Influence Of Imposed Magnetic Field On Flows In Some Geometries:A Numerical Approach

Magnetic field effects are encountered in numerous engineering applications which include but are not limited to metal casting,nuclear reactor coolers,and geothermal energy extraction.On the other hand,the heat transfer in Magnetohydrodynamic flows has been potentially investigated due to its applications in several systems such as heat exchangers,electromagnetic casting,adjusting blood flow,X-rays,magnetic drug treatment,cooling of nuclear reactors,and magnetic devices for cell separation.The purpose of this dissertation is to understand how the imposed magnetic field interacts with the underlying structure of Newtonian fluids/non-Newtonian fluids(for example,nanofluid/micropolar fluid)in some important(from an application point of view)problems.We employed a numerical approach for solving the complex nonlinear ordinary or partial differential equations,resulting from the mathematical modeling of the fluid flow in the presence of a magnetic field,as the analytical solution for such complicated problems either does not exist or is very difficult to find.The governing partial differential equation(Navier-Stokes equations or some variant)is solved in dimensionless form using the Stream-Vorticity formulation.We present a numerical solution to a variety of problems dealing with different fluids in several geometries like a cavity with moving lids,a channel having porous walls,and a vertical duct with the dipole placed nearby.Unlike most of the works found in the scientific literature,we do not assume a uniform magnetic field everywhere in the flow regime,which seems to be more realistic.In a single-phase approach to modeling the nanofluid,we know that the nanofluid particles are very small,uniformly distributed,and easily fluidized.It is a more practical approach to assessing the heat transport of nanofluids.So,we will consider the single-phase model(SPM)whereas the flow,as well as fluid properties,would be grouped together to give rise to dimensionless parameters.In chapter 4,we investigate the nanofluid flow in the cavity under the influence of the variable magnetic field,we opt to choose a relatively simple finite difference-based numerical approach for solving the stream-vorticity formulation of the governing NavierStokes equation,resulting from the mathematical modeling of the problem.Due to spatially varying magnetic field,we prefer to solve the complete set of Navier stokes equations and even those the bulkier and complicated ones than the original Navier stokes equation.We have to solve the modified Navier Stokes equations without complexity.So that we used the Pseudo-Transient approach in which we incorporated fictitious time derivative terms into the governing partial differential equations and we adopted a time margin technique in which we kept on solving the equation till we reach the time level where we reach a steady state.That is the system no longer depends upon the time.So,it means that we have solved a system and whereas for the derivative we used central differences because of their accuracy and simplicity.The finding shows that the magnetic field creates new vortices nearby the dipole and these new vortices also seem to be rotating in the opposite direction due to dipole movement.Increasing the strength of the dipole results in distorting the symmetry of the streamlines by first enhancing the size of the lower vortex;some vortices near the dipole also join and merge together.In the presence of dipole,the skin friction becomes more effective compared to the Nusselt number along the lower wall of the cavity while the Nusselt number enhances around the dipole and reduces along the left adiabatic wall.The applied magnetic field moves the region of a higher thermal gradient to the location of a dipole while also making the temperature field non-symmetric.The Reynolds number reduces the Nusselt number along the lower wall while affecting the strength of the vortices near the dipole location.It also enhances the flow velocity but does not remarkably distort the symmetry of the problem.There is a substantial decrease in the Nusselt number and negligible effect on skin friction as the nanoparticle volume fraction increases.The problem of micropolar flow in a porous channel has been studied in chapter 5,where we have to solve the system of nonlinear ordinary differential equations.We have presented that there are many numerical methods and usually,a shooting-like approach is used to tackle such types of equations but we have used the Quasi-linearization technique because of its minimum requirement of human input while obtaining the solution.Using a quasi-linearization technique for numerically solving the nonlinear system of dimensionless ordinary differential equations arising due to the incorporation of similarity variables into the governing partial differential equations.The results show that the permeability parameter and the material parameters tend to enhance the microrotation,but these parameters depreciate the normal velocities.The Reynolds number inserts a low effect on couple stresses while it yields a significant effect on skin friction and heat transport rates.The permeability parameter substantially enhances shear stresses,couple stresses,and the rates of heat transfer on both the channel walls.The impact of the Eckert and the Prandtl number is to uplift the temperature curves.The micropolar constants intensively affect the microrotation rather than the streamwise and the normal velocities.The micropolar fluid causes an escalation in couple stresses and a reduction in the shear stresses.Finally,in chapter 6,a finite volume approach has also been incorporated for the numerical study of the problem of flow in a vertical duct under the thermal boundary condition of axially uniform wall heat flux with a constant peripheral temperature.We came across the system of coupled Poisson equation and for that,we use the vortex-based finite volume method where a control volume is considered around a point on the domain,and the governing equations are integrated over that control volume.As a result,the system of algebraic equations is obtained which are solved numerically.This chapter sheds light on the changes in the velocity and temperature fields,as well as on the friction factor,and the Nusselt number in the fully developed flow of a nanofluid in a vertical rectangular duct due to a dipole placed near a corner of the duct.A significant rise in the Nusselt number is noted in the presence of a strong magnetic field due to the dipole and when the duct shape is transformed from rectangular to square,the number is reduced remarkably.In the case of free convection,the impact of the presence of dipole and the aspect ratio on the skin friction and the Nusselt number is the same qualitatively,as compared to the forced convection case.However,forced convection is far more effective quantitatively on a rectangular duct.For both free and forced convection,an opposite trend of f Re is recorded in the presence of a magnetic field compared to the Nusselt number.The temperature distribution is not significantly affected by the dipole but it only affects the velocity and temperature distributions at the duct corner near the dipole.With the help of our self-developed computer codes in the MATLAB language,we intend to understand the way for which the pertinent parameters affect the various aspects of the flows in different geometrical configurations.For validating our computational schemes,the numerical results for the Nusselt number and skin friction are compared with the ones available in the literature and are found to be in good agreement.The impact of the relevant governing parameters on the streamlines,isotherms,Nusselt number,shear stresses,couple stresses,skin friction,microrotation,temperature,and velocity profiles are interpreted and discussed through tables or graphs whichever is appropriate.