RESEARCH ARTICLE

 

Blasius and Sakiadis flow of magneto hydrodynamic Tangent hyperbolic fluid with exponentially decaying heat source or sink

 

P. Priyadharshini ¹, S.Vijay Babu ², C.S.K. Raju ³

^1,2 Department of Mathematics, Sri Shakthi Institute of Engineering and Technology, Coimbatore-641 005, India.

³Department of Mathematics, GITAM University, Bangalore-562123, India.

 

ABSTRACT:

In this article, we have studied the flow and heat transfer in magneto hydrodynamic Tangent hyperbolic fluid with exponentially heat source/sink. The partial differential equations governing the problem have been reduced by similarity transformations into the ordinary differential equations. Numerical solutions are transferred out by using Runge-Kutta Shooting technique. The effects of various governing parameters on the flow quantities are demonstrated graphically. The solution depends on various interesting parameters including the friction factor coefficient and local Nusselt numbers for the Sakiadis and Blasius flow cases. It is determined that the rate of heat transfer is extremely high in Blasius flow case when compared with Sakiadis flow case.

 

 

 

LITERATURE REVIEW:

Due to tremendous practical significance as well as to meet the demands of the modern technological needs, the Blasisus and Sakiadis flow problems, which involve the combination and interaction of various kinds of phenomena, have attracted the attention of many researchers in this area. A comparable problem transpires when the plate passes with constant velocity in a calm fluid. This problem was employed for the first time by Sakiadis [2]. Tsou et al. [3] recorded that the Sakiadis flow is corporally realizable authenticating his data. The problem of a moving plate inside a free stream (both Blasius and Sakiadis flow) with suction or injection has been examined by Chen [4].

 

The investigation of Sparrow et al. [5] is associated to the convection flow about an inclined surface in which the consolidated compelled and free boundary layer problem has been explained using the similarity method. For a likely surface, the buoyancy force generating motion has an ingredient in both the tangential and normal directions.

 

This produces a pressure gradient transversely the boundary layer, pointing to a theoretical interpretation more complex than that for a horizontal or a vertical surface.

 

Erickson et al. [6] stretched the work to bear blowing or suction at the stretched sheet surface on a continuous solid surface under constant speed and reviewed its results on the heat and mass transfer in the boundary. layer. MHD nano-fluid flow over an exponentially stretching sheet with radiative dual solutions of heat generation has been considered by Sandeep et al. [7]. Heat transfer and magnetic range properties of a Maxwell fluid due to Flakner-Skan flow have been illustrated by Qasim et al. [8]. Gaffar et al. [9] studied heat transfer boundary layer slip flow of a non-Newtonian fluid from a cylinder stimulated to Keller box finite-difference method. An announcement on the Blasius flow of a non-Newtonian power-law fluid in a constant transverse magnetic field has been described by Khan et al. [10]. Numerous specimens have been tested by many authors to study the fundamental papers in References [11-15]

 

BASIC EQUATION:

We consider the steady two-dimensional boundary layer flow over a fixed Blasius and Sakiadis Maxwell fluid with exponentially decaying heat generation or absorption. The flow is assumed to be incompressible and laminar. The flow is considered in x-direction and y-axis is normal to it.  It is assume that the flow takes place at. It is also assumed that the constant temperature of the flat plate isand that of the ambient fluid is. Under the boundary layer approximations, the fact is that the flow is of zero pressure gradients. Based on the assumptions the basic equations

                                                                                                                                                                    (1)

                                                                            (2)

                                                                                              (3)

We assume that these equations are subject to the boundary conditions

i)         Blasius problem                                                                                                              (4)

ii)       Sakiadis problem                                                                                                          (5)

Where the velocity components along the x and y-directions, is the Casson fluid parameter, is the electric conductivity, is the permeability parameter, is the acceleration due to gravity, is the thermal expansion coefficient, is thermal diffusivity and is the heat generation coefficient. U is the constant velocity of the free stream (inviscid flow) or that of a moving flat plate. The boundary conditions for the energy equation (3) are

                                                                                                                        (6)

 

We look for a similarity solution of Eqs. (1-3) with the boundary conditions (4), (5) and (6) of the following form:

                                                                (7)

 

Where is the kinematic viscosity of the fluid fraction and is the stream function that is defined in the usual way as. On substituting Eqs. (7) into (2) and (3), we obtain the following uncoupled ordinary differential equations:

                                                       (8)

                                                                                                                                   (9)

 

Subject to the boundary conditions

i)         Blasius problem                                                                                          (10)

ii)       Sakiadis problem                                                                                       (11)

and the energy conditions are

                                                                                                                                                        (12)

 

 

Where a prime denotes the differentiation with respect to the similarity variableand is the Prandtl number,is heat generation parameter, is the magnetic field parameter,is porosity parameter and is the buoyancy parameter.

Quantities of practical interest in this study are the skin friction coefficientand local Nusselt number, which are defined as

                                                                                                                           (13)

whereis the skin friction or the shear stress andis the heat flux from the plate which are given by

                                                                                                                     (14)

Substituting Eqs. (7) into (13) and (14), we obtain

                                                                                                                   (15)

 

whereis the local Reynolds number.

 

RESULTS AND DISCUSSION:

The non-dimensional governing equations (8) and (9) subjected to boundary conditions (10), (11) and (12) are solved numerically employing the shooting technique. The numerical solutions are found for several values of non-dimensional governing parameters on the Blasius and Sakiadis flow of magneto hydrodynamic Tangent hyperbolic fluid with exponentially decaying heat source or sink.

 

It is visible from Fig. 1 that Porosity parameter subsides with momentum boundary layer thickness both the Blasius and Sakiadis flow. It is evident from Fig. 2 that the effect of Porosity parameter K on temperature fields. We have noticed that rise in the thermal  boundary layers for positive values of Blasius and Sakiadis flow in the flow characterization. It is obvious from Figs. 3 and 4 show that the impact of magnetic field parameter M decreases the thickness of momentum boundary layer due to the domination of drag force and intensify the thermal boundary layer due to the matter of Tangent hyperbolic fluid. It is clear from Figs. 5 and 6 that heat generation parameter Q_H reduce the velocity and temperature profiles.

 

 

Fig.1 The effects of K on velocity profiles

 

Fig.2 The effects of K on temperature profiles

 

 

Fig.3 The effects of M on velocity profiles

 

Fig.4 The effects of M on temperature profiles

 

 

Fig.5 The effects of Q_H on velocity profiles

 

Fig.6 The effects of Q_H on temperature profiles

 

CONCLUSION:

We found that the flow and heat transfer in magneto hydrodynamic Tangent hyperbolic fluid with exponentially heat source/sink. The Numerical solutions are worked out by using Runge-Kutta based Shooting technique. The effects of various governing parameters on the flow quantities are demonstrated graphically. Also computed the friction factor coefficient and local Nusselt number in terms of heat transfer for the Sakiadis and Blasius flow cases. It is found that the rate of heat transfer and friction between the fluid phases to boundary phase is very high in Blasius flow case when compared with Sakiadis flow case.

 

REFERENCES:

1.        H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Math. Phys. 1908; 56: 1–37.

2.        B.C. Sakiadis, Boundary-layer behaviour on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow, AIChE J.1961; 7: 26–28.

3.        F.K. Tsou, E.M. Sparrow, R.J. Goldstein, Flow and heat transfer in the boundary layer on a continuous moving surface, Int. J. Heat Mass Transfer. 1967; 10: 1 281–288.

4.        C.H. Chen, Forced convection over a continuous sheet with suction or injection moving in a flowing fluid, Acta Mech. 1999; 138: 1–11.

5.        E.M. Sparrow, R. Eichhorn, J.L. Grigg, Combined forced and free convection in a boundary layer, Phys. Fluids  1959; 2: 319–320.

6.        L.E. Erickson, L.T. Fan, V.G. Fox, Heat and mass transfer on a moving continuous flat plate with suction or injection , Ind. Eng. Chem. vol.   1966; 5: 19-25.

7.        S. Sandeep and C. Sulochana, Dual solutions of radiative MHD nanofluid flow over an exponentially stretching sheet with heat generation/absorption, Applied Nanoscience. 2016; 1: 131-139.

8.        M.Qasim and S.Noreen, Falkner-Skan flow of a Maxwell fluid with heat transfer and magnetic field, International journal of Engineering Mathematics, 2013; 7:  13-20.

9.        S. A. Gaffar, V. Ramachandra Prasad and E. Keshava Reddy, Computational study of non-Newtonian Eyring-powell fluid from a horizontal circular cylinder with Biot number effects, Int. J. Math. Archv. 2015;  6: 114-132.

10.      W.A. Khan, R. Culham and O.D. Makinde, Hydromagnetic blasius flow of power‐ law nano-fluids over a convectively heated vertical plate, The Canadian Journal of Chemical Engineering, 2015; 93(10): 1830-1837.

11.     C.S.K. Raju, N.Sandeep, Heat and mass transfer in MHD non-Newtonian bioconvection flow over a rotating cone/plate with cross diffusion, Journal of Molecular Liquids, 2016; 215: 115-126.

12.     C.S.K. Raju, N.Sandeep, A comparative study on heat and mass transfer of the Blasius and Falkner-Skan flow of a bio-convective Casson fluid past a wedge, European Physical Journal Plus, 2011; 131: 405-411.

13.     K. Gangadhar, Radiation, Heat Generation and Viscous Dissipation Effects on MHD Boundary Layer Flow for the Blasius and Sakiadis Flows with a Convective Surface Boundary Condition, Journal of Applied Fluid Mechanics, 2015;  5598(3): 562 -570.

14.     K. R. Sekhar, G. Viswanatha Reddy and S. V. K. Varma, Mixed convection coquette flow of a nano-fluid through a vertical channel, Elixir International Journal, 2016; 99: 43237-43241.

15.     C.S.K. Raju, K. R. Sekhar, S. M. Ibrahim, G. Lorenzini, G. Viswanatha Reddy and E. Lorenzini, Variable viscosity on unsteady dissipative Carreau fluid over a truncated cone filled with titanium alloy nano-particles, Continuum Mech.Thermodyn., 2017; DOI 10.1007/s00161-016-0552-8.

 

 

 

Accepted on 28.09.2017      ©A&V Publications All right reserved

Research J. Science and Tech. 2017; 9(3): 484-488.