OSCILLATORY FLOW OF A VISCOUS CONDUCTING FLUID THROUGH A UNIFORMLY MOVING VERTICAL CIRCULAR CYLINDER UNDER PRESSURE GRADIENT

A study of a transient MHD mass transfer flow of a viscous incompressible and electrically conducting Newtonian non-Gray optically thin fluid through a vertical circular cylinder influenced by a time dependent periodic pressure gradient subject to a magnetic field applied in azimuthal direction, in the presence of a frequency parameter and periodic wall temperature is provided. The flow, heat and mass transfer governing equations are converted into ordinary differential equations by imposing some suitable transformations and solved in closed form using Bessel functions of order zero. Graphs depict the effects of various physical parameters on concentration, velocity, temperature field and on the coefficient of the skin friction, mass flux across a normal suction of the cylinder and the rates of heat and mass transfer at surface of the cylinder in the issue.


INTRODUCTION
MHD viscous fluid flow through pipes is important in many fields of Science and Technology, including Bio-mechanics, Petroleum Industry, Drainage and Irrigation Engineering and so on.
Many authors have investigated the problems of steady and unsteady flows of viscous incompressible fluids through cylinders with various cross sections under various flow geometries and physical aspects. Some of them are Kumari and Bansal [1], Drake [2], Varma et al. [3], Antimirov and Kolyshkin [4], Hughes and Young [5], Sankar et al. [6], Jain and Mehta [7] and Globe [8.] Several scholars, on the other hand, have looked into MHD flows and heat transfer in channels and circular pipes. Chamkha [9] looked at how two different applied pressure gradients (ramp and oscillating) affected unsteady laminar MHD flow and heat transfer in channels and circular pipes. Singh [10] gave an exact solution to the problem of MHD mixed convection periodic flow in a rotating vertical channel in the presence of heat radiation.
Problems with MHD heat transfer and periodic wall temperature are also common.
Israel-Cookey et al. [11] looked into MHD free convection and oscillating flow of an optically thin fluid surrounded by two horizontal porous parallel walls in the presence of periodic wall temperature. In addition, Reddy et al. [12] added periodic wall temperature to their problem.
In the context of geothermal power generation and drilling operations, studies of free convection flow along a vertical or horizontal cylinder are important because the free stream and buoyancy induced fluid velocities are of approximately the same order of magnitude. In the presence of a first-order chemical reaction, Machireddy [13] found a numerical solution to investigate the effects of radiation on MHD heat and mass transfer flow past a moving vertical cylinder.
Ganesan and Loganathan [14] looked at the effects of radiation and mass transfer on the movement of an incompressible viscous fluid through a moving vertical cylinder. The study of flow and heat transfer in the vertical circular cylinder is a focus of investigation due to its wide range of practical applications such as solar collectors, electrical machineries, cooling system for electronic devices and other rotating system.
The effects of thermal radiation on heat and mass transfer are more significant in many system, 4185 OSCILLATORY FLOW OF A VISCOUS CONDUCTING FLUID and they play a key role in the filtrations processes, design of spacecraft, nuclear reactors and the drying of porous material in textiles industries solar energy collector. Raju et al. [15] recently investigated the effect of thermal radiation on an unsteady free convection flow of water near four degree Celsius through a vertically moving porous plate by taking into account the effect of suction/injection. Gundagani et al. [16] investigated the effects of radiation on an unsteady MHD two-dimensional laminar mixed convective boundary layer flow along a vertically moving semi-infinite permeable plate with suction, embedded in a uniform porous medium, thus accounting for viscous dissipation. Rao et al. [17] looked at how radiation influenced the unsteady mass transfer flow of a chemically reacting fluid through a semi-infinite vertical plate in the presence of viscous dissipation. Recently, Ahmed and Dutta [18] carried out an analytical study of a laminar unsteady MHD flow past a vertical annulus in presence of thermal radiation under the influence of periodic wall temperature and pressure gradient. N. Ahmed [19] made a theoretical analysis of a steady MHD free convective flow on a vertical circular cylinder with Soret and Dufour effects.
The present investigation is concerned with the study of unsteady, MHD flow of Newtonian non-Gray optically thin fluid through a vertical circular cylinder in the presence of frequency parameter, time dependent pressure gradient and periodic temperature and concentration maintained at the wall of the cylinder. In the current work, such an attempt has been made.

BASIC EQUATIONS
The equations defining the motion of an incompressible, electrically conducting, viscous and radiating fluid in the presence of magnetic field are: Ohm's law for an electrically conducting fluid: Species continuity equation: The nomenclature specifies all physical quantities.

MATHEMATICAL ANALYSIS
Consider a vertical circular cylinder of radius a with an unsteady laminar radiative flow of a viscous incompressible electrically conducting fluid. In Figure 1 Our investigation is limited to the following assumptions in order to idealize the mathematical model of the problem: i. Except for the density in the buoyancy force term, all fluid properties are constant.
ii. Energy dissipation due to viscous and ohmic dissipation is insignificant.
iii. The radiative heat flux in the vertical direction is insignificant in comparison to that in normal direction.
iv. The induced magnetic field can be ignored because the magnetic Reynolds number is so small.
v. The flow is parallel to the cylinder's axis.
From equations (2) and (6), we get Where, Momentum equation (7) on application of the equations (8) and (9), finally reduces to In view of the assumptions, the following reduced form of the equations (3) is obtained: Additionally, since the cylinder is infinite in both directions and so equation (5) becomes According to Cogley et al. [20] result, the rate of radiative heat flux in the optically thin limit for a non-Gray gas near equilibrium is given by the formula below: Using equation (13) in the equation (11), we arrive at The pertinent boundary conditions are as follows: 11 ,, The following non-dimensional quantities are added to normalize the mathematical model: The non-dimensional forms of equations (10), (12) and (14) are as follows:

METHOD OF SOLUTION
Consider the concentration as ( ) ( ) Hence the solution of equation (22) is Where o J is the Bessel function of first kind, of order 0.
As a result, for the concentration field, we get the following expression: Where o I is zeroth order modified Bessel function of first kind.  GmA A  = − .

MASS FLUX
The mass flux over every normal section of the cylinder can be calculated using the formula: The mass flux coefficient f M is calculated using equation (27)  I is modified Bessel functions of first kind, of order 1.

SKIN FRICTION
The Newton's law of viscosity, as shown below, gives the viscous drag per second area on the cylinder's surface: In Figures 2 and 3, the temperature profiles are illustrated. These diagrams show how Pr and  affect the temperature field. Figure 2 depicts the effect of changing the value of Pr on the temperature field. This figure shows that when the fluid's thermal diffusivity is decreased, the flow accelerates, which may be due to the fact that low thermal diffusivity causes a corresponding increase in the kinetic energy of the fluid's molecules. In addition, as shown in Figure 3, the frequency parameter has a tendency to lower fluid temperature. It is worth nothing that the fluid temperature drops as the temperature on the cylinder's surface oscillates. As a result, the frequency parameter serves as a useful regulatory mechanism for preserving the desired temperature field. The concentration profiles for  and Sc variance are shown in Figures 6 and 7. Figure 6 shows that as the frequency parameter's magnitude increases, so does the fluid concentration. Figure 7 depicts the increase in concentration field caused by an increase in Sc. It is worth noting that as Sc rises, mass diffusivity decreases. As a result, high mass diffusivity causes fluid concentration to slow down. These figures exhibit a common feature that the behaviour of mass flux under the effects of these parameters is periodic due to the pressure gradient being periodic function of time. Furthermore, we can see from these graphs that the magnitude of mass flux, which measures the rate at which mass is transmitted, increases as Gm and Pr increase caused by an increase in solutal buoyancy force and thermal diffusivity. Figures 10 and 11 show how changes in Gm and Gr affect skin friction on the cylinder's surface as times goes on. Figures 10 and 11 shows that the direction of skin friction changes on a regular basis. This could be explained by the fact that the pressure gradient is periodic. Figure 11 show that as Gr rises, skin friction rises in the direction of fluid flow, which may be due to the buoyancy force acting on the fluid. However, due to the increase in solutal buoyancy force, the magnitude of skin friction increases as Gm increases (see Figure 10).

CONCLUSIONS
The following are the key findings of the previous investigation: • The fluid temperature falls under the influence of thermal diffusivity and frequency parameter.
• The fluid velocity increases as the solutal Grashof number rises, but the opposite is true as the magnetic parameter rises.
• Increase the mass diffusivity associated with the fluid flow to decrease species concentration.
• The mass flux is increased by the solutal buoyancy force, while themal diffusivity has the opposite effect.
• With increasing solutal and thermal Grashof numbers, the level of viscous drag in the fluid increases.
• The rate of heat transfer and mass transfer increases as the frequency parameter is increased.