Effects of viscous dissipation and transverse magnetic field heat transfer over a stretching cylinder under convective boundary condition

 

Sasikala M.^{1,2 *} , Ramachandraprasad V. ¹,  Bhuvanavijaya R.²

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

²Department of Mathematics, Jawaharlal Nehru Technological University Anantapur, Ananthapuramu 515002, India

 

ABSTRACT:

A numerical treatment for axi-symmetric flow and heat transfer due to a stretching cylinder under the influence of a uniform magnetic field, viscous dissipation and convective condition is presented. The governing partial differential equations are converted into nonlinear, ordinary, and coupled differential equations and are solved using Keller-Box method. The effects of important parameters such as magnetic parameter, curvature parameter, Prandtl number, Eckert number and the local Biot number are described through graphs. The numerical results are compared with the published data and are found to be in good agreement.

 

 

 

INTRODUCTION:

Magneto-hydronomics (MHD) boundary layers through heat and mass transfer over flat surfaces are established in numerous engineering and geophysical applications such as geothermal reservoirs, thermal insulation, enhanced oil recovery, packed-bed catalytic reactors, cooling of nuclear reactors. Numerous chemical engineering processes like metallurgical and polymer extrusion processes engross cooling of a molten liquid being stretched into a cooling system. The fluid mechanical properties of the last but one product depend primarily on the cooling liquid worn and the rate of stretching. A number of polymer liquids like polyethylene oxide and polyisobuylene solution in cetin, having better electromagnetic properties are normally worn as cooling liquid as their flow container be keeping up by outside magnetic fields in order to progress the eminence of the final product. Rahman et al. investigated the effects of joule heating and magneto-hydro dynamics mixed convection in an obstructed lid-driven square cavity¹. Olanrewaju et al. investigated the stagnation point flow of micro polar fluid over a vertical plate with MHD and thermal radiation².

 

Gangadhar investigated the effects of radiation and viscous dissipation on MHD boundary layer flow of heat and mass transfer through a porous vertical flat plate³. Mohammed Ibrahim et al. studied the effect of MHD on oscillatory flow of heat and mass transfer⁴. Rawat et al. studied the influence of Non Darcy porous medium and MHD on micro polar fluid over a non-linear stretching sheet and they concluded that on mounting the material parameter leads to a falling skin-friction coefficient as well as wall couple stress⁵.”

 

Outstanding to the measured viscous dissipation effect and convection of heat transfer along a flat plate, the energy equation is customized by including a term instead of the viscous dissipation effect⁶. Many investigations have been through by the different authors to study effect of viscous thermal dissipation on the fluid flows and heat transfer. Studies reveals that in the fluid flows through high Eckert number, generated heat owed to the viscous thermal dissipation dominates the fluid temperature and the Eckert number cannot be zero in the investigation of convection heat transfer. The work of Brinkman materializes to be the first theoretical work dealing with viscous dissipation⁷. Tyagi executed an extensive study on the effect of viscous dissipation on the completely developed laminar forced convection in cylindrical tubes with an arbitrary cross-section and uniform wall temperature⁸. Khan et al. studied the effects of viscous dissipation and joule heating on MHD bio-convection flow over a porous wedge in the presence of nanoparticles and gyrotactic microorganisms⁹. Viscous dissipation and non-uniform heat source/sink on MHD non-Newtonian fluid with Cattaneo-Christor heat flux is studied by Ramadevi et al.¹⁰.

 

In recent years, researches on the boundary layer flow problem with a convective surface boundary condition have achieved to a great extent importance amid researchers, since first established by Aziz, who considered the thermal boundary layer flow over a flat plate in a uniform free stream with a convective surface boundary condition¹¹. This problem was then extended by Bataller by considering the Blasius and Sakiadis flows, both under a convective surface boundary condition and in the presence of thermal radiation¹². Gangadhar et al. investigated the hydrodynamic effect on heat and mass transfer past a vertical plate in addition to the effects of convective boundary condition and chemical reaction¹³. Ishak find the similarity solutions for the steady laminar boundary layer flow over a permeable plate with a convective boundary condition¹⁴. Makinde and Aziz investigated numerically the effect of a convective boundary condition on the two dimensional boundary layer flows past a stretching sheet in a nanofluid¹⁵.

 

So far as we are aware, no attempt has ever been made to study the impact of magnetic field, suction/injection, viscous dissipation and prescribed convective heating on axisymmetric boundary layer flow along a stretching cylinder. In this paper, the governing partial differential equations of the flowand temperature fields are reduced to ordinary differential equations, which are then solved numerically using Keller – Box method.

 

MATHEMATICAL FORMULATION:

Consider a steady, axisymmetric boundary layer flow of an incompressible viscous fluid over a circular stretching cylinder of radius with a constant temperature. The x-axis is measured along the tube, and the r-axis is calculated in the radial direction. A uniform magnetic field of strength  is implicit to act in the radial direction, while the induced magnetic field is negligible, which can be justified for MHD flow at small magnetic Reynolds number¹⁶. Further, the cylinder is assumed to be axially stretched with velocity , where  is constants, and L is the characteristics length as shown in Fig. 1. With these assumptions, the boundary layer equations govern the flow and heat transfer.

 

                                                                                                                                                (1)

                                                                                          (2)

                                                                                                (3)

Where and are the fluid velocity components along x – , r – directions respectively. Here  is the kinematic viscosity, is electrical conductivity,  is the transverse magnetic field, is the fluid density, is the thermal diffusivity, T is the fluid temperature in the boundary layer.

 The hydrodynamic boundary conditions of this problem are:

                               at  and  as                                                      (4)

While the thermal (surface heat flux) boundary conditions are

                                at  and  as                                                         (5)

The continuity equation (1) is satisfied by introducing the stream function such that: and . Following Mukhopadhyay ¹⁶.  and can be defined as

        ,,

Where f is the dimensionless stream function and  is the similarity variable. By defining  in this manner, the boundary condition at  is reduced to the boundary condition at , which is  further suitable for numerical computations¹⁷. From Eq. (6),

                                   ,                                                              (7)

Where prime denotes differentiation with respect to .

Substituting  Eqs. (6)- (7) into  Eqs. (2) - (5), the governing equations are boundary conditions reduce to

                                  ,                                     (8)

                                                         (9)

Subject to the boundary conditions

                                      , , , at                                                           (10)

 and

                                         as                                                                                           (11)

Where  is the curvature parameter, is the magnetic parameter,  is suction and injection parameter,  is Prandtl number. Note that for a cylinder  and for a plate. (without magnetic field) and (without suction/injection), the problem under consideration reduces to that considered by Ishak et al.¹⁸ (with) and Liu¹⁹  (with ) in those papers. Therefore, for particular cases, the solutions of the present stretching cylinder model contest well with those accounted by ^18,19 for stretching plate.

 

Quantities of physical interest for the phenomena we are studying the skin friction coefficient  and the local Nusselt number . Physically, represents the wall shear stress and defines the heat transfer rate and these can be written as

 

Table 1: Comparison of skin friction coefficient for different values of M when γ = 0.

                                       , and                                                                         (12)

Here  is the skin friction and  is heat flux from the cylinder which are given by

                                     , and                                                                      (13)

Substituting Eq. (6) into Eqs. (12)-(13), we obtain

                                      and                                                              (14)

Where  is the local Reynolds number.

 

SOLUTION OF THE PROBLEM:

As Equations (8)-(9) are nonlinear, it is not possible to obtain the closed form solutions. as a result, the equations through the boundary conditions (10) & (11) are solved numerically by means of a finite-difference scheme recognized as the Keller-box method. This technique has been described succinctly in viscoelastic flows^{20, 21} and Subba Rao et al.²² and for non-Newtonian flows with Biot number effects^{23, 24} . The major steps in the Keller-box method to obtain the numerical solutions are the following:

i). Decrease the specified ODEs to a system of first order equations.

ii). write down the condensed ODEs to finite differences.

iii). Linearized the algebraic equations by using Newton’s method and write down them in vector form.

iv). Solve the linear system through the block tridiagonal elimination technique.

 

One of the factors so as to be affecting the correctness of the method is the suitability of the preliminary guesses. The accurateness of the method depends on the alternative of the preliminary guesses. The choices of the primary guesses depend on the convergence criteria and the boundary conditions (10) & (11). The subsequent primary guesses are chosen

In this study, a consistent grid of size Δ𝜂 = 0.006 is found to be convince the convergence and the solutions are obtained through an error of tolerance  in all cases. In our study, this gives regarding six decimal places perfect to the majority of the agreed quantities.

 

RESULTS AND DISCUSSION:

Numerical computation was carried out for several non-dimensional parameters, namely magnetic parameter M, the transverse curvature parameter γ , suction/injection parameter f_w, Eckert number Ec, local Biot number Bi and the Prandtl number Pr. For comparison purpose we have incorporated results for a plate. Figures 2, 3, 4, 5 and 6 have been plotted to demonstrate the effect of parameters on the flow field and heat transfer characteristics.

 

Fig. 2: Dimensionless velocity distribution for different values of M and γ.

 

Fig. 3: Dimensionless velocity distribution for different values of Bi and  f_w.

 

Figure 2 plotted for the influence of magnetic field parameter and curvature parameter on non – dimensional velocity profiles. It is evident that the figure that a raise in magnetic field parameter decelerates the velocity profile. This is due to fact that an enhance in the magnetic field parameter develops the reverse force to the flow, is called Lorentz flow. This force has propensity to reduce the velocity boundary layer.

 

Fig. 4: Dimensionless temperature distribution for different values of M and γ.

 

Fig. 5: Dimensionless temperature distribution for different values of Bi and f_w.

 

Figure 3 show that the influence of suction/injection parameter on velocity profiles. It is noticed that the velocity profile show a fall in with suction, where as contradictory trend is observed for the injection case for cylinder. This is appropriate to truth that more fluid is withdrawn from the wall when injection increases.   represents the case of non-porous stretching sheet. Figure 4 exhibit the effects of magnetic field parameter and curvature parameter  on dimensionless temperature profile. Since the magnetic parameter is inversely proportional to the density by , hence increase in M causes decrease in density and consequently the temperature of the fluid rises. Further move the temperature of the fluid diminishes with an enlarged in curvature parameter. The influence of local Biot number and suction / injection parameter on temperature distribution is shown in figure 5. It is observed that fluid temperature increases with an enhanced in. From figure 5 that, rising  for the case of suction  decrease the temperature within the boundary layer whereas increasingfor the case of injection  enhances the temperature profiles. That is, the burden of suction on the surface causes reduction in the thermal boundary layer thickness and the injection causes an increase in the thermal boundary layer thickness. It can be seen from figure 6 that the non- dimensional temperature increases with an increase in Ec whereas the temperature profile decreases with raise in Pr. Since the Prandtl number represents the ratio of momentum diffusivity to thermal diffusivity as an increase in the Prandtl number decreases the thermal boundary layer of the cylinder. Therefore the figures show that an increase in Pandtl number Pr, reduces in the temperature profile at a given point of flow parameter.

 

Fig. 6: Dimensionless temperature distribution for different values of Ec and Pr.

 

From the figure 7 that skin friction coefficient increases with increase in magnetic and curvature parameters. It can be seen directly forwardly from figures 8 and 9 that the magnitude of local Nusselt number increases with increasing values of the curvature parameter and local Biot number. Further, the magnitude of Nusselt number is a decreasing function of M, Ec. Physically a large Biot number simulates a strong surface convection which as a result provides more heat to the surface of the sheet. 

 

In order to regulate the method used in the present study and to make a decision the accuracy of the present analysis and to compare with the results available (Butt and Ali²⁵ , Fang et al.²⁶ , Mukhopadhyay²⁷  and Maboob et al. ²⁸) relating to the local skin-friction coefficient and found in an agreement (see table 1).

 

 Fig. 7: Skin friction coefficient for different values of M and γ.

 

Fig. 8: Local Nusselt number for different values of M and γ.

 

Fig. 9: Local Nusselt number for different values of Bi and Ec.

 

CONCLUSIONS:

In the present work, the MHD boundary layer flow and heat transfer over permeable stretching cylinder with viscous dissipation under convective heating have been investigated. The present results are in good concurrence with those reported in open literature for some special cases. From the study, the following remarks can be summarized.”

1.      On growing the magnetic parameter, the resultant dimensionless velocity distribution reduces within the boundary layer but the dimensionless temperature distribution increases within the boundary layer.

2.      The local skin friction coefficient decreases and local Nusselt number increases by increasing the magnetic parameter.

3.      Viscous dissipation and convective heating are strongly influenced the temperature of the boundary layer flow.

4.      On increasing the Eckert number is to reduce the local Nusselt number but local Biot number is to increase the local Nusselt number.

 

REFERENCES:

1.     Rehman A, Nadeem S. Mixed convection heat transfer in micropolar nanofluid over a vertical slender cylinder.   Chin Phys Lett. 2012; 29(12): 124701.

2.     Olanrewaju PO, Okedayo GT,  Gbadeyan JA. Effects of thermal radiation on magneto hydro dynamic (MHD) flow of a micro polar fluid towards a stagnation point on a vertical place. Int.J. Of App.Sci. and Tech 2011;  6:  219-230.

3.     Gangadhar K.  Radiation and viscous dissipation effects on chemically reacting MHD boundary layer flow of heat and mass transfer through a porous vertical flat plate. Journal of Energy, Heat and Mass Transfer. 2012;  34:  245-259.

4.     Mohammed Ibrahim S, Gangadhar K,  Bhaskar Reddy N. Radiation and Mass Transfer Effects on MHD Oscillatory Flow in a Channel Filled with Porous Medium in the presence of Chemical Reaction. Journal of Applied Fluid Mechanics. 2011; 8(3):  529-537.

5.     Rawat S, Kapoor S, Bhargava R. MHD Flow and heat  and Mass Transfer of Micropolar Fluid over a Nonlinear Stretching Sheet with Variable Micro Inertia Density,Heat Flux and Chemical Reaction in a Non-Darcy Porous Medium. Journal of Applied Fluid Mechanics. 2016; 9(1):  321-331.

6.     Patrick Oosthuizen H,  David N. An Introduction to Convective Heat Transfer Analysis.  WCB McGraw-Hill, Singapore. 1999.

7.     Brinkman HC. Heat transfer in capillary flow I. Applied Sci Res. A2. 1951; 120–124.

8.     Tyagi VP.  Laminar forced convection of a dissipative fluid in a channel.  J. Heat Transfer. 1996;  88:  161–169.

9.     Khan U, Ahmed N, Mohyud‑Din ST. Influence of viscous dissipation and Joule heating on MHD bio‑convection flow over a porous wedge in the presence of nanoparticles and gyrotactic microorganisms. SpringerPlus. 2016; 5: 20-43. DOI 10.1186/s40064-016-3718-8.

10.  Ramandevi B, Ramana Reddy JV, Sugunamma V, Sandeep N. Combined influence of viscous dissipation and non-uniform heat source/sink on MHD non-Newtonian fluid flow with Cattaneo-Christov heat flux. Alexandria Eng. J. 2017.  http://dx.doi.org/10.1016/j.aej.2017.01.026

11.  Aziz A. A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simulat. 2009; 14: 1064-1068.

12.  Bataller RC. Radiation effects for the Blasius and Sakiadis flows with a convective surface boundary condition. Appl. Math. Comput. 2008;206: 832-840.

13.  Gangadhar K, Bhaskar Reddy N, Kameswaran PK. Similarity Solution of Hydro-Magnetic Heat and Mass Transfer over a Vertical Plate with a Convective Surface BoundaryCondition and Chemical Reaction. International Journal of  Nonlinear Science. 2012; 13(3): 298-307.

14.  Ishak A. Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition.  Appl. Math. Computation. 217: 837-842.

15.  Makinde OD, Aziz A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int. J. Therm. Sci. 2011; 50: 1326-1332.

16.  Mukhopadhyay S. MHD boundary layer slip flow along a stretching cylinder.  Ain Shams Eng. J. 2013; 4:  317–324.

17.  Bachok N, Ishak A. Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malays. J. Math. Sci. 2010; 4: 159–169.

18.  Ishak A, Nazar R, Arifin NM, Ali FM, Pop I. MHD stagnation-point flow towards a stretching sheet with prescribed surface heat flux. Sains Malays. 2011; 40:  1193–1199.

19.  Liu IC. A note on heat and mass transfer for a hydromagnetics flow over a stretching sheet.  Int. Commun. Heat Mass Transf. 2011; 32: 1075–1084.

20.  Amanulla CH, Nagendra N, Suryanarayana Reddy M. Numerical Study of Thermal and Momentum Slip Effects on MHD Williamson Nanofluid from an Isothermal Sphere. Journal of Nanofluids. 2017; 6(6): 1111–1126.

21.  Amanulla CH, Nagendra N, Suryanarayana Reddy M. MHD Flow and Heat Transfer in a Williamson Fluid from a Vertical Permeable Cone with Thermal and Momentum Slip Effects: A Mathematical Study. Frontiers in Heat and Mass Transfer. 2017; 8 (40).

22.  Subba Rao A, Amanulla CH, Nagendra N, Bég OA, Kadir A. Hydromagnetic flow and Heat Transfer in a Williamson Non-Newtonian fluid from a Horizontal Circular Cylinder with Newtonian Heating.  Int. J. Appl. Comput. Math. 2017;  1-21.

23.  Gaffar SA, Prasad VR, Reddy EK. Magnetohydrodynamic free Convection flow and heat transfer of non-Newtonian Tangent Hyperbolic fluid from horizontal circular cylinder with Biot number effects. International Journal of Applied and Computational Mathematics. 2017; 3(2):  721-743.

24.  Abdul Gaffar S, Ramachandra Prasad V. Computational study of non-Newtonian Eyring–Powell fluid from a vertical porous plate with biot number effects. Journal of the Brazilian Society of Mechanical Sciences and Engineering.

25.  Butt AS, Ali A. Computational study of entropy generation in magnetohydrodynamic flow and heat transfer over an unsteady stretching permeable sheet. Eur. Phys. J. Plus. 2014; 129(1): 13.

26.  Fang T, Zhang J, Yao S. Slip MHD viscous flow over a stretching sheet-an exact solution. Commun. Nonlinear Sci. Numer. Simul. 2009;  14: 3731–3737.

27.  Mukhopadhyay S. Chemically reactive solute transfer in a boundary layer slip flow along a stretching cylinder.  Front. Chem. Sci. Eng. 2011; 5:  385–391.

28.  Mabood F, Lorenzini G, Pochai N, Muhammad Ibrahim S.  Effects of prescribed heat flux and transpiration on MHD axisymmetric flow impinging on stretching cylinder. Continuum Mech. Thermodyn. DOI 10.1007/s00161-016-0519-9.

 

 

 

 

Accepted on 29.11.2017      ©A&V Publications All right reserved