# Research

Research work

To design, an efficient magneto-hydrodynamic (MHD) device or to calculate the performance of existing device requires an accurate analysis of the behavior of the working fluid in the device. Therefore the study of interaction of the conducting fluid with electromagnetic fields, separation, stability and physical parameters (shear stress, heat transfer rate to the wall, boundary layer thickness etc.) of the flow problem is very much needed. The governing equations of such a flow problem reduced to nonlinear partial differential equations.

As the boundary layer is subjected to maximum thermal and gas dynamic stresses, the analysis is confined to the entrance region of the flow. Many authors have obtained similar solutions with unrealistic assumptions. However, I have obtained non-similar solutions for compressible flows of subsonic and supersonic cases with more realistic assumptions. The effect of electromagnetic boundary layer heat transfer rate is analyzed and a parametric survey has been made for most practical problems with the inclusion of Hall and ionslip currents. Boundary layer separation has been studied for incompressible flow, using unconditionally stable Crank-Nicolson scheme. Using linear stability theory, the analysis of fully developed MHD flows in channels has been studied. The governing equations of disturbed flow are reduced to a modified classical Orr-Sommerfeld type equation. This problem is an eigen value problem with homogeneous boundary conditions. As this system of equations is of peculiar nature, a special method of complete ortho-normalization is used and analyzed the flow thoroughly. The emphasis of the above research work was on formulating some MHD flow problems, mathematical modeling, application of different numerical techniques (initial value method, quasi-linear technique, finite difference schemes, complete ortho-normalization etc.), developing the corresponding computer codes and comparison of results where ever possible.

In the recent work, aim is to introduce a new Computational method to obtain approximate solution to one phase Stefan problems. Several methods exist, but each of them is mostly specific problem oriented and is not general enough to be applicable to a wide range of problems. The work developed a front tracking finite difference method with variable time step. This variable time step method was suggested earlier; but without a well-defined complete methodology. For a fixed space step, first two time steps are obtained using collocation and/or Green’s theorem of vector calculus. Subsequent step sizes are obtained by an iterative process with assured convergence. For a non-thermal diffusion, Stefan condition is of implicit nature. For such class of two point boundary value problems, method of bisection is efficient to obtain their solutions. The methods are illustrated by presenting three examples one of which is much discussed oxygen diffusion problem, which is published. The procedure is general enough to be applicable to a broad class of moving boundary problems.