polar fluids.

 

Motivated by the above analysis, this paper outlines the numerical solutions of boundary layer flow of Carreau dusty fluid towards a stretching sheet in the presence of magnetic field, exponentially decaying space dependent internal heat source and convective boundary condition. The governing nonlinear partial differential equations are reduced into a set of nonlinear ordinary differential equations by using local similarity transformation. These nonlinear ordinary differential equations then solved numerically by Runge-Kutta based shooting process for different values of parameters of interest. Influence of governing parameters on velocity, temperature, Skin friction coefficient and local Nusselt number are analyzed and discussed in detail with help of plotted graphs and tables.

 

MATHEMATICAL FORMULATION:

A steady two-dimensional, incompressible, magneto hydrodynamic boundary layer flow of a Carreau fluid with suspended uniform size dust particles over a stretching sheet is considered in the present study. The stretching sheet is coincident with the axis and the flow confined to  . The sheet is stretched with the velocity   because of the application of two equal and opposite forces along the axis and the origin is considered fixed. Magnetic field of strength is imposed in a direction transverse to the stretching plate. Electrical conductivity of the fluid is considered to be small hence the induced magnetic field is neglected. Convective surface temperature is characterized by and heat transfer coefficient by  .The temperature  at the surface of the sheet is considered to be more than the ambient fluid temperature  i.e.   The extra stress tensor for Carreau fluid given by Raju and Sandeep [11-12].

                                                                                                                            (1)

Here is the extra stress tensor, is the zero shear rate viscosity,   is the characteristic time constant,  is the dimensionless power law index,   is the infinite shear rate and is defined as

                                                                                                                           (2)

Here  is the second invariant strain tensor.

By considering the above assumptions the governing boundary layer equations are given by [2].

                                                                                                                                                          (3)           (4)

                                                                                                                                                      (5)

                                                                                                                                       (6)

               (7)

                                                                                                               (8)

Where  and  are the velocity components along theand direction of the fluid and dust particle phase ,  and  are the density of the fluid and dust particle phase, and are the mass and number density of the dust particles per unit volume,  is the dynamic viscosity of the fluid,is the electrical conductivity of the fluid, is the uniform magnetic field,  is the stokes resistance (drag coefficient) and  is the radius of the dust particle.and  are the temperature of the fluid and dust particle phase,  and are the specific heat of fluid and dust particles, represents the thermal equilibrium time i.e. the time required by the dust cloud to adjust its temperature to that of fluid,  represents the relaxation time of the dust particles i.e. the time required by the dust particle to adjust its velocity relative to the fluid.  is the thermal conductivity of the fluid.

Corresponding boundary condition for the physical are given by

                                                                                          (9)

To convert the governing equations into a set of similarity equations, we introduce the following transformations as,

                                                                               (10)

 Where a prime denote differentiation with respect to . Continuity equations (3) and (5) are identically satisfied. Equations (4), (6)-(10) are transformed as follows.

               (11)

 

                                                                (12)

               (13)

                                                                                                              (14)

The boundary conditions defined in equation (8) will be transformed to:

                                                                                (15)

 the mass concentration parameter of dust particles,   is the magnetic parameter,   is the relaxation time of the dust particles, i.e, the time required by a dust particle to adjust its velocity relative to the fluid, is the Weissenberg number, is the fluid particle interaction parameter. Distinctive measures of practical interest are skin friction coefficient and local Nusselt number .  Which are defined as;

                                                                                                              (16)

Where the surface shear stress  , surface heat flux and surface mass flux  are given by;

                                                                    (17)

Using the non-dimensional variables, we obtain

                                                                                   (18)

Whereis the local Reynolds number.

 

RESULTS AND DISCUSSION:

For analysing the approximate solutions of velocity as well as temperature fields, the non-linear ordinary differential Eqs. (12)- (15) with reference to the boundary conditions (15) are solved numerically with the assistance of Runge-Kutta and Newton’s methods. For numerical solutions we have considered the values of non-dimensional parameterthese values are taken constant in this study besides the varied parameters as mentioned in the figure. The same parameters are used in this study to examine skin friction coefficient and heat transfer rate with the assistance of Table 1and 2.

Fig.1and2 depicts the effect of magnetic parameter on velocity of fluid and dust phase and temperature of fluid and dust phase, by these graph it is clear that Magnetic parameter enhances the temperature but decrease the velocity of both fluid and dust phase. Fig.3 depicts the effect of biot number on temperature profile from this it is clear that the biot number increases the temperature profile of fluid phase. Fig.4and5 depicts the effect of porous on velocity of fluid and temperature of fluid, from these two figures it is clear that the effect of porous decreases the velocity of fluid and dust phase and increase the temperature profile of fluid phase. Fig 6and7 depicts the effect of Weissennberg number on velocity of fluid and temperature profile, by these two graphs it is clear that The Weissennberg number reduces the temperature profile and enhances the velocity of both fluid and dust phase.Fig.8 depicts the effect of heat generation parameter on temperature profile from figure it is clear that heat generation parameter reduces the temperature profile of fluid phase. Fig 9 and 10 depicts the effect of power law index on velocity of fluid and temperature profile by these graphs it is clear that the power law index enhances the temperature profile and reduces the velocity of fluid and dust phase.

 

 

Fig - 1: Velocity profile for different values of magnetic parameter.

 

Fig-2: Temperature profile for different values of magnetic parameter.

 

Table-1 Comparison of the results for local Nusselt number with.

Pr	Ishak[9]	Krupa Lakshmi et al.[10]	Present Study
0.72	0.8086	0.808630	0.8075
1.0	1.0000	1.0000	1.0000
3.0	1.9237	1.92367	1.9146
10.0	3.7207	3.72067	3.6516
100	12.2941	12.294087	12.2936

 

Table-2: The variations of friction factor and local Nusselt number for various values of non-dimensional governing parameters.

 

 Fig-3: Temperature profile for different values of biot number.

 

Fig-4: Velocity profile for different values of porous.

 

Fig-5: Temperature profile for different values of porous.

 

Fig-6: Velocity profile for different values of Weissenberg

Number.

 

 

Fig - 7: Temperature profile for different values of Weissenberg number

 

Fig-8: Temperature profile for different values of heat generation parameter

 

Fig-9: Velocity profile for different values of power law index.

 

Fig-10:Temperature profile for different values of power law index.

 

 

 

CONCLUSIONS:

This study reports the flow and heat transfer characteristics of MHD Carreau Dusty fluid over a stretching sheet with exponentially decaying heat generation with a convective boundary condition. The boundary value problem was solved numerically with the assistance of Runge-Kutta and Newton’s methods.  The conclusions of this study are as follows:

• Magnetic parameter enhances the temperature of fluid phase but decreases the velocity of both fluid and dust phase.

• By the increase the value of heat generation parameter the temperature profile decreases.

• The power law index enhances the temperature and reduces the velocity of fluid and dust phase.

• The Weissennberg number reduces the temperature profile and enhances the velocity of both fluid and dust phase.

 

REFERENCES:

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11.     C.S.K. Raju, K.R. Sekhar, S.M. Ibrahim, G. Lorenzini, G. Viswanatha Reddy, E. Lorenzini, Variable viscosity on unsteady dissipative Carreau fluid over a truncated cone filled with titanium alloy nanoparticles, Contin. Mech. Thermodyn.; 2017. doi: 10.1007/s00161-016-0552-8.

12.     Raju, C. S. K., Hoque, M. M., Anika, N. N., Mamatha, S. U., and Sharma, P. Natural convective heat transfer analysis of MHD unsteady Carreau nanofluid over a cone packed with alloy nanoparticles. Powder Technology, 2017; 317, 408-416.

 

 

 

 

Accepted on 29.11.2017      ©A&V Publications All right reserved