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Temperature dependent viscosity and thermal conducting heat and mass transfer flow with chemical reaction and periodic magnetic field past an isothermal oscillating cylinder have been considered. The partial dimensionless equations governing the flow have been solved numerically by applying explicit finite difference method with the help Compaq visual 6.6a. The obtained outcome of this inquisition has been discussed for different values of well-known flow parameters with different time steps and oscillation angle. The effect of chemical reaction and periodic MHD parameters on the velocity field, temperature field and concentration field, skin-friction, Nusselt number and Sherwood number have been studied and results are presented by graphically. The novelty of the present problem is to study the streamlines by taking into account periodic magnetic field.

In recent, the study of unsteady free convective flows through an oscillating cylinder evinces a vital role in chemical engineering, turbo machinery, and aerospace technology. The occurrence of heat and mass transfer is too general in chemical process industries such as the production of polymer and food processing. The study of MHD incompressible viscous flows has several vital engineering applications in devices such as MHD power generators, electrical components transmission lines, cooling of nuclear reactors, geothermal systems, forming metal, crystal growing, aerodynamic processes, and heat exchange designs. The focus of a huge number of researchers has drowned by MHD due to its various applications such as pumps, bearings. Implicit finite-difference scheme of Crank-Nicolson type has been used to solve radiation and mass transfer effects on unsteady MHD free convection flow of an incompressible viscous fluid past a moving vertical cylinder which has been analyzed by Reddy et al. (2009) [

Unlike other fluid, Cintaginjala et al. (2014) [

By considering temperature dependent viscosity and thermal conducting heat and mass transfer flow with chemical reaction and magnetic field past an isothermal oscillating cylinder have been studied. Unlike other researchers, we have used magnetic field periodically. The main aim of this paper is to investigate the effects of chemical reaction, periodic magnetic field on velocity field, temperature field and concentration field, skin-friction, Nusselt number, Sherwood number and stream-lines with different time steps, oscillation angle and also compared with the absence of a non-periodic magnetic field. The partial dimensionless equations governing the flow are solved numerically by using explicit finite difference method with the help of Compaq visual 6.6a.

In the presence of periodic magnetic field unsteady two-dimensional free convective flow of a viscous incompressible electrically conducting fluid past a semi-infinite oscillating cylinder of radius r_{0} have been investigated. Here, the x-axis is taken along the axis of cylinder in the vertical direction and the radial coordinate r is considered as normal to the cylinder. Initially the fluid and the cylinder are at the same temperature _{0}.

The temperature of the surface of the oscillating cylinder is increased to _{0}) is imposed to the oscillating cylinder which is presented in

With boundary conditions,

It is necessary to make the Equations (1) to (4) with boundary conditions (5) dimensionless. For this intention we introduce the following dimensionless quantities

If γ and ε denotes the non-dimensional viscosity variation parameter and thermal conductivity then

The corresponding boundary conditions in terms of non-dimensional variables are

Skin friction coefficient, the rare of heat transfer rate and Sherwood number are expressed as follows

An explicit finite difference method has been devoted to solve the nonlinear partial differential Equations (7)-(10) along with boundary condition (11). The finite difference equations for the Equations (7)-(10) have been recounted by the Equations (15) to (18) respectively

To get the finite difference equations the region of the periodic MHD flow is divided into the grids or meshes of lines parallel to X and R is taken normal to the axis of the oscillating cylinder. Here we consider that the height of the cylinder is X_{max} = 20.0 i.e. X varies from 0 to 20 and regard R_{max} = 50.0 as corresponding to R → ∞. In the above Equations (15) to (18) the subscripts i and j designate the grid points along the X and R coordinates, respectively, where X = iΔX and

In order to obtain the corporal insight of the problem of the study, the velocity profile, temperature profile and concentration profile are reveal by assigning numerical values to different parameters encountered into the corresponding equations. The value of Schmidt number (Sc) are chosen for Hydrogen gas diffusing in electrically-conducting Air (Sc = 0.20), Helium (Sc = 0.30), steam (Sc = 0.60), Oxygen (Sc = 0.66), NH_{3} (Sc = 0.78),CO_{2} at 250C (Sc = 0.94), Methanol (Sc = 1.00) and high viscous fluid (Sc = 5.00). The value of Prandtl number (Pr) are chosen for air (Pr = 0.71), Nitrogen (Pr = 0.78), water (Pr = 7.0) and water at 40C (Pr = 11.62).

With the increases of viscosity variation parameter (γ) the velocity decreases which elucidates in the

different time steps. The velocity of the fluid is increasing with the increasing of time steps and low periodic magnetic field indicates more smooth curves than the higher magnetic field. Due to the increasing values of Prandtl number (Pr) and Scmidth number (Sc) increases the viscosity of the fluid and decrease the molecular diffusivity which decreases the velocity of fluid. For the increasing values of Prandtl number (Pr), Scmidth number (Sc) and magnetic parameter (M) the velocity curves let on a different shape and its decreasing which have been indicated in

71.512%, 19.57% respectively.

apparent that the curves are upward direction with the decreasing values of Scmidth number (Sc). By descending order of Scmidth number (Sc) at same point R = 2.333, the difference of temperature between two curves are 6.9225%, 38.56% and 23.18% respectively.

Increase of chemical reaction (K) significantly alerts the concentration boundary layer thickness but does not alerts momentum boundary layer. Due to the increasing values of Sc, Pr and K the concentration curves let on different curves for fixed values rest parameters as shown in

equivalent to increase the thermal conductivities and therefore heat is able to diffuse from the cylinder more rapidly than higher values of Prandtl number. For the increasing values of Prandtl number the values of Nusselt number increases which is displayed in

Finally, a comparison of the present results with the published results (Machireddy [

Previous results given by Machireddy [ | Present results | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Increased Parameters | U | T | C | Skin fri. | Nusl. Num. | Sher. Num. | U | T | C | Skin fri. | Nusl. Num. | Sher. Num. |

M | Dec | Dec | Dec | Dec | Dec | Dec | ||||||

Pr | Dec | Dec | Not Given | Not Given | Dec | Dec | Dec | Inc | ||||

N | Dec | Inc | Not Given | Not Given | ||||||||

K | Dec | Dec | Dec | Dec | ||||||||

Sc | Not given | Dec | Dec | Inc | Dec | Dec | Dec | Inc |

An elaborated numerical analysis has been performed for the effects of the chemical reaction of the first order on the periodic MHD free convective flow for a gas past a moving semi-infinite oscillating cylinder with variable kinematic viscosity and thermal conductivity. The partial dimensionless equations governing the flow have been solved numerically by applying explicit finite difference method with the help Compaq visual 6.6a. The obtained outcome of this inquisition has been discussed for different values of well-known flow parameters with different time steps and oscillation angle. By analyzing the problem, the concluding remarks have been carried out as follows:

・ The velocity decreases with an increase of Scmidth number (Sc), Prandtl number (Pr) and periodic magnetic field (M) also higher magnetic field indicate more non- smooth curves than the lower periodic magnetic field (M). i.e. the wavy curves occurs only when we impose the magnetic field (M) periodically.

・ Higher oscillation angle (ϕ) indicates the lower point on the wall than lower oscillation angle (ϕ) at which the initial velocity starts.

・ With the decreasing of chemical reaction parameter (K), viscosity variation parameter (γ), result to increasing the velocity profiles while velocity increases with the increases of thermal conductivity (ε).

・ For the decreasing values of Scmidth number (Sc) and Prandtl number (Pr), the temperature increases while temperature increases for increasing values of thermal conductivity (ε).

・ The concentration increases with the decreasing values of Scmidth number (Sc), Prandtl number (Pr) and chemical reaction parameter (K).

・ Nusselt number increases for the increasing values of Prandtl number (Pr), Scmidth number (Sc) and skin-friction decreases for the increasing values of the periodic magnetic field (M), Prandtl number (Pr).

・ Sherwood number increases with the increasing values of Scmidth number (Sc).

・ With the increases of viscosity variation parameter (γ) and thermal conductivity (ε) increases the values of stream-lines also lower periodic magnetic field (M) indicates the more smooth streamlines than the higher periodic magnetic field (M).

We acknowledge the ministry of National Science and Technology (NST), Dhaka, Bangladesh and thank for providing financial support for this research work as a fellowship in 2015-2016.

Ahmed, R., Uddin, R., Jewel Rana, B.M., Selim, R. and Ahmmed, S.F. (2016) Study on Periodic MHD Flow with Temperature Dependent Viscosity and Thermal Conductivity past an Isothermal Oscillating Cylinder. World Journal of Mechanics, 6, 419-440. http://dx.doi.org/10.4236/wjm.2016.611030