The flow of magnetohydrodynamic flow over cylinder with heat source or sink

 

Ch. Murali Krishna^1*, K. R. Sekhar², C.S.K. Raju³, G. V. Reddy² , P. Prakash⁴

¹Assistant Professor, Department of BS&H (Mathematics), Sree Vidyanikethan Engineering College (Autonomous), A. Rangampet, Tirupati-517102, (A.P), India.

²Research Scholar, Dept. of Mathematics, S.V. University, Tirupati (A.P). India.

³Assistant Professor, Dept. Of Mathematics, GITAM University, Bangalore (K.A) India

⁴ Professor, Dept. of Mathematics, S.V. University, Tirupati (A.P) India

⁵Research Scholar, Dept. of Mechanical Engineering, NIT Warangal, Warangal (Telangana), India

 

ABSTRACT:

A theoretical analysis performed for investigating steady boundary layer flow of magnetohydrodynamic flow over cylinder with heat source/sink. Proposed mathematical model has a tendency to characterize the effect of magnetohydrodynamic flow over cylinder heat source/sink. The non-linear ordinary differential equations are solved using the Runge-Kutta method. The characteristics of velocity and temperature boundary layers for different physical parameters such as heat source parameter, Reynolds number, the Prandtl number, the magnetic field parameter and power law index parameter. Moreover, the local friction factor coefficients, Nusselt number are also estimated and discussed for aforesaid physical parameters. It is observed that heat transfer rate increases with in power law index parameter and magnetic field parameter while decrease in power law index parameter and Reynolds number.

 

 

INTRODUCTION:

Non-Newtonian fluids are imperative in boundary layer flows because of their engineering and technology related applications. Paints, ketchup, polymeric liquids, apple sauce, jellies, tomato sauce, glues, soaps, blood, cosmetic products are examples of non-Newtonian fluids. As there are diversity of non-Newtonian fluids so it is reasonably complicated to form an equation expressing the viscous and elastic properties of these fluids. In comparison to viscous fluids, mathematical modeling of non-Newtonian fluids is considerably complex and challenging. The Navier-Stokes expressions are proved inadequate for examining rheological characteristics of non-Newtonian materials. Generally the differential, integral and rate types classification have been made in this direction.

 

Further, the phenomenon of stratification is very important in many engineering and manufacturing processes involving non-Newtonian materials. It arises in the flow fields due to change in concentration differences and temperature or fluids with different densities. Double stratification occurs through simultaneous effects of heat and mass transfer. The relevant examples include stratification of lakes and oceans, salinity stratification in rivers, ground water reservoirs, industrial, heterogeneous mixtures in atmosphere, manufacturing food processes and several others. Density differences through gravity are useful for dynamics and involvement of heterogeneous fluid. For instance in lakes, thermal stratification diminishes the blending of oxygen to the base water to develop anoxic from the effort of biological methods. The biomedical aspects are pathology, ophthalmological, public health and nursing etc. The Stratification has a key role in ponds and lakes since it influences the difference between concentration and temperature of oxygen and hydrogen.

 

Crane [1] was first to introduce the boundary layer flow of viscous fluid through a stretching sheet. Further, Cortell [2] proposed the effects of suction or blowing and heat generation or absorption through porous a stretching surface. Ibrahim and Makinde [3] explained the MHD stagnation point flow and heat transfer of Casson nanofluid past a stretching sheet with partial slip and convective boundary layer condition. Ibrahim and Makinde [4] illustrated that the MHD stagnation point flow of a power-law nanofluid towards a convectively heated stretching sheet with slip. Nadeem and Akram [5] studied theperistaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. In another paper Nadeem and Akram [6] have presented the effects of partial slip on the peristaltic transport of a hyperbolic tangent fluid model in an asymmetric channel. Nadeem et al. [7] examined the boundary layer flow of second grade fluid in a cylinder with heat transfer. Nadeem et al. [8] studied the axisymmetric stagnation flow of a micropolar nanofluid in a moving cylinder.

 

Various aspects of such problem have been investigated by many authors such as [9-10]. Wang [11] proposed non-Newtonian fluids for mixed convection heat transfer from a vertical plate. Wang [12] examined the natural convection on a vertical stretching cylinder. An important branch of the non-Newtonian fluid models is the hyperbolic tangent fluid model. Raju et al. [13] reported the flow of magnetohydrodynamic Maxwell nanofluid over a cylinder with Cattaneo-Christov heat flux model. Raju et al. [14] analyzed the transpiration effects on magnetohydrodynamic (MHD) flow over a stretched cylinder with Cattaneo-Christov heat flux with suction/injection. Raju and Sandeep [15] portrayed the magnetohydrodynamic slip flow of a dissipative Casson fluid over a moving geometry with heat source or sink. Numerical study of convective heat transfer on the power law fluid over a vertical exponentially stretching cylinder

 

Recently, investigated by many authors such as [16-18] on MHD flow over a stretching sheet and stretching cylinder with non-uniform heat source or sink.Naseer et al. [19] reported the hyperbolic tangent fluid is used extensively for different laboratory experiments. Naseer et al. [20] have explained the boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder.

 

In view of these facts the present study focuses on the numerical investigation of boundary layer flow of magnetohydrodynamic flow over cylinder with heat source/sink. The boundary layer equations given as a set of partial differential equations (PDEs) are first changed into non-linear ordinary differential equations (ODEs) ahead being solved numerically via Runge-Kutta method. The effects of the governing flow parameters on the velocity and temperature profiles are discussed and presented in table and graphs.

 

FLUID MODEL:

For the hyperbolic tangent fluid the continuity and momentum equations are given as

Where  is the density, is the velocity vector, is the Cauchy stress tensor, b represents the specific body force vector and  represents the material time derivative. The constitutive equations for hyperbolic tangent fluid model is given by [20].

 

We consider the case with    and. Therefore, the component of extra stress tensor can be written as

FORMULATION:

Consider the problem of natural convection boundary layer flow of a hyperbolic tangent fluid flowing over a vertical circular cylinder of radius a. The cylinder is assumed to be stretched exponentially along the axial direction with velocity . The temperature at the surface of the cylinder is assumed to be   and the uniform ambient temperature is taken as . Under these assumptions the boundary layer equations of motion and heat transfer are 

                                                                                                                                                 (1)

                            (2)

                                                                                                  (3)

where the velocity components along the axes are ,   is density , is the kinematic viscosity, p is pressure, g is gravitational acceleration along the z-direction , is the coefficient of thermal expansion ,T is the temperature ,  is the  infinite shear rate viscosity,  is the zero shear rate viscosity ,  is the time constant , n is the power law index,  is the thermal diffusivity and  is the dimensional heat absorption coefficient, is the magnetic field of strength.  The corresponding boundary conditions for the problem are

 

                                                                         (4)

                                                                      (5)

where  (k is dimensional constant) is the field velocity at the surface of the cylinder.

 

SOLUTION OF  THE PROBLEM:

Introduce the following similarity transformations:

                                                                                                         (6)

                                   (7)

where the characteristic temperature difference is calculated from the relations    With the help of transformations (6) and (7) ,Eqs. (1)-(3) take the form

                          (8)

                              (9)

In which  is the natural convection parameter,    is the Prandtl number   is the Weissennberg number and      is the Reynolds number and is heat source parameter, is the magnetic field parameter.  The boundary conditions in non-dimensional form become

                                                                                              (10)

                                                                                                                 (11)

The important physical quantities such as the shear stress at the surface  , the skin friction coefficient  , the heat flux at the surface of the cylinder   and the local Nusselt number  are

 ,  

                                                                                     (12)

The solution of the present problem is obtained numerically by using the Runge-Kutta method.

 

RESULTS AND DISCUSSION:

In this paper an analysis is carried out for natural convection boundary layer flow of a hyperbolic tangent fluid over an exponentially stretched cylinder with non-uniform heat source/sink parameter. It is assumed that the cylinder is stretching exponentially along it axial direction. Expression is the assumed exponential stretching velocity at the surface of the cylinder. For the solution of the problem Runge-Kutta method is used. The impact of the different parameters such as heat source parameter, Reynolds number, the Prandtl number , the magnetic field parameter and power law index parameter over the non-dimensional velocity and temperature profiles are presented graphically.

 

 

Fig.1 Temperature field for different values of

 

Fig.2 Velocity field for different values of

 

Fig.3 Temperature field for different values of

 

Fig.4 Temperature field for different values of

 

Fig.5 Velocity field for different values of

 

Fig.6 Temperature field for different values of

 

Fig.7 Velocity field for different values of

 

Fig.8 Temperature field for different values of

 

 

Fig.1 depicts the effect of Pr on temperature field; an increase in Prandtl number enhances the temperature profiles. Figs. 2 and 3 represent the typical velocity profiles for various values of magnetic field parameter. From these graphs, it is obvious that the velocity of the fluid decelerates with an increase in the strength of magnetic field. The effects of a transverse magnetic field on an electrically, conducting fluid give rise to a resistive-type force called the Lorentz force. This force has the tendency to slow down the motion of the fluid in the boundary layer. These results quantitatively agree with the expectations, since magnetic field exerts retarding force on natural convection flow. Also, it is clear that the decreases the temperature of the fluid. 

 

Fig. 4 is graphical representation of temperature profiles for different values of. It is observed from this figure that the temperature profile increase with increase of. This is due to the fact that, when heat is absorbed, the buoyancy force accelerates the flow. Also, it is observed that thermal boundary layer thickness increases.

 

The power index parameter is sketched (Figs.5-6) on dimensionless distributions on velocity and temperature suppress the momentum and thermal boundary layer thicknesses increasing the values of power index parameter. Fig. 7 depicts that the effects of Reynolds number decelerates the velocity and accelerates in temperature profiles (Fig.8).

 

Table 1 shows the deviation in local friction coefficient, local Nusselt number with various values of Reynolds number, the Prandtl number, heat source parameter and the magnetic field parameter  and power law index parameter, and magnetic field parameter. It is observed that the magnetic field parameter and Reynolds number decrease the local friction factor rate and increase in Prandtl number, heat source parameter and power law index parameter. It is observed that heat transfer rate increases with in power law index parameter and magnetic field parameter while decrease in power law index parameter and Reynolds number.

Table-1: Variation of local friction factor coefficient and Nusselt number for different non-dimensional parameters for.

 

Closing Remarks:

The present computational results characterize the effects of steady boundary layer flow of magnetohydrodynamic flow over cylinder with heat source/sink. Proposed mathematical model has a tendency to characterize the effect of magnetohydrodynamic flow over cylinder heat source/sink. The resultant non-linear governing partial differential equations (PDEs) are solved using the robust Rune-Kutta method. Based on the present computational investigation the following observations are made:

·        It is observed that heat transfer rate increases with in power law index parameter and magnetic field parameter while decrease in power law index parameter and Reynolds number.

·        The momentum and thermal boundary layer thicknesses increasing the values of power index parameter.

·        The dimensional heat source parameter  enhances the thermal boundary layer thickness.

 

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Accepted on 10.12.2017      ©A&V Publications All right reserved