**1. Introduction and definition of terms**

Magneto-hydrodynamics (MHD) is the study of the motion of electrically conducting fluid. The study of magneto-hydrodynamic plays an important role in agriculture, engineering and petroleum industries. For instance, it may be used to deal with problems such as cooling of nuclear reactor by liquid sodium. The importance of MHD cannot be over emphasized. MHD has applications in many areas like the earth, sun, industry, fusion etc.

To appreciate the importance of fluid dynamics in life demands little more than just a glance around us. In general, life as we know would not exist if there are no

fluids and the behavior they exhibit. The water and air we respectively drink and breathe are fluids. In addition, our body fluids are mostly water based. As a matter of fact, our body system is made up of about 75% of fluid which helps in regulating the activities of the body system ranging from body temperature control to waste removal. In a more practical setting, like in our transportation systems, recreation, entertainment (sound from radio speakers) and our sleep (water beds), fluids greatly influence our comfort. It is clear to see from this that engineers need a clear knowledge of fluid behavior to handle many systems of their encounter.

Over the past decades, studies have been carried out on magneto-hydrodynamic natural convection in an annulus under different physical situations and geometry. This is because of its applications in nature, engineering, industries and technologies. These applications include but not limited to underground disposal of radioactive waste materials, storage of foodstuffs, exothermic and/or endothermic chemical reactions, heat removal from nuclear fuel debris, dissociating fluids in packed bed reactors, aerodynamics, geothermal energy extraction, purification of crude oil and spacecraft, MHD generators, MHD flow meters and MHD pump.


The presence of magnetic field on a Couette flow induces a Lorentz force which either accelerates or decelerates the flow element between the planes which depend on the electrical properties of the plane. The need to control the motion of the boundary layer is one motivation for this study. In many technological phenomena, *Magneto-Hydrodynamic Natural Convection Flow in a Concentric Annulus with Ramped… DOI: http://dx.doi.org/10.5772/intechopen.100827*

such as earth core, aeronautics etc. the motion of a system initially start with an accelerated velocity and then after some time moves with almost constant velocity. This prompted us to consider the ramped like motion of a concentric cylinder and analyze the flow formation.

Chandran *et al.* [1] studied natural convection near a vertical plate with ramped wall temperature and they obtained two different solutions, one valid for Prandtl number different from unity and the other for which the Prandtl number is unity. They concluded that the solutions for dimensional velocity and temperature variables depend upon the Prandtl number of the fluid and the expression of the fluid velocity is not uniformly valid for all values of Prandtl number. In their work, heat generating/absorption is absent. However, when the temperature differences are appreciably large, the volumetric heat generation/absorption term may exert strong influence on the heat transfer and as a consequence on the fluid flow as well. Jha *et al.* [2] studied natural convection flow of heat generating or absorbing fluid near a vertical plate with ramped temperature and consider two cases, plate with continuous ramped temperature and the other with isothermal temperature. They concluded that the isothermal case is always higher than the ramped case. The above mentioned works are carried out in the absence of magnetic field. Seth and Ansari [3] considered hydro-magnetic natural convections flow past an impulsively moving vertical plate embedded in a porous medium with ramped wall temperature in the presence of thermal diffusion with heat absorption. Nandkeolyar and Das [4] studied unsteady MHD free convection flow of a heat absorbing dusty fluid past a flat plate with ramped wall temperature. Seth and Nandkeolyar [5] studied MHD natural convection flow with radiative heat transfer past an impulsively moving plate with ramped wall temperature. Again, Seth et al. [6] investigated hydromagnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid past an accelerated moving plate with ramped temperature and ramped surface concentration through a porous medium. Recently, Khadijah and Jibril [7] studied Unsteady MHD natural convection flow of heat generating/absorbing fluid near a vertical plate with ramped temperature and motion. In the same year, Khadijah and Jibril [8] investigated Time dependent MHD natural convection flow of a Heat generating/absorbing fluid near a vertical porous plate with ramped boundary conditions.

Jha and Jibril [9] investigated hydro-magnetic flow due to ramped motion of the boundary. In their work they studied the effect of magnetic field on velocity and skin friction, due to ramped motion of the horizontal plate and it was concluded that the ramped motion are less compared to the constant motion. However, the effect of ramped temperature profile was not considered. Kumar and Singh [10] studied the transient magneto hydrodynamic Couette flow with ramped velocity. The velocity of the magnetic field, applied perpendicular to the plate is taken to be different from the velocity of the lower plate (the lower plate is moving with ramped velocity). It was concluded that the effect of the velocity on the magnetic field is to increase the velocity of the fluid from the upper plate to the lower plate Jha and Jibril [11] studied the effects of transpiration on the MHD flow near a porous plate having ramped motion. In their work they compare flow formation due to ramped motion of porous plate with the flow formation due to constant motion of the porous plate. Jha and Jibril [12] studied the time dependent MHD Couette flow due to ramped motion of one of the boundaries. It was found that velocity and skin friction increases with an increase of Hartman number when the magnetic field is fixed with respect to the moving plate. While the reverse when it is fixed with respect to the fluid. Jha and Jibril [13] investigated the unsteady hydromagnetic Couette flow due to ramped motion of the porous plate. The aforementioned works were carried out on a horizontal plate.

Jha and Apere [14] on the other hand investigated Unsteady MHD Couette flow in an annulus, by applying Riemann-sum approximation approach to obtain the Laplace inversion of their solution in time domain. Jha and Apere [15] studied unsteady MHD two-phase Couette flow of fluid particles suspension in an annulus. In their work, they employed the D'Alermbert method used by Reccebli and kurt in conjunction of Riemann sum approximation method for both cases of the magnetic field being fixed to either the fluid or the moving cylinder to obtain the solution of the problem. Anand [16] investigated the Effect of radial magnetic field on free convective flow over ramped velocity moving vertical inner cylinder with ramped type temperature and concentration. In the same year, Anand [17] studied the effect of radial magnetic field on natural convection flow in alternate conducting vertical concentric annuli with ramped temperature. Anand [18] studied the effect of velocity of applied magnetic field on natural convection over ramped type moving inner cylinder with ramped type temperature solved numerically by using implicit finite difference Crank–Nicolson method. They found out that, when velocity is employed to magnetic field, then effect of magnetic field gets reversed and the effect of velocity of magnetic field get more pronounced with radii ratio. Also, Hartmann number and time parameter have increasing effects on the skin-friction profile. Taiwo [19] investigated the exact solution of MHD natural convection flow in a concentric annulus with heat absorption. It is found that the magnitude of maximum fluid velocity is greater in the case of isothermal heating compared with the constant heat flux case when the gap between the cylinders is less or equal to radius of the inner cylinder. More also, the various values of the non-dimensional heat absorption parameter (A) and the corresponding values of annular gap are almost the same conditions. Other important researchers [20–28] investigated MHD flow under different physical geometry and thermal conditions of the boundaries.

To the best of the authors' knowledge, no studies have been reported concerning the combined effect of constant temperature, heat generating/absorbing parameter, MHD and ramped like motion of the inner cylinder and temperature in a concentric cylinder. The condition involving ramped like cylindrical motion appears in aerodynamics and oil refinement industry. Therefore, it is important to analyze the flow processes and try to understand the function of related mechanics of ramped moving vertical cylinder and ramped temperature.

#### **2. Mathematical formulation**

This research considers the time dependent natural convection flow of viscous, incompressible and electrically conducting fluid formed by two cylinders of infinite length with radius *a* and *b* such that *a*<*b* under the influence of transverse magnetic field. The motion as well as the temperature of the inner cylinder is ramped while the motion together with the temperature of the outer cylinder is fixed. The z - axis is taken along the axis of the cylinder in the vertical upward direction and *r*<sup>0</sup> -axis is in the radial direction. A magnetic field of strength *B*<sup>0</sup> is assumed to be uniformly applied in the direction perpendicular to the direction of flow. In the present physical situation, a constant isothermal heating of *Tw*is applied at the outer surface of the inner cylinder such that *Tw* > *T*0. When the time is greater than zero that is *t* <sup>0</sup> > 0, the temperature of the cylinder is increased or decreased to *T*<sup>0</sup> <sup>0</sup> þ *T*<sup>0</sup> *<sup>w</sup>* � *T*<sup>0</sup> *<sup>t</sup>* 0 *t*0 , and it begins to move with a velocity proportional to *f*(*t* 0 ) when *t* <sup>0</sup> ≤ *t*<sup>0</sup> and thereafter *t* <sup>0</sup> > *t*<sup>0</sup> is maintained at constant temperature *Tw* as presented in **Figure 1**.

The momentum and energy equations governing the present physical situation are given by;

*Magneto-Hydrodynamic Natural Convection Flow in a Concentric Annulus with Ramped… DOI: http://dx.doi.org/10.5772/intechopen.100827*

**Figure 1.** *Physical configuration.*

$$\frac{\partial u'}{\partial t'} = \nu \left[ \frac{\partial^2 u'}{\partial r'^2} + \frac{\mathbf{1}}{r'} \frac{\partial u'}{\partial r'} \right] - \frac{\sigma B\_0^{-2} u'}{\rho} + \mathbf{g}\beta (T' - T\_0) \tag{1}$$

This is valid when the magnetic lines of force are fixed relative to the fluid. If the magnetic field is also having ramped motion with the same velocity as the moving cylinder, the relative motion must be accounted for. In this case the Eq. (1) above is replaced by:

$$\frac{\partial u'}{\partial t'} = \nu \left[ \frac{\partial^2 u'}{\partial r'^2} + \frac{1}{r'} \frac{\partial u'}{\partial r'} \right] - \frac{\sigma B\_0}{\rho} (u' - V\_0 f(t')) + \mathbf{g} \beta (T' - T\_0) \tag{2}$$

This is valid when the magnetic lines of force are fixed relative to the moving cylinder. Eqs. (1) and (2) can be combined to obtain the momentum and energy equation respectively.

$$\frac{\partial u'}{\partial t'} = \nu \left[ \frac{\partial^2 u'}{\partial r'^2} + \frac{1}{r'} \frac{\partial u'}{\partial r'} \right] - \frac{\sigma B\_0}{\rho} \left( u' - K\xi(t') \right) + \mathbf{g}\beta (T' - T\_0) \tag{3}$$

$$\frac{\partial T'}{\partial t'} = \frac{k}{\rho c\_p} \left[ \frac{\partial^2 T'}{\partial r'^2} + \frac{1}{r'} \frac{\partial T'}{\partial r'} \right] + \frac{Q}{\rho c\_p} \tag{4}$$

The relevant dimensional boundary conditions are;

$$t' \le 0 \qquad u' = 0, \quad T' = T\_0 \quad \text{For } a \le r' \le b$$

$$t' > 0 \qquad \left\{ \begin{array}{ll} u' = U\_0 f(t'), & T' = T\_0' + (T\_w' - T\_0) f(t') \\ & u' = 0, \quad T' = T\_w \quad \text{ at} \quad r' = b \end{array} \right. $$

Where,

K ¼ 0 if the magnetic field is fixed relative to the fluid 0*:*5 if the velocity of the magnetic field is less than the velocity of the moving cylinder 1 if the magnetic field is fixed relative to the moving cylinder 8 >>>>< >>>>:

Anand and Kumar [10].
