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Engineering Science and Technology, an International Journal

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Full Length Article

Effect of radial magnetic field on natural convection flow in alternate conducting vertical concentric annuli with ramped temperature

Vanita, Anand Kumar *

Department of Mathematics, Central University ofRajasthan, Ajmer, India

ARTICLE INFO

Article history: Received 18 February 2016 Revised 29 April 2016 Accepted 30 April 2016 Available online xxxx

Keywords:

Radial magnetic field Vertical cylinder Induced magnetic field Hartmann number Matlab software

ABSTRACT

In this article, a numerical analysis has been performed to study the effect of radial magnetic field on natural convection of a viscous incompressible and electrically conducting fluid within the concentric vertical annuli containing a ramped type temperature profile at the surface of inner cylinder taking into account the induced magnetic field. Two cases are considered which are case 1, when inner cylinder is perfectly conducting and outer cylinder is non-conducting and case 2, has opposite configuration. The governing partial differential equations have been solved by using Matlab software. The influence of Hartmann number and magnetic Prandtl number on different profiles has been presented graphically. It is found that the effect of Hartmann number and magnetic Prandtl number is to decrease the velocity profile in both the cases. This study suggest that the heat transfer can be controlled by using ramped temperature and conductivity of the cylinder.

© 2016 The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Free convective flow along vertical cylinders is one of the recent advances in the field of MHD. In recent years, researchers have shown their interest in this emerging field due to its wide range of applications in astrophysics, geophysics and engineering. These types of studies are important in nuclear power plants, drilling operations, geothermal power generation and space vehicles technology etc. Shercliff [1] investigated the flow of conducting fluids in circular pipes under the transverse magnetic field and found that when the induced field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall. The flow of an electrically conducting fluid between two infinite long vertical cylinders under the influence of magnetic field was analyzed by Globe [2]. Globe [2] show for the limiting case where the radii of annulus become infinite, but their difference remains finite then the solution behaves as a flow between infinite planes. Ramamoorthy [3] scrutinized the flow between two concentric rotating cylinders with a radial magnetic field. He compared the classical hydrodynamic velocity with the magneto hydrodynamic velocity between two rotating co-axial cylinders in presence of a radial magnetic field

* Corresponding author. E-mail addresses: vanitacuraj@gmail.com ( Vanita), aanandbhu@gmail.com (A. Kumar).

Peer review under responsibility of Karabuk University.

and by neglecting the induced magnetic field. Hughes and Young [4] introduced the Electromagnetodynamics of Fluids which proved to be useful for better understanding of electromagnetism.

Sparrow and Lloyd [5] have considered combined forced and free convective flow of a viscous incompressible fluid on vertical surface. Powe et al. [6] studied free convective flow patterns in cylindrical annuli. They presented graphs which helps in prediction of type of flow that occur for a wide range of cylindrical combinations and annulus operating conditions. Dube [7] analyzed the steady laminar flow of a viscous incompressible electrically conducting fluid between infinite long concentric porous cylinders under the influence of a radial magnetic field and found that the shearing stress at the inner cylinder decreases as cross flow Reynolds number increases. Kuehn and Goldstein [8] had carried out an experimental and theoretical study of natural convection in annulus between horizontal concentric cylinders and showed that the comparison between the present experimental and numerical results under similar conditions are in good agreement. Ghosh [9] presented a note on steady and unsteady hydrodynamic flow in a rotating channel in the presence of inclined magnetic field. Lee and Kuo [10] investigated the laminar flow in annular ducts with constant wall temperature at the boundaries. Harris et al. [11] have made the analysis of unsteady mixed convection from a vertical flat plate embedded in a porous medium and obtained the occurrence of transients when the buoyancy parameter is positive and negative. Chandran et al. [12] made a unified approach to analytic solution of a hydromagnetic free convection flow. Pal and

http://dx.doi.org/10.1016/j.jestch.2016.04.010

2215-0986/® 2016 The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

a radius of inner cylinder T dimensionless temperature

b radius of outer cylinder T'a temperature of inner cylinder

Cp specific heat at constant pressure T'f temperature of the fluid

g acceleration due to gravity u dimensionless velocity component along axis of cylin-

H0 constant magnetic field der

Ha Hartmann number u' velocity component along axis of cylinder

h' induced magnetic field component along axis of cylinder dimensionless induced magnetic field component along u characteristics velocity

h Greek symbols

axis of cylinder b coefficient of thermal expansion

J induced current density K fluid thermal conductivity

Jh induced current flux k ratio of radii

Nu1 Nusselt number at inner cylinder le magnetic permeability

Nux Nusselt number at outer cylinder Vh magnetic diffusivity

Pm magnetic Prandtl number V kinematic viscosity of fluid

Pr Prandtl number p density of fluid

r dimensionless radial distance a electrical conductivity

r' radial distance Sl skin friction at inner cylinder

t' time Sk skin friction at outer cylinder

t dimensionless time

t0 characteristics time

Biswas [13] applied perturbation analysis to MHD oscillatory fluid over an infinite moving permeable plate saturated in porous medium in presence of chemical reaction. They deduced that with increase in the chemical reaction the skin-friction coefficient decreases at the wall while opposite behavior is observed by increasing the permeability parameter of the porous medium.

There raised some problems which were difficult to solve analytically. For solving those type of problems the Computational Techniques for Fluid Dynamics introduced by Fletcher [14] proved to be very useful. The mixed convection in vertical annulus under a radial magnetic field was considered by Mozayyeni and Rahimi [15] and it was found that velocity and temperature of fluid can be suppressed more effectively by using external magnetic field. Prasad and Kulacki [16] studied the free convective heat transfer in a liquid filled vertical annulus. They analyzed that in the laminar flow regime, the Nusselt number is weakly dependent on the aspect ratio when Nu and Ra are considered in terms of the annulus height and the starting of laminar flow regime is delayed with an increase in radius ratio. Joshi [17] has investigated the fully developed free convection flow in vertical annuli with isolated boundaries in which inner boundary is maintained at higher temperature and examined that the resulting flow and Nusselt number are function of annulus gap and non-dimensional temperature ratio. Ali et al. [18] presented the flow profiles for heat and mass transfer in presence of magnetic field oevr a inclined plate in the porous medium. Javaherdeh et al. [19] performed numerical investigation to study the heat and mass transfer in MHD fluid flow past a moving vertical plate in a porous medium and presented that the Grashof number enhances the flow velocity while the porosity parameter reduces the velocity profile. Also, the rate of heat and mass transfer decreases with increase in transverse magnetic field. Sandeep and Sulochana [20] examined the effect of nonuniform heat source/sink on an unsteady mixed convection boundary layer flow of a magneto micropolar fluid past a stretching/ shrinking sheet. They gave conclusion that the mass transfer rate get enhanced with positive values of non-uniform heat source/sink parameters and the velocity and temperature boundary layers increase with increase in micropolar parameter.

Singh et al. [21] considered the study of steady fully developed laminar natural convection flow in open ended vertical concentric annuli in presence of radial magnetic field by considering the case of isothermal and constant heat flux on inner cylinder of concentric annuli. Sankar et al. [22] have presented the effect of magnetic field on natural convection in a vertical cylindrical annulus. Their computational results reveal that in shallow cavities the flow and heat transfer are suppressed more effectively by an axial magnetic field while in tall cavities a radial magnetic field is more effective. Sheik-holeslami et al. [23] have done the numerical investigation of the effect of magnetic field on natural convection in a curved shape enclosure and concluded that Hartmann number can be a control parameter for heat and fluid flow. Deka et al. [24] have studied transient free convection flow past an accelerated vertical cylinder in a rotating fluid. They concluded that axial velocity component decreases with increase in Prandtl number but increases with increase in Grashoff number while transverse velocity component increases with increase in Grashoff number but decreases with increase in Prandtl number. Afrand et al. [25] presented the numerical simulation by using multi-objective particle swarm optimization algorithm on natural convection in a cylindrical annulus mold under magnetic field. Seth et al. [26] shown the effect of hall current, radiation and rotation on free convective flow past moving vertical plate. Hussanan et al. [27] have studied the free convective flow in presence of magnetic field with Newtonian heating and constant mass diffusion of heat and mass transfer. Afrand et al. [28] have done the numerical simulation of natural convection in presence of a magnetic field in an inclined cylindrical annulus and found that the effect of Hartmann number on the average Nusselt number is not noticeable in the case of horizontal annulus as compared to the case of vertical annulus.

These all studies were restricted to the case in which induced magnetic field has been neglected. Induced magnetic field strongly influences the flow formation and heat transfer. To control the flow formation rate more accurately, it is necessary to take induced magnetic field into consideration, Since, the induced magnetic field has many important applications in the experimental and theoretical studies of MHD flow due to its use in many industrial

and technological phenomena. It has applications in metal coating, crystel growth, reactor cooling, planetary magnetosphere, chemical engineering and electronics. Therefore, as an advanced step in progress of magnetohydromagnetic flow, Kumar and Singh [29] investigate the effect of induced magnetic field on unsteady MHD free convective flow past a semi-infinite vertical wall. They deliberated that the influence of magnetic parameter is to decrease the velocity, temperature and induced magnetic fields while that of magnetic Prandtl number is to increase them. Further, Kumar and Singh [30] investigated the fully developed free-convective flow of a viscous incompressible fluid between two concentric annuli by taking into account the induced magnetic field. They discussed the case when boundaries at inner and outer cylinder are of mixed kind. Gireesha et al. [31] analyzed the effect of induced magnetic field on the heat transfer in boundary stagnation-point flow of nanofluid past a stretching sheet. They concluded that with increase in strength of hydromagnetic field, the induced magnetic field and the temperature profiles are enhanced.

Afrand et al. [32] have presented the effect of an applied magnetic field in the enclosure in presence of free convection heat transfer and showed that as radii ratio increase the Nusselt number increases. Further, Afrand et al. [33,34] extended the work by considering the tilted cylinder in presence of magnetic field and by considering vertical cylinder in presence of induced magnetic field respectively. Sarveshanand and Singh [35] have analyzed the free convection between vertical concentric annuli with induced magnetic field when inner cylinder is electrically conducting. They obtained results for various physical parameters and found that Hartmann number decreases the velocity profile while increases the induced magnetic field profile. Recently, Vanita and Kumar [36] have numerically investigated the effect of induced magnetic field over a vertical cone in free convective flow and found that the steady-state time for average skin-friction and average Nusselt number reduces with rise in magnetic parameter while magnetic Prandtl number and the semi-vertical angle have opposite effect.

In nature, motion starts with constant temperature i.e. T = 1, that means when time is increased from zero, the fluid gain maximum temperature. Therefore, some researchers start motion by taking T = t which shows that temperature increases with increase in time. But in practical situations, it is not possible. When a fluid starts heating, temperature gradually increases but after boiling point temperature becomes constant. This phenomenon is commonly seen in cooling systems like air-conditioner, refrigerators. A significant contribution in study of ramped temperature was given by Kumar and Singh [37] by investigating transient MHD natural convection past a vertical cone having ramped temperature on the curved surface. They presented numerical results for the velocity, temperature, skin-friction and Nusselt number with help of graphs. Singh and Singh [38] examined the transient MHD free convective flow near a semi-infinite vertical wall having ramped temperature. They have found that the effect of the magnetic field is to delay the steady state time of the velocity and temperature fields.

Seth et al. [39] have considered the heat transfer on free convection past an impulsively moving plate with ramped type wall temperature. Further, Seth and Sarkar [40,41] extended the work by considering the effect of nth order chemical reaction, induced magnetic field and effect of applied magnetic field, radiation respectively past a moving vertical plate with ramped temperature. Seth et al. [42] showed the effect of ramped temperature on natural convection with radiative heat transfer in a porous medium. Rajesh and Chamkha [43] have studied the effect of thermal radiation and chemical reaction past a vertical plate with ramped type temperature at the plate. Recently, Vanita and Kumar [44] numerically studied the effect of radial magnetic field on free convective flow over ramped velocity moving vertical cylinder with ramped type temperature and concentration. Vanita and Kumar [44] have

found that the average value of skin-friction decline with increase in magnetic parameter and Schimdt number but thermal Grashof number and mass Grashof number have opposite effect.

In this paper, we have discussed the transient fully developed MHD free convective flow of an electrically conducting and viscous incompressible fluid between two co-axial vertical circular cylinders with ramped type temperature distribution at inner cylinder. Here, we have considered the induced magnetic field into the account. According to the nature of conductivity of cylinders, we have discussed two cases on boundary condition of induced magnetic field at the cylinders i.e.

(a) When inner cylinder is perfectly conducting and outer cylinder is non-conducting, case 1 and

(b) When inner cylinder is non-conducting and outer cylinder is perfectly conducting, case 2.

Further, the governing partial differential equations which describe the flow formation have been solved numerically using MATLAB software. Finally, the effects of non-dimensional parameters on the velocity, induced magnetic field, skin-friction coefficients, and heat transfer rate at both the cylinders are shown by graphs and tables.

2. Mathematical formulation

Here, we have considered the transient fully developed natural convective flow of a viscous incompressible and electrically conducting fluid in vertical concentric circular cylinders of infinite length. The z'-axis is considered along the axis of cylinder in vertical upward direction and r0 is in radial direction measured outward from the axis of cylinders. The radii of inner and outer cylinders are taken as a and b respectively as illustrated in Fig. 1. Initially, for time t' 6 0, the temperature of the fluid as well as temperature of the cylinders are same Tf. At time t' > 0, the temperature of the cylinder is raised or lowered to Tf + (T'a - Tf) ^ upto time t' 6 t0 and thereafter for t' > t0, is maintained at the constant temperature Ta. An applied radial magnetic field of form is taken in radially outward direction. Under the above assumptions, physical variables in the considered model become function of time and space variables. By using the Boussinesq approximation, the governing momentum, induced magnetic field and energy equations for the considered model are derived as follows [21,29,30,38,44]:

Fig. 1. Physical geometry of considered model.

du' d2u' 1 du'\ ieH0a dh' _

ÄF = V ÄF +1 dû + pH7" dh+ gß(T- Tf )

dh' id2h' 1 dh*

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 r

(a) Casel

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 r

(b) Casel

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

(c) Casel and 2 Fig. 2. Grid independency test at steady-state values.

dt' M dr'2 + r' dr'

H0 a du

r' dr'

' 1 ffA

dr2 + r dr)

We have obtained the initial and boundary conditions for velocity, induced magnetic field and temperature fields as:

t' 6 0 : u' =h' = 0, T' = T'f for a 6 r 6 b

t P 0 :

[u! = 0, dh — 0 or h' — 0, T' — Tf +(T'a - Tf ) t0 T T

at r' — a

at r' — a and t' 6 to; at r' — a and t' > t0;

u' — 0, dh — 0 or h' — 0, T' — Tf at r' — b

By defining the non-dimensional parameters

Sßa3 (T'a - Tf )

T' - Tf

(TO-Tf) '

aieH0aU'

Eqs. (1)-(3) in non-dimensional form can now be expressed as follows:

du _ d2u 1 du Ha2 dh T dt — dr2 + r dr + ~T dr +

1 (tfh 1 dh

Pm\ dr2 r dr

dT _ 1 id2T 1 dT\ dt — Pr ^dr2 + r drj

1 1 du Pm r dr

The initial and boundary conditions in non-dimensional form becomes

t 6 0 : u = h = 0, T = 0, for 1 6 r 6 k

(u — 0, f = 0 or h — 0,

t P 0 :

= 0, dh -t T —1, T —1,

r -0 äh-

at r - 1

at r — 1 when t 6 1 at r = 1 when t > 1

u = 0, dh = 0 or h = 0, T = 0 at r = k

where U is the characteristic velocity, k = a is the ratio of radii.

The non-dimensionalisation process has suggested some non-dimensional parameters which are Hartmann number (Ha), Prandtl number (Pr), magnetic Prandtl number (Pm) and characteristic time (t0) and can be defined as:

Ha — 1eH0a.

Pr = ICL k '

According to the considered two cases of boundary conditions on the induced magnetic field, desired case can be obtained by re-deriving all boundary conditions for induced magnetic field into a combined form as follows :

Table 1

Comparison of steady state numerical values of non-dimensional Nusselt number (R = 0).

Nu Kumar and Singh [30] Present results

Nu, 1.44269504 1.4427

Nuk 0.72134752 0.7213

Table 2

Comparison of steady state numerical values of skin-friction at inner and outer cylinders and induced current flux (R = 0).

Ha Parameters Kumar and Singh [30] Present results

1 Si 0.3273 0.3264

Si -0.1154 -0.1194

J 0.0240 0.0237

2 Si 0.2962 0.3032

Sk -0.1008 -0.1145

J 0.0211 0.0225

3 Si 0.2579 0.2695

Si -0.0832 -0.1072

J 0.0175 0.0208

Al + Bi h = 0 at r = 1

A2 d- + B2 h = 0 at r = k dr

The above equation clearly shows that a desired case can be obtained by assigning suitable values to Ai, A2, Bi and B2 while A] and A2 or B1 and B2 can not be vanish simultaneously. If inner cylinder is perfectly conducting, we assign A1 = 1 and A2 = 0 while in case of outer cylinder perfectly conducting, we assign A1 = 0 and A2 = 1. Similarly when the inner cylinder is non conducting we consider the values of B1 and B2 as 1 and 0 respectively while when outer cylinder is non-conducting the values of B1 and B2 are reversed.

3. Numerical solution procedure

The coupled partial differential Eqs. (6)-(8) with appropriate initial and boundary conditions (9) clearly suggest that the solution must be obtained numerically. For this purpose, we have used the Matlab software. The tool pde solver in Matlab software is used for solving the initial and boundary value problems for the systems of parabolic partial differential equations in one space variable and

(a) Ha = 0

(c) Ha = 4

(d) Ha = 8

Fig. 3. Velocity profile for different values of Ha in case 1 for Pm = 0.1, Pr = 0.71.

(c) Ha = 4 (d) Ha — 8

Fig. 4. Velocity profile for different values of Ha in case 2 for Pm = 0.1, Pr = 0.71.

time. The PDE resulting from discretization in space are integrated to obtain the approximate solutions on each point of time as specified At in time space. The tool pde solver returns values of the solution on a mesh provided in space variable. For the computational procedure, we have converted the physical domain under consideration into computational domain and then restricted the computational domain to finite dimensional rectangle by taking rmin = 1 and rmax = 2 and tmin = 0 and tmax = 2.5. To obtain more efficient results in simulation process, a grid independency test has been performed for steady-state values, which is shown in Fig. 2. The values obtained in the iteration process with grid system 20 x 50 differ in the fourth decimal place from the grid system of 10 x 40 and 30 x 60. Hence, for obtaining better results, we have considered 21 x 51 grid points in the numerical computation with the fixed value of Pr = 0.71. The mesh sizes in the r and t direction are taken as Ar = 0.01 and At = 0.01 in entire numerical computations.

During any one time step At the computed values of previous time step have been used for evaluation of coefficients of T, u and h appearing in Eqs. (6)-(8). At the end of each time step, first the temperature field is computed and then the evaluated values are used to obtain the velocity and induced magnetic field components. The unsteady values corresponding to a particular time have been obtained through required iterations and the computational procedure is advanced until we get the steady state. The steady state numerical solutions have been obtained for the temperature, velocity and induced magnetic fields when the following convergence criterion is satisfied

where ®n stands for either the temperature, velocity or induced magnetic fields. The superscripts denote the values of the

(b) Ha = 4

(c) Ha = 8

Fig. 5. Velocity profile for different values of Ha in case 1 for Pm = 0.1, Pr = 7.0.

(c) Ha = 8

Fig. 6. Velocity profile for different values of Ha in case 2 for Pm = 0.1, Pr = 7.0.

dependent variable after the nth and (n + 1)th iterations of the time t(= nDt) respectively, whereas the subscript i indicate grid lacation in the r direction.

By using the computed values of velocity and temperature field, the skin frictions coefficients and the Nusselt number in non-dimensional form at outer surface of inner cylinder and inner surface of outer cylinder are calculated by using following relations:

(b) Pm = 0.5

(b) Pm = 0.5

(c) Pm = 1.0 (c) Pm = 1.0

Fig. 7. Velocity profile for different values of Pm in case 1 for Ha = 2, Pr = 0.71. Fig. 8. Velocity profile for different values of Pm in case 2 for Ha = 2,Pr = 0.71.

(c) Ha = 8 (c) Ha = 8

Fig. 9. Induced magnetic field profiles for different values of Ha in case 1 for Fig. 10. Induced magnetic field profile for different values of Ha in case 2 for Pm = 0.1, Pr = 0.71. Pm = 0.1, Pr = 0.71.

(a) Ha — 2

(a) Ha = 2

(b) Ha = 4

(c) Ha = 8 (c) Ha = 8

Fig. 11. Induced magnetic field profile for different values of Ha in case 1 for Fig. 12. Induced magnetic field profile for different values of Ha in case 2 for Pm = 0.1, Pr = 7.0. Pm = 0.1, Pr = 7.0.

(b) Pm = 0.5

(b) Pm = 0.5

(c) Pm = 1.0 (c) Pm = 1.0

Fig. 13. Induced magnetic field profile for different values of Pm in case 1 for Fig. 14. Induced magnetic field profile for different values of Pm in case 2 for Ha = 2, Pr = 0.71. Ha = 2, Pr = 0.71.

0.4 -,

-1— 0.5

~1— 1.0

No. Ha Pm

1 2 0.1

2 3 0.1

3 5 0.1

4 2 0.5

5 3 0.5

6 5 0.5

7 2 1.0

8 3 1.0

9 5 1.0

— 2.0

(a) Skin-friction at inner cylinder

No. Ha Pm

1 2 0.1

2 3 0.1

3 5 0.1

4 2 0.5

5 3 0.5

6 5 0.5

7 2 1.0

8 3 1.0

9 5 1.0

(a) Skin-friction at inner cylinder

No. Ha Pm

1 2 0.1

2 3 0.1

3 5 0.1

4 2 0.5

5 3 0.5

6 5 0.5

7 2 1.0

8 3 1.0

9 5 1.0

0.6 0.5-1 0.4 0.3-1

fcT* 0.2

(b) Skin-friction at outer cylinder

Fig. 15. Variation of Skin-friction profiles in case 1 for different values of Ha and Pm at (a) r = 1, (b) r = k.

The induced current density and induced current flux respectively are also calculated as follows:

No. Ha Pm

1 2 0.1

2 3 0.1

3 5 0.1

4 2 0.5

5 3 0.5

6 5 0.5

7 2 1.0

8 3 1.0

9 5 1.0

~1 2.5

(b) Skin-friction at outer cylinder

Fig. 16. Variation of skin-friction profiles in case 2 for different values of Ha and Pm at (a) r = 1, (b) r = k.

'dr r=1

dr r=k

Jh = "

dh dr '

J = ^ 'i@dr

4. Results and discussions

In this study, we have discussed the effects of various physical parameters on the velocity and induced magnetic field profiles with the help of graphs. In this study, we have focused our attention on the parameters Ha, Pm and t while taking Pr as 0.71 (air) and k = 2.0 for all calculations. Here, we have discussed two cases, in case 1, we have taken inner cylinder as perfectly conducting and outer cylinder as non-conducting and in the case 2, we have taken inner cylinder as non-conducting and outer cylinder as perfectly

(a) Nusselt number at inner cylinder

(a) Nusselt number at innner cylinder

(b) Nusselt number at outer cylinder

Fig. 17. Variation of Nusselt number for different values of Pr when constant temperature at inner cylinder.

■ Pr = 0.03

i I i 1 i I i 1 i I

(b) Nusselt number at outer cylinder

Fig. 18. Variation of Nusselt number for different values of Pr when ramped temperature at inner cylinder.

conducting. For result validation, we have compared our results with that of Kumar and Singh [30] for Nusselt number by Table 1 at steady state time and buoyancy force distribution (R = 0 in paper Kumar and Singh [30]). We can notice from Table 1 that at steady state time Nu1 and Nuk are same as given by Kumar and Singh [30]. Also a comparative study of steady state numerical values of skin-friction at inner and outer cylinders and induced current flux at buoyancy force distribution (R = 0) have been shown in Table 2. The result is in good agreement for different values of parameter Ha = 1, 2,3 and buoyancy force distribution (R = 0).

In Figs. 3 and 4, we have shown the variation in velocity profile for different values of Hartmann number for the case 1 and 2 respectively. Figs. 3(a) and 4(a) show variation in velocity profile for Ha = 0, and we notice that velocity profile is same for both

the cases. From Figs. 3(b) and 4(b), it is observed that velocity decreases when Ha increases from 0 to 2. Initially velocity profile increases with time and then attains steady state. In Figs. 3(c) and 4(c), we have shown the variation in velocity profiles for Ha = 4 and the result is qualitatively similar to the previous one. When Ha is further increased, we get an interesting behavior of velocity profile in both the cases. The velocity profile becomes negative with increase in time and steady state is achieved in the same direction which are shown by Figs. 3(d) and 4(d). This is due to the fact that with increase in strength of applied magnetic field, a resistive type of body force known as Lorentz force, is generated in an electrically conducting fluid, which opposes the fluid motion. This force has the property to reduce the velocity boundary layer. From

the comparative study of Figs. 3 and 4, we observe that influence of Hartmann number on the velocity profiles is more in case 1 as compared to the case 2. Further, the steady state time also reaches later with increase in Hartmann number. Figs. 5 and 6 present the variation in velocity profiles for different values of Hartmann by taking Pr = 7.0 for the case 1 and 2 respectively. From Figs. 5 and 6, we find the qualitatively similar results as that in Figs. 3 and 4. But here the maximum value of velocity is less as compared to previous cases (Figs. 3 and 4).

In Figs. 7 and 8, we have shown the effect of magnetic Prandtl number {Pm) on velocity profile for case 1 and 2 respectively. As the magnetic Prandtl number is increased the velocity first increases and then slight decreases and after that becomes steady with time t, we can see by Figs. 7(a) and 8(a). It can be observed that the velocity becomes negative when Pm = 1 in case 1 by Fig. 7(c) whereas in case 2, with the same value of Pm there is slight decrease in velocity profile by Fig. 8(c). By Figs. 7 and 8, it is also observed that the magnitude of velocity decreases with increase in Pm. Physically, it is true due to the fact that on increasing Pm, the kinematic viscosity will increase and therefore resistance to the flow will increase and as a resultant the velocity boundary layer reduces.

Figs. 9 and 10 show the effect of Hartmann number on induced magnetic field profiles for case 1 and case 2 respectively by taking Pr=0.71. It is clear from the Fig. 9 that in case 1, on increasing Ha, the induced magnetic field decreases whereas in case 2, the induced magnetic field decreases on increasing Ha from 5 to 6 but when Ha is further increased from 6 to 8, then induced magnetic field starts increasing which is clearly visible from Fig. 10. As we know that the Lorentz force reduces the velocity boundary layer consequently the induced magnetic boundary layer reduces. In case 1 by Fig. 9, it is observed that the value of induced magnetic field is maximum at inner cylinder and it goes on decreasing towards outer cylinder while in case 2, the value of induced magnetic field is minimum at inner cylinder and it gradually increases towards outer cylinder and attains maximum value at outer cylinder which has been shown in Fig. 10. Figs. 11 and 12 represent the variation in induced magnetic field for different values of Hartmann number by taking Pr = 7.0 for the case 1 and 2 respectively. From Figs. 11 and 12, we obtained the qualitatively similar results as that in Figs. 9 and 10. From Figs. 13 and 14, we have shown the effect of magnetic Prandtl number on induced magnetic field profiles for the case 1 and 2 respectively. The induced magnetic field increases with increase in magnetic Prandtl number in both the cases. Since, as the magnetic Prandtl number increases, the magnetic diffusivity will decease and consequently the magnetic boundary layer will increase. From comparative study of Figs. 13 and 14, we conclude that the maximum magnitude of induced magnetic field is high in case 1.

Fig. 15 illustrate the sketches for skin-friction profiles at r = 1 and r = k respectively in case 1 (when inner cylinder is perfectly conducting and outer cylinder is non-conducting). Fig. 15(a), reveals that skin-friction has decreasing nature for increasing values of Ha and Pm. From Fig. 15(b) we can easily estimate that the skin-friction show exactly opposite behavior at r = k to at r = 1. Fig. 16(a) illustrate the skin-friction profiles in case 2 at r = 1 and reveals that skin-friction decreases for increasing values of Ha and Pm. Fig. 16(b) presents the skin-friction profiles at r = k in case 2. The skin-friction profiles increases for increasing values of Ha and Pm and finally becomes steady-state with time t. This is due to the reason that Ha and Pm both reduces the velocity boundary layer.

Figs. 17 and 18, present the graphs for Nusselt number at constant temperature and ramped temperature respectively at the surface of inner cylinder with respect to time for different values of Pr. We can clearly notice from Fig. 17(a), that at inner cylinder,

-0.04-

-0.06-

-0.08-

-0.12-

No. Ha Pm

1 2 0.1

2 3 0.1

3 5 0.1

4 2 0.5

5 3 0.5

6 5 0.5

7 2 1.0

8 3 1.0

9 5 1.0

—I-'-1-1-1-1-r~

0.5 1.0 1.5 2.0

(a) induced current density at inner cylinder

0.26-, 0.240.220.200.180.160.140.12 0.100.08 0.06 0.04 0.02 0.00 -0.02

—I— 0.5

~l— 1.0

—I 2.5

No. Ha Pm

1 2 0.1

2 3 0.1

3 5 0.1

4 2 0.5

5 3 0.5

6 5 0.5

7 2 1.0

8 3 1.0

9 5 1.0

—I 2.5

(b) induced current density at outer cylinder

Fig. 19. Variation in induced current density in case 1 for different values of Ha and Pm at (a) r = 1, (b) r = k.

with constant temperature, Nuj has maximum magnitude at beginning and with increase in time t, it decreases monotonically and finally becomes steady while with ramped type temperature, the behavior of Nuj profiles is noticeable. Initially, these start with minimum magnitude, increases with time, attain its maximum value at t = 1 and finally becomes steady. This due to the fact that as time increases the heat transfer rate increases and at time t = 1, the temperature of inner cylinder becomes constant and therefore the Nusselt number will merge with steady state. Figs. 17(b) and 18(b) show Nu profiles at outer cylinder. From these figures, we found that with constant temperature at surface of inner cylinder, maximum magnitude is achieved earlier in comparison to that with ramped type temperature at surface of inner cylinder. Also, from Figs. 17 and 18, we conclude that Nuj increase with Pr while Nuk decreases with Pr. In Figs. 19(a) and 19(b) we have presented the graph for Induced current density for cases 1. The induced

current density has increasing trend with increase in Hartmann number at inner cylinder while at outer cylinder, it shows opposite behavior. The reason behind this is that as Ha increases, it generates the induced current in the fluid flow and therefore the induced current density will increase.

5. Conclusions

The effect of radial magnetic field on natural convection between two concentric vertical cylinders with ramped temperature at the inner cylinder is numerically studied. In this study, two phenomenon have been discussed, in first inner cylinder is taken as perfectly conducting and outer is taken as nonconducting and in second inner is taken as non-conducting and outer is taken as perfectly conducting. A computer code is developed to solve the governing equations in MATLAB software. By comparing the two cases, we observed several noteworthy features as:

(i) The influence of Hartman number is to suppress the velocity for both cases and also, the steady-state time is reached later with increase in Ha.

(ii) For case 1, the steady-state is achieved for much less value as compared to that of case 2.

(iii) In case 1, Ha plays an important role in decreasing the magnitude of induced magnetic field

(iv) The maximum magnitude of induced magnetic field occurs on inner cylinder in case 1 while in case 2, it is maximum on outer cylinder which indicates towards conductivity of cylinders. Further, role of Pm is to enhance the magnitude of induced magnetic field in both the cases.

(v) Skin-friction profiles decreases with increasing value of Ha and Pm at inner cylinder while at outer cylinder these show exactly opposite behavior in both the cases.

(vi) Induced current density increases with Ha at inner cylinder while it shows qualitatively contrary behavior at outer cylinder in case 1.

(vii) Nusselt number profiles show decreasing nature with increasing value of Pr at inner cylinder while at outer cylinder these increases with Pr.

An important result of this study is that the heat transfer can be controlled by taking ramped like temperature profile on boundary and by the conductivity of the cylinder.

Acknowledgments

Both the authors are thankful to the Department of Mathematics, Central University of Rajasthan, Ajmer, India to provide the MATLAB Software. Also, we would like to express our thanks to the anonymous editor and referees for their very expertise remarks and kind suggestions to improve the quality of manuscript.

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