Transient development of MHD natural convection flow in vertical concentric annulus

Received Mar 29, 2019 Revised Apr 20, 2019 Accepted Mar 12, 2020 This paper examines the role of magnetic field on fully developed natural convection flow in an annulus due to symmetric of surfaces. The transport equations concerned with the model under consideration are rendered non-dimensional and transformed into the ordinary differential equation using Laplace transform technique. The solution obtained is then transformed to time domain using the Riemann-sum approximation approach. The governing equations are also solved using implicit finite difference method so as to establish the accuracy of the Riemann-sum approximation approach at transient as well as at steady state solution. The solutions obtained are graphically represented and the effects of pertinent parameters on the flow formation are investigated in detail. The Hartmann number (M), is seen to have a retarding effect on the velocity, skin-frictions and the mass flow rate. Also, skin-friction at both surfaces and the mass flow rate within the annulus are found to be directly proportional to the radii ratio (λ).

 ISSN: 2252-8792 Int J Appl Power Eng, Vol. 9, No. 1, April 2020: 58 -66 60 the radial direction. A magnetic field of strength 0 is assumed to be uniformly applied in the direction perpendicular to the direction of flow. In the present physical situation, the inlet fluid temperature is maintained at 0 , while a constant uniform heating of is applied at the outer surface of the inner cylinder and at the inner surface of the outer cylinder such that > 0 as presented in Figure 1. The flow is assumed to be fully developed both thermally and hydrodynamically, and the viscous dissipation, radiation, and compressibility effects are neglected. Following the work of Jha et al. [24] the momentum and energy equations governing the present physical situation is given by (1) and (2).
The relevant dimensional initial and boundary conditions are; Introducing the following dimensionless quantities in (1) Equations (1) and (2) in dimensionless form are obtained as follows: The initial and boundary conditions in dimensionless form are: The physical quantities used in (1) to (6) are defined in the nomenclature. The solution of (5) and (6) with the associated initial and boundary conditions (7) and (8) can be obtained by using the Laplace transform technique. Defining the following transform variables.
Where the Laplace parameter, ( > 0) in (5) and (6) are transformed into the Laplace domain using the initial condition (7) to obtain.
Applying Laplace transform technique (9) on the boundary conditions (8), we have: The solution of (10) and (11) in Laplace domain subject to the boundary conditions (12) are: Equation (13) and (14) are to be inverted in order to obtain their solutions in the time domain. Due to the complex nature of these inversions, we adopt a numerical procedure used in Jha and Yusuf [25] as well as Jha and Apere [26] which is based on the Riemann-sum approximation. According to this technique, any function in the Laplace domain can be inverted to the time domain as follows: where Re refers to the real part of = √−1 the imaginary number. N is the number of terms used in the Riemann-sum approximation and is the real part of the Bromwich contour that is used in inverting Laplace transforms. The Riemann-sum approximation for the Laplace inversion involves a single summation for the numerical process its accuracy depends on the value of and the truncation error dictated by M. According to Tzou [27], the value of that best satisfied the result is 4.7.
The solutions are as follows: where 0 , 0 , 1 , 1 are the modified Bessel function of first and second kind of order 0 and 1 respectively. In the same manner, the solutions are inverted to the time domain by applying the Riemann-sum approximation stated in (15).

Validation of the method
The accuracy of the Riemann-sum approximation approach in (14) is validated by computing the steady-state solution for the velocity field. This is obtained by taking (5) and (6) which then reduces to the following ordinary differential equations. The implicit finite difference method has also been used to validate the Riemann-sum approximation approach, the advantage of this numerical procedure over others is that comparison can be made at both steady and transient state solution of the transport equations.
These are solved under the boundary conditions (8) to obtain the expressions for the steady-state velocity field, steady-state temperature field, steady-state skin frictions as well as the mass flow rate of the fluid. The solutions are respectively; The constants 5 6 in (25) are stated by: The numerical values of the velocity obtained using the Riemann-sum approximation approach, implicit finite difference method and those obtained from the exact solution of the steady-state choosing value of = 2, = 2. Is presented in Table 1. The comparison between the results, shows that at large time (steady-state) there is an excellent agreement between the Riemann-sum approximation approach and the implicit finite difference.

RESULTS AND DISCUSSION
In order to have a clear insight of the physical problem under consideration, a numerical computation is performed using the mathematical laboratory software (MATLAB) to compute and generate graphs for the velocity field, temperature field, skin-frictions and mass flux for different values of the governing parameters, so as to comment on their relative contribution to the flow formation. In this work, two different cases of fluid are been examined these include air with = 0.71 and water with = 7.0. The effect of variation of the governing parameters , Pr on the flow formations are presented in Figures 2-9. Unless otherwise stated, the value = 0.2, = 2 = 2 are selected arbitrarily to study the effect of various parameters on the flow behavior. The influence of Prandtl number and time on the temperature profiles is shown in Figure 2. It is revealed in Figure 2 that fluid temperature increases as and increases. It is concluded from Figure 2 that the reduction in fluid temperature is directly proportional to the decrease in thermal diffusivity. Figure 3 shows that an increase in the values of the Hartmann number causes retardation to the fluid flow indicating the fact that the imposition of magnetic field slow down the flow.  This remark is consistent with the physical fact that the Lorentz force that appears due to the interaction of the magnetic field and the fluid velocity resists the corresponding fluid flow, resulting in the velocity to decrease gradually. Figure 3 also indicates how the velocity field is affected corresponding to an increase in the values of . We recall that an increase in in Figure 2 signifies a fall in thermal diffusivity for the model under consideration. It is learnt from Figure 3 that when the thermal diffusivity of the fluid is reduced, the flow gets decelerated largely which may be attributed to the fact that a low thermal diffusivity leads to a corresponding decrease in the kinetic energy of the molecules of the fluid, which in turn  Figure 3 also revealed that the fluid velocity increases with increase in time till it attains steady state. It is worthy to note that has no effect on the velocity and temperature profiles at steady state. Figures 4 and 6 present variation of skin-friction at outer surface of the inner cylinder ( = 1) and the inner surface of the outer cylinder ( = ) respectively for different values of the Hartmann number ( ). It is obvious from these Figures 4 and 6 that skin-friction decreases with increase in Hartmann number ( ) on both surfaces for both cases of . Fluid with = 0.71 is observed to induces higher friction at both surfaces of the cylinders in comparison with fluid with = 7.0 which is physically true, since higher velocity results in friction at the walls. In addition, it is clear that skin friction at both cylinders for air attains steady state faster than water. This suggests that if one considers to reduce the friction at the surfaces, fluids with higher Prandtl numbers like water ( = 7) should be considered. Variation of skinfriction profiles at the outer surface of the inner cylinder and the inner surface of the outer cylinder for different values of radii ratio ( ) are shown respectively in Figures 5 and 7. It is obvious from both Figures 5  and 7 that skin-friction on both cylinder increases with ( ). A keen scrutiny of the Figures 5 and 7 reveal that the skin-friction is independent of time for = 0.71 except as tends to zero.  Figure 9 illustrations variation of mass flow rate for different values of radii ratio ( ). It is evident that mass flow rate increases with increase in radii ratio ( ) for both cases of ( ). It is worthy to note that mass flow rate is constant for different values of radii ratio ( ) in the case = 0.71, but increases with time ( ) for = 7.0. Figures 8 and 9 lead us to conclude that the parameters and have significant contributions in regulating the amount of total discharge of fluid through the annulus and they may be suitably chosen to control the mass flux.

CONCLUSION
A semi-analytical study is conducted to examine the role of magnetic field on an incompressible and an electrically conducting fluid filled within two coaxial cylinders. The Laplace transform technique and Riemann-sum approximation method have been used to obtain the solution of the governing equations. The influence of Hartmann number ( ), Prandtl ( ), radii ratio ( ), and time ( ) on the velocity field, temperature distribution, skin frictions and mass flow rate have been extensively discussed. The main findings in the present research are: a. It is found that an increase in Hartmann number ( ) has a retarding effect on the velocity field, mass flow rate and skin-friction at both surfaces. b. It is worthy to conclude that an increase in the radii ratio ( ) increases the skin-friction at both surfaces. c. Air is established to have higher fluid velocity, mass flow rate and skin-friction on both surfaces in comparison with water. d. Generally, Air ( = 0.71) attains steady state temperature faster, due to its higher thermal diffusivity (See Figure 3) in comparison with water ( = 7.0). e. Skin-friction at both surfaces and Mass flow rate are seen to be independent of time except at small value of time ( ).