Magnetic Field Effect on Tangent Hyperbolic Fluid towards Porous Plate with Suction/Injection

 

L. Nagaraja^1,2*, M. Sudhakar Reddy¹, M. Suryanarayana Reddy²

¹Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle, Andhra Pradesh, India

²Department of Mathematics, JNTUA College of Engineering, Pulivendula, Kadapa, Andhra Pradesh, India

 

ABSTRACT:

Present study investigates the convective flow and transfer of heat in non-Newtonian Tangent Hyperbolic fluid (incompressible) from a vertical porous plate with suction/injection effects. The transformed governing equations are solved subject to physically appropriate boundary conditions using Keller Box (finite-difference) scheme. The influence of emerging non-dimensional parameters discuss graphically, to be precise the Weissenberg number , the power law index , magnetic parameter , Prandtl number , suction/injection parameter  and dimensionless tangential coordinate in the boundary layer regime are examined the effects of these parameters on surface heat transfer rate and local skin friction are also investigated tabular form.

 

 

1.      INTRODUCTION:

A significant branch of the non-Newtonian flowing models is the tangent hyperbolic fluid model. The tangent hyperbolic fluid is used far for different laboratory experiments. Friedman et al. (1) have used the tangent hyperbolic fluid model for large-scale magneto-rheological fluid damper coils. The number of researchers worked on non-Newtonian fluids. Examples of such fluids include coal-oil slurries, grease, shampoo, paints, custard, cosmetic products, and physiological liquids. The classical equations employed in simulating Newtonian viscous flows i.e. the Navier–Stokes equations fail to simulate a number of characteristics of non-Newtonian fluid. Recent investigations have implemented, Muhammad Naseer et al., (2) analyzed the boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder. MHD on tangent hyperbolic fluids also investigated M. Ali Abbas et. al, (3), Recently Subbarao et al (4) discussed Non-Similar Computational Solution for Boundary Layer Flows of Non-Newtonian Fluid from an Inclined Plate.

 

A motivating non-Newtonian model developed for chemical engineering systems is the Tangent Hyperbolic fluid representation. This rheological model has definite advantages over the other non-Newtonian formulations, including simplicity, physical robustness and ease of computation. In addition it is deduced from kinetic theory of liquids rather than the empirical relation. Several communications utilizing the Tangent Hyperbolic fluid model have been presented in the logical literature. There is no single non-Newtonian model that exhibits all the properties of non-Newtonian fluids. Among several non-Newtonian fluids, hyperbolic tangent model is one of the non- Newtonian models presented by Nadeem and Akram (5) (2011) investigated the peristaltic flow of a MHD hyperbolic tangent fluid in a vertical asymmetric channel with heat transfer. Akbar et al. (6) (2013) analyzed the numerical solutions of MHD boundary layer flow of tangent hyperbolic fluid on a stretching sheet.  Nadeem et al. (7) (2009) made a detailed study on the hyperbolic tangent fluid in an asymmetric channel.

 

With above analysis in mind, we are interested in analytical approximation of convective flow (non-Newtonian) in vertical porous plate. The reduced highly nonlinear partial differential equations are solved with help of accurate Keller-Box Technique (finite difference) (5-10). The impact of various pertinent parameters, like Prandtl number (Pr), Weissenberg parameter (We), Power law index (m), Magnetic parameter (M) and Suction/Injection parameter are plotted and discussed.

 

2.      MATHEMICAL FORMULATION:

Consider a steady, two dimensional free convective heat transfers along the vertical porous plate embedded in non-Newtonian Tangent Hyperbolic fluid. The x-coordinate (tangential) is measured along the surface of the porous plate from the lowest point and the y-coordinate (radial) is directed perpendicular to the surface. The equations for continuity, momentum and energy can be written as follows:

 

A steady, laminar, two dimensional boundary layer flows and heat transfer of an incompressible tangent hyperbolic fluid over a vertical porous plate. Both the plate and Tangent Hyperbolic fluid are maintained primarily at the equivalent temperature. straight away they are raised to a temperature , the ambient temperature of the fluid which remains unchanged.

 

                                                                                                                                                             

                                                               (2)         

                                                                                                                                            

 

where u and v are the velocity components in the x- and y-directions respectively, ν=μ/ρ is the kinematic viscosity of the Tangent Hyperbolic fluid, β is the coefficient of thermal expansion, α  is the thermal diffusivity, β is the temperature, and ρ is the density of the fluid. The Tangent Hyperbolic fluid model therefore introduces a mixed derivative (second order, first degree) into the momentum boundary layer Eq. (2). Boundary conditions are:

                                                                                                              

Here  is the free stream temperature, k is the thermal conductivity, is the convective heat transfer coefficient, and  is the convective fluid temperature. The stream functionis defined by and , and therefore, the continuity equation is automatically satisfied.

In order to provide the governing equations and the boundary conditions in dimensionless form, the following non-dimensional quantities are introduced.

                                                                                 (5)

In view of the transformations defined in (5). The boundary layer equations (2)-(3) are reduced to the following nonlinear, dimensionless partial differential equations for momentum, energy for the regime.

                                                       (6)       

                                                                                                                (7)

The transformed dimensionless boundary conditions are as follows:

                                                                                                             

The skin friction coefficient (shear stress at the sphere surface) and Nusselt number (heat transfer rate) can be defined using the transformations described above with the following expressions.

 

                                                                                              

                                                                                                                                     

 

3.      NUMERICAL SOLUTION WITH KELLER-BOX IMPLICIT METHOD:

The Keller-Box implicit difference method is implemented to solve the nonlinear boundary value problem defined by eqns. (7)–(8) with boundary conditions (10).This technique despite recent developments in other numerical methods as elaborated by Keller, 1978. The Keller Box Scheme comprises four stages.

 

1.      Decomposition of the Nth order partial differential equation system to N first order equations.

2.      Finite Difference Discretization.

3.      Quasilinearization of Non-Linear Keller Algebraic Equations and finally.

4.      Block-tridiagonal Elimination solution of the Linearized Keller Algebraic Equations

 

4.      RESULTS AND DISCUSSIONS:

In order to obtain a physical approaching into the problem, a representative set of numerical results are presented in Figs. 1-5. The numerical problem comprises of two independent variables , two dependent fluid dynamic variables  and thermo-physical and body force control parameters, viz., We, n, Pr, M, f_w. The following default parameter values i.e., We = 0.3, n = 0.1, Pr = 1.0, M = 1.0, f_w = 0.3 are discussed.

 

 

Fig. 1(a) Velocity profiles for different values of Pr

 

Fig. 1(b) Temperature profiles for different values of Pr

 

Fig. 2(a) Velocity profiles for different values of We

 

Fig. 2(b) Temperature profiles for different values of We

 

Fig. 3(a) Velocity profiles for different values of n

 

Fig. 3(b) Temperature profiles for different values of n

 

Fig. 4(a) Velocity profiles for different values of M

 

Fig. 4 (b) Temperature profiles for different values of M

 

 

Fig. 5(a) Velocity profiles for different values of fw

 

Fig. 5(b) Temperature profiles for different values of fw

 

Figure 1(a)-1(b) depict the velocity , temperature  distributions for various values of Prandtl number Pr. It is observed that an increase in the Prandtl number significantly decelerates the flow i.e., velocity and temperature decreases. Figure 2(a)-2(b) shows the velocity, temperature  distributions, the influence of Weissenberg number, absorbed that velocity increases, temperature and concentration are decreases.

 

Figures 3(a) - 3(b) illustrates the effect of the power law index, n, on the velocity  and temperature  distributions through the boundary layer regime. Increasing n velocity is increased. Conversely temperature is consistently reduced with increasing values of n.

 

Figures 4(a)-4(b) the dimensionless velocity, temperature for various values of magnetic parameter M are shown. Fig. 4(a) represents the velocity profile for the different values of magnetic field parameter M. It is observed that velocity of the flow decreases significantly with increasing values of magnetic parameter M. In Fig. 3(b), the temperature distribution increases with increasing magnetic values.

 

Figure 5(a)-5(b) this plots indicates the influence of Suction/ Injection parameter. We observed that increase the Suction/ Injection parameter throughout the region decreases the velocity  and temperature.

 

5.      CONCLUSIONS:

Mathematical solutions have been obtained for the convection heat transfer boundary layer flow through the vertical porous plate the presence of Suction/Injection effects, using the local non-similarity finite difference method.

1.      Increasing Weissenberg number, decreases velocity, whereas increases temperature.

2.      Increasing power law index, n, increases velocity, whereas, decreases temperature.

3.      Increasing Suction/Injection f_w, decreases velocity, temperature.

4.      Increasing magnetic parameter M, increases velocity, whereas, decreases temperature.

 

6. REFERENCES:

1.       A.J. Friedman, S.J. Dyke, B.M. Phillips, Over-driven control for large-scale MR dampers, Smart Mater. Struct. 22 (2013) 045001, 15pp.

2.       Muhammad Naseer, Muhammad Yousaf Malik, Sohail Nadeem, Abdul Rehman, 2014, “The Boundary Layer Flow of Hyperbolic Tangent Fluid Over a Vertical Exponentially Stretching Cylinder” Alexandria Engineering Journal, 53, 747–750. http://dx.doi.org/10.1016/j.aej.2014.05.001

3.       Ali Abbas. M., Bai. Y.Q., Bhatti. M.M., Rashidi. M.M., 2016, “Three Dimensional Peristaltic Flow of Hyperbolic Tangent Fluid in Non-uniform Channel Having Flexible Walls” Alexandria Engineering Journal, 55, 653–662. http://dx.doi.org/10.1016/j.aej.2015.10.012

4.       Subba Rao. A., Ramachandra Prasad V., Nagendra. N, Bhaskar Reddy. N and Anwar Bég O. 2016, “Non-Similar Computational Solution for Boundary Layer Flows of Non-Newtonian Fluid From an Inclined Plate with Thermal Slip”, J. Applied Fluid Mechanics, 9, 795-807.

5.       Nadeem, S. and S. Akram (2011). Magneto hydrodynamic peristaltic flow of a hyperbolic tangent fluid in a vertical asymmetric channel with heat transfer. Acta Mech. Sin. 27(2), 237–250.

6.       Akbar, N. S., S. Nadeem, R. U. Haq and Z. H. Khan (2013). Numerical solution of Magneto hydrodynamic boundary layer flow of tangent hyperbolic fluid towards a stretching sheet. Indian J. Phys 87(11), 1121-1124.

7.       Nadeem, S. and S. Akram (2009). Peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. ZNA 64a, 559–567.

8.       Subba Rao. A, V.R. Prasad, N. Bhaskar Reddy, O. Anwar Beg, 2015, “Modelling Laminar Transport Phenomena in a Casson Rheological Fluid from a Semi-Infinite Vertical Plate With Partial Slip”, Heat Transfer-Asian Research, 44(3), 272-291, http://10.1002/htj.21115

9.       Subba Rao. A, V.R. Prasad, N. Bhaskar Reddy and O. Anwar Beg,2015, “Modelling Laminar Transport Phenomena In A Casson Rheological Fluid From An Isothermal Sphere With Partial Slip”, Thermal science, 19(5), 1507-1519; http://10.2298/TSCI120828098S

10.     Subba Rao. A, V.R.Prasad, K. Harshavalli and O. Anwar Beg, 2016, “Thermal Radiation Effects on Non-Newtonian Fluid in a Variable Porosity Regime with Partial Slip”, J. Porous Media, 19(4), 313-329.

11.     Subba Rao. A, V. R. Prasad, V. Nagaradhika, O. Anwar Beg,2017, “Heat Transfer in Viscoplastic Boundary Layer Flow from a Vertical Permeable Cone with Momentum and Thermal Wall Slip: Numerical Study”, Heat Transfer Research; (In Press).

12.     L. Nagaraja, M. Sudhakar Reddy, Suryanayana Reddy, “Magneto hydrodynamic effects on non-Newtonian Eyring-Powell fluid from a circular cylinder with Soret and Dufour effects”, International Journal of Mathematical Archives, 8(6),153-166, 2017

13.     Subba Rao A., V. R Prasad , O. Anwar Beg and M. Rashidi, 2017, “Free Convection Heat and Mass Transfer of a Nanofluid Past a Horizontal Cylinder Embedded in a Non-Darcy Porous Medium,  Journal of Porous Media; (In Press).

 

 

 

 

 

Accepted on 04.12.2017      ©A&V Publications All right reserved