Heat Transfer Analysis of MHD non-Newtonian fluid over a Horizontal Circular Cylinder with Biot number effect

 

K. Madhavi^1,2, N. Nagendra¹*, G.S.S. Raju², V. Ramachandra Prasad¹

¹Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle-517325, A.P.India.

²Department of Mathematics, JNTUA College of Engineering Pulivendula, Pulivendula-516390, A.P. India.

 

ABSTRACT:

In the proposed work deals with the heat transfer analysis of Magneto hydrodynamic third grade fluid flow over a horizontal circular cylinder with Biot-number effect. Using the Non – Similarity transformations, the governing partial differential equations are converted into ordinary differential equations and these equations are solved numerically by using an implicit finite difference Keller - Box method. The influence of emerging non-dimensional parameters, namely the third grade fluid parameter, the material fluid parameters, Magneto hydrodynamic parameter, Biot number effect and the Prandtl number, on velocities and temperatures are evaluated in the boundary layer regime in detail. Increasing the third grade fluid parameter and material fluid parameters are found to increase the velocity and opposite behavior is observed on temperature profiles. Increasing the Magneto hydrodynamic parameter is found to decreases the velocity and increases the temperature. An increasing the Biot number whereas the velocity and temperature are enhances and increasing the Prandtl number, the velocity and temperature profiles are in depresses .The study is relevant to chemical materials processing applications.

 

 

INTRODUCTION:

Many important industrial fluids are Non-Newtonian in their flow characteristics. These include paints, various suspensions, glues, printing inks, food materials, soap and detergent slurries, polymer solutions and many others. Because such fluids have more complicated equation that relate the stress to the velocity gradient than is the case with Newtonian fluids, new branches in the fields of fluid mechanics and heat transfer are developed. Another important characteristic of such fluids, because of their large apparent viscosities, is that they have a tendency towards low Reynolds and Grashof numbers and high Prandtl numbers.

 

Thus laminar flow situations are encountered more often in practice than with Newtonian fluids. The non-Newtonian fluids can in turn be divided into purely viscous and visco-elastic fluids. The purely viscous time independent fluids are defined as those whose shear stress depends only upon some function of the shear rate, and sometimes an initial yields stress. Visco-elastic fluids are those which possess properties of both viscosity and elasticity. Notable non-Newtonian models which have emerged as useful in modern engineering materials processing include Giesekus fluids [1], viscoelastic Maxwell upper-converted models [2] and Casson models [3].

 

These models capture the different fluid mechanical characteristics of non-Newtonian liquids. The differential type fluid model has the simplest sub-class known as second grade fluid, which describes the normal stress differences but cannot predict the shear thinning / thickening phenomena. Third grade fluid is a subclass of differential type non-Newtonian fluid. However, the third grade fluid model is capable of predicting both the normal stresses and the shear thinning / thickening phenomena. Appreciable progress has been made on the mechanics of third grade fluid in recent times. This is due to its numerous applications in geology, petro-chemical engineering, pharmaceutical industries, lubricants, confectioneries etc. Some studies on the third grade fluid can be seen in references [4-8]. Hayat and Mustafa [9] studied the analysis of a third grade fluid over a stretching surface with heat and mass transfer. Yurusoy and Pakdermirli [10] analyze heat transfer flow of third grade fluid in a pipe. Massoudi and Chiristie [11] studied the fully developed flow of an incompressible, thermo dynamically compatible fluid of grade three in a pipe. Anwar Beg et al [12] explained the numerical solution presented for the natural convective dissipative heat transfer an incompressible third grade non-Newtonian fluid flowing past an infinite porous plate embedded in Darcy – Forcheimer porous medium. Oluwole Daniel Makinde [13, 14] explained to investigate the thermal criticality for a reactive third-grade liquid flowing steadily between two parallel isothermal plates. BikashShaoo [15] analysis the two-dimensional stagnation point flow and heat transfer analysis of the thermodynamically compatible third grade. The study of magneto hydrodynamics (MHD) plays an important role in agriculture, engineering and petroleum industries. MHD has won practical applications, for instance, it may be used to deal with problems such as cooling of nuclear reactors by liquid sodium and induction flow water which depends on the potential differencing the fluid direction perpendicular to the motion and goes to the magnetic field. For instance, Hayat et al. [16] have examined MHD on the boundary layer of axisymmetric flow of third grade fluid over stretching cylinder. Zeeshan Khan et. al [17] Effect of thermal radiation and MHD on non-Newtonian third grade fluid in wire coating analysis with temperature dependent viscosity. Mhd flow and heat transfer in a williamson fluid from a vertical permeable cone with thermal and momentum slip effects and Biotnumber effect on hydromagnetic convection boundary layer flow of a Williamson non-Newtonian fluid was analysedAmanulla et al. [18]- [19].

 

The objective of the present study was to investigate the” heat transfer analysis of mhd non-newtonian flow over a horizontal circular cylinder with biot number effect”. Numerical solutions for the velocity and temperature distributions are gained by using a Keller-Box finite difference method.

 

MATHEMATICAL MODEL:

The regime under investigation is illustrated in Fig. 1. Represents steady, incompressible hydromagnetic Third grade fluid boundary layer flow and heat transfer from a cylindrical body under radial magnetic field is considered.

 

 

Figure 1: Physical model and coordinate system

 

Here x-coordinate is measured along the circumference of the horizontal cylinder from the lowest point and the y-coordinate is measured normal to the surface, with ‘a’ denoting the radius of the horizontal cylinder. is the angle of orientation of the y-axis with respect to the vertical. The gravitational acceleration, g acts downwards. Both the horizontal cylinder and the fluid are maintained initially at the same temperature. Instantaneously they are raised to a temperature  i.e. the ambient temperature of the fluid which remains unchanged.

The corresponding velocities in the x and y directions are u and v respectively. The governing conservation equations can be written as follows:

                                                                                                                                                               (1)

                                              (2)

                                                                      (3)  

Where u and v are the velocity components in x- and y- directions respectively. The boundary conditions are prescribed at the surface and the edge of the boundary layer regime, respectively as follows:

 

                                                                                                  (4)

Here is the free stream temperature, is the thermal conductivity, is the Convective heat transfer coefficient and  is the convective fluid temperature.

 

The stream function is defined by and, and therefore, the continuity equation (1) is automatically satisfied. In order to render the governing equations and the boundary conditions in dimensionless from, the following non-dimensional quantities are introduced.

                                                                                               (5)

In view of the transformation defined in eqn. (8) the boundary layer equations (5) - (6)  are reduced to the following coupled, nonlinear, dimensionless partial differential equations for momentum and energy for the regime:

                        (6)

                                                                                                (7)

 

Where E₁, E₂ and A are fluid parameters, M is the magnetic parameter, Pr is the Prandtl number and F is the radiation parameter .These quantities are defined as follows.

,,,   and

The transformed dimensionless boundary conditions are:

                                                                                                      (8)

 

Here primes denote the differentiation with respect to

The skin friction coefficient (shear stress at the cylinder surface) and Nusselt number (heat transfer ate) can be defined using the transformation described above with the following expressions

                                                                                                                                        (9)

                                                                                                                                                                                                                         (10)

 

NUMERICAL SOLUTION:

In this study, the efficient Keller-Box implicit difference method has been employed to solve the general flow model defined by equations (6) – (7) with boundary conditions (8).This method was originally developed for low speed aerodynamic boundary layers and this system is developed by Keller [20].and This technique has been described briefly in Cebeci and Bradshaw [21].This method has been employed in a diverse range of industrial multi-physical fluid dynamics problems. This method remains among the most powerful, versatile and accurate computational finite difference schemes employed in modern viscous fluid dynamics simulations.

 

RESULTS AND DISCUSSIONS:

 

Comprehensive solutions have been obtain and are presented in Figs.2-7. The numerical problem comprises two independent variables, two dependent fluid dynamic variables  and 7 thermo-physical and body force control parameters, namely are prescribed. The following default parameter values i.e. are considered throughout the problem. Due to space problemherei choose some important parameters are discussed through the graphs.

 

 

Fig -2: Effect of A on Velocity Profiles

 

 

Fig -3: Effect of A on Temperature Profiles

Figs. 2 to 3 explain the influence of Third grade fluid parameter (A) on velocity and temperature sketches, it is observed that, the boundary layer flow is enhanced with increasing Third grade fluid parameter A, a week development in the velocity throughout the boundary layer and opposite behaviour is detected in temperature profiles.

 

Fig -4: Effect of M on Velocity Profiles

 

 

 

Fig -5: Effect of M on Temperature Profiles

 

Figs. 4 to 5 demonstrates the effect of the Magneto hydrodynamic parameteron the velocity and the temperature. When increases the Magneto hydrodynamic parameter with declines the velocity and rises the temperature profiles.

 

 

Fig -6: Effect of on Velocity Profiles

 

Fig -7: Effect of on Temperature Profiles

 

Figs. 6 to 7 illustrate the influence of the Biot numberon the transient velocity and temperature. As increases the velocity and temperature are also increases throughout the boundary layer.

 

CONCLUSIONS:

A theoretical study has been conducted to simulate the heat transfer analysis of Magneto hydrodynamic third grade fluid flow over a horizontal circular cylinder with Biot-number effect. The transformed momentum and heat boundary layer equations have been solved computationally with Keller box finite difference method. The present study has shown

 

Increasing viscoelastic Third grade fluid parameter     (A) increases the velocity flow and decreases thermal boundary layer thickness throughout the boundary layer.

 

Increasing Magnetohydrodynamic parameter (M) on velocity profiles decelerates the flow whereas it enhances temperature.

 

 

Increasing Biotnumber reduces velocity and the temperatures.

 

ACKNOWLEDGEMENT:

The authors are thanks to DST-WOS-A [Ref: N0. SR/WOS-A/MS-09/2014(G)], New Delhi, for financial support and Management of Madanapalle Institute of Technology and Science, Madanapalle for providing research facilities in the campus.

 

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Accepted on 08.08.2017      ©A&V Publications All right reserved