Heat Source or Sink on MHD Tangent Hyperbolic Dusty Fluid in Suspension of Convective Conditions

P. Durga Prasad^1*, N. Sivakumar², B. Rushi Kumar³, S.V.K. Varma⁴, C.S.K. Raju⁵

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

²Assistant Professor, Dept. of Mathematics, SRM University, Kattankulathur (T.N) India.

³Associate Professor, SAS, VIT University, Vellore (T.N) India.

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

⁵Assistant Professor, Dept. Of Mathematics, GITAM University, Bangalore (K.A) India

 

ABSTRACT:

This paper comprehensively analyses the momentum, heat and mass transfer behavior of a heat source or sink on magnetohydrodynamic tangent hyperbolic dusty fluid in suspension of convective conditions. The governing partial differential equations of the flow, heat transfer are transformed into non-linear ordinary differential equations by using self-similarity transformations, which are further solved numerically using the Runge-Kutta and Newton’s method. The effects of various non-dimensional governing parameters on velocity and temperature distributions are discussed with the help of graphs. Furthermore, the effects of these parameters on local friction factor coefficient and heat transfer rate are also discussed and presented through tables.

 

 

INTRODUCTION:

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 the peristaltic 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. So many researchers on dusty fluids also proved that an inclusion of micrometer sized dust particles to base fluids helps to enhance the thermal conductivity. In addition to this, dusty fluids have a various types of industrial applications, and also in science and technology. Gireesha et al. [9] reported the MHD dusty fluid flow over stretching surface in heat transfer effects. The heat transfer flows along with magnetohydrodynamic effects have excellent applications in solar energy systems and power transformer electronics, etc. Abu Bakaret al. [10] examined that the magnetohydrodynamic (MHD) boundary layer fluid flow of a Maxwell nanofluid in a vertical plate with suction or injection effects. From this they concluded that suction parameters increase in the flow of velocity distributions.

 

Further, all the researchers of fluid dynamics have concentrated their work on the flow of different types of fluids, such as Williamson fluid, Cattaneo-Christov heat flux, Jeffrey fluid,  Oldroid B-fluid and Maxwell fluid etc. To understand the mechanism of convective heat transfer, it is important to study the flow behavior past axisymmetric structures, like horizontal cylinders, tangent hyperbolic, vertical cylinders, sphere and cones etc. Nadeem and Saleem [11] developed an unsteady Eyring Powell nanofluid flow over a rotating cone. Ramesh [12] illustrated the Jeffrey fluid along a stretching surface in a stagnation point flow.  He observed that the dusty fluids are improving the thermal conductivity. Gorla et al. [13] analyzed the magnetohydrodynamic flow of a dusty fluid over an exponentially stretching sheet with non-uniform source or sink effects. Through this study, they concluded that increase in heat source parameter causes enhancement in both base fluid and temperature profiles. Ramana Reddy et al. [14] discussed the magnetohydrodynamic nanofluid flow in vertical cone with non-uniform heat source or sink. Gireesha et al. [15] investigated heat transfer in MHD flow of a dusty fluid over a stretching sheet with viscous dissipation.Sandeep and Sulochana [16] explained the MHD flow of dusty nanofluid over a stretching surface with volume fraction of dust particles. Gireesha et al. [17] melting phenomenon in MHD stagnation point flow ofdusty fluid over a stretching sheet in the presence ofthermal radiation and non-uniform heat source/sink. Naseer et al. [18] reported the hyperbolic tangent fluid is used extensively for different laboratory experiments. Naseer et al. [19] have explained the boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder. Mamatha et al. [20] reported the Cattaneo-Christov on heat and mass transfer of unsteady Eyring Powell dusty nanofluid over sheet with heat and mass flux conditions.

 

In view of these facts the present study focuses on the numerical investigation of heat source or sink on magnetohydrodynamic tangent hyperbolic dusty fluid in suspension of convective conditions. 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-Feldberg integration method. The reduced governing equations of the flow are solved numerically. Further, the effects of various physical parameters involved in the governing equations are discussed through graphs. Also, the influence of different parameters on skin friction and Nusselt number in terms of heat transfer are presented through Table 2

 

MODELING:

A steady two-dimensional, incompressible, magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid with suspended uniform size dust particles over a stretching sheet is developed. The stretching sheet is aligned with the axis and the flow is confined to. The surface is stretched with the velocity  in direction with rate . Magnetic field of strength is imposed normal to the fluid flow as shown in Figure 1. By assuming electrical conductivity of the fluid to be small we have neglected the induced magnetic field. 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. The associated governing boundary layer equations are:

 

 

Fig. 1 Geometry of the problem

 

                                                                                                                                                          (1)

                                        (2)

                                                                                                                                                      (3)

                                                                                                                         (4)

                                           (5)

                                                                                                                 (6)

With boundary conditions 

                                                                                (7)

Here - velocity components along  direction of the fluid and dust particle phase, velocity components along direction of the fluid and dust particle phase,are density of the fluid, density of the dust particle phase, mass of the dust particles, number density of the dust particle, kinematic viscosity, thermal conductivity, electric conductivity, uniform magnetic field, specific heat of the fluid and dust phase. -permeability of porous medium, A is power law index, denotes stokes drag coefficient. -radius of the dust particle. whereis the drag coefficient,represents temperature of the fluid, temperature of the dust particle phase, ambient fluid temperature, convective fluid temperature, and convective heat transfer coefficient. andrepresents relaxation time of dust particles and thermal equilibrium timecoefficient of the dimensionless space-dependent internal heat generation.

 

Following similarity transformation were used to convert PDEs into set of ODEs.

                                                                               (8)

Continuity equations (1) and (3) are identically satisfied. Equations (2), (4)-(7) are transformed as follows.

(9)

                                                                     (10)

        (11)

                                                                                                 (12)

The boundary conditions defined in equation (7) will be transformed to:

                                                                         (13)

 

in the above expressions Weissennberg number, power law index parameter, magnetic parameter, mass concentration of dust particles, fluid-particle interaction parameter, inertia coefficient, Prandtl number, heat source parameter, Eckert number, specific heat ratio, fluid particle interaction parameter and Biot number.

 

                                                   (14)

Distinctive measures of practical interest are skin friction coefficient   and local Nusselt number  Which are defined as ;

Where the local Reynold’s number.

RESULTS AND DISCUSSION:

For analyzing the approximate solutions of velocity as well as temperature fields, the non-linear ordinary differential Eqs. (9)-(12) with reference to the boundary conditions (13) are solved numerically with the assistance of Runge-Kutta and Newton’s method. For numerical solutions we have considered the values of non-dimensional parametersthese values are taken constant in this study besides the varied parameters as mentioned in the respective figures.

 

Fig.2 Temperature profiles for different values of heat source sink parameter	Fig.3 Velocity profiles for different values of magnetic field parameter

 

Fig.4 Temperature profiles for different values of magnetic field parameter	Fig.5 Temperature profiles for different values of Biot number
Fig.6 Temperature profiles for different values of Eckert number	Fig.7 Temperature profiles for different values of power law index parameter
Fig.8 Velocity profiles for different values of power law index parameter	Fig.9 Temperature profiles for different values of Prandtl number

Fig.10 Velocity profiles for different values of fluid interaction parameter

Fig. 2 explains the effect of heat source/sink  on temperature field for both the fluid phase and dusty phase cases. It is evident that increasing values of  depreciate the temperature profiles. The heat source parameter absorbs the internal heat energy from the surface. Because of this we have seen decrement in the temperature profiles.

 

Figs. 3 and 4portrayed that the magnetic field parameter it is observed that an increase in the magnetic field parameter decelerates the velocity distributions of both the fluid and dust phases. This is due to the fact that an increase in the magnetic field parameter develops an opposite force to the flow, called the Lorentz force. This force has the tendency to slow down the motion of the fluid in the boundary layer. Hence, it leads to enhanced deceleration of the flow. The reverse phenomena can be finding in temperature distributions. The influence of Biot number on Fig.5 displays that the thermal boundary layer thickness increases with increase in Biot number.

 

Fig. 6 is plotted for the temperature profiles for fluid phase and dust phase cases respectively, for different values of. One can observe that the effect of increasing values of Eckert number is to enhance the temperature at a point which is true for both cases of the fluid phase as well as dust phase. It is observed that the effect of viscous dissipation is to amplify the temperature both cases. Also it is observed that the fluid phase temperature is higher than the dust phase temperature and also it indicates that the fluid particle temperature is parallel to that of dust particle.

The impact power law index parameter  is illustrated in Figs.7 and 8 for fluid phase and dust phase cases. It is observed that the thermal boundary layer thickness decreases with increase in values of power law index parameter. While the momentum boundary layer thickness increases in both fluid phase dust phase cases.

 

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

 

Table 2 The variation in friction factor and local Nusselt number.

 

 

The effect of Prandtl number on the heat transfer is shown in Fig. 9. By analysing these graph it reveals that the effect of increasing the Pr is to decreases the temperature distribution in the flow region in both Fluid and dust phase cases, it is evident that large values of Prandtl number results in thinning of thermal boundary layer. This is in contrast to the effects of other parameters on heat transfer. Finally, the velocity distribution decreases in both fluid phase and dust fluid cases for the fluid interaction parameter as shown in Fig.10.

 

Table 1shows the validation of the present results with the published work under some special limited cases. We observed good accuracy of the present results with the existing results. This proves that the present results are valid. It can be seen from Table 2that an enhancement in the physical parameters depreciates the profiles of the friction factor coefficient and the local Nusselt number, but the parameter helps to enhance the heat transfer rate.

 

CONCLUSIONS:

The present computational results characterize the heat source or sink on magnetohydrodynamic tangent hyperbolic dusty fluid in suspension of convective conditions. The resultant non-linear governing partial differential equations (PDEs) are solved using the robust RKF integration method. Based on the present computational investigation the following observations are made:

 

·        The skin friction coefficient decelerates with increasing values of heat source/sink parameter.

·        The heat transfer rate and skin friction factor decreases with increase in values of magnetic field parameter.

·        The skin friction rate decreases in Eckert number.

 

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Accepted on 07.12.2017      ©A&V Publications All right reserved