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The numerical solutions of unsteady hydromagnetic natural convection Couette flow of a viscous, incompressible and electrically conducting fluid between the two vertical parallel plates in the presence of thermal radiation, thermal diffusion and diffusion thermo is obtained here. The fundamental dimensionless governing coupled linear partial differential equations for impulsive movement and uniformly accelerated movement of the plate were solved by an efficient Finite Element Method.Computations ware performed for a wide range of the governing flow parameters, viz., Thermal diffusion (Soret) and Diffusion thermo (Dufour) parameters, Magnetic field parameter, Prandtl number, Thermal radiation and Schmidt number. The effects of these flow parameters on the velocity (u), temperature (θ) and Concentration (φ) are shown graphically. Also the effects of these pertinent parameters on the skin-friction, the rate of heat and mass transfer are obtained and discussed numerically through tabular forms. These are in good agreement with earlier reported studies. Analysis indicates that the fluid velocity is an increasing function of Grashof numbers for heat and mass transfer, Soret and Dufour numbers whereas the Magnetic parameter, Thermal radiation parameter, Prandtl number and Schmidt number lead to reduction of the velocity profiles. Also, it is noticed that the rate of heat transfer coefficient and temperature profiles increase with decrease in the thermal radiation parameter and Prandtl number, whereas the reverse effect is observed with increase of Dufour number. Further, the concentration profiles increase with increase in the Soret number whereas reverse effect is seen by increasing the values of the Schmidt number. NOMENCLATURE List of Variables: h Distance between two parallel plates ( m ) o H Magnetic field component along   y axis ( 1  m A ) p C Specific heat at constant pressure   K Kg J 1  s C Concentration susceptibility ( 1  mole m ) Gr Grashof number for heat transfer Gc Grashof number for mass transfer g Acceleration of gravity ( 2  s m ) M Hartmann number Pr Prandtl number Sc Schmidt number R Thermal radiation parameter F Accelerating parameter D Chemical molecular diffusivity ( 1 2  s m ) T  Temperature of fluid   K w T  Temperature of the fluid near to moving plate   K h T  Temperature of the fluid near to the stationary plate   K C Concentration of fluid ( 3  m mol ) w C Concentration of the fluid near to moving plate ( 3  m mol ) h C Concentration of the fluid near to the stationary plate ( 3  m mol ) t Time in x , y coordinate system ( s ) t Time in dimensionless co-ordinates ( s ) u Velocity component in   x direction ( 1  s m ) u Dimensionless velocity component in   x direction ( 1  s m ) o Nu Nusselt number at the moving hot plate 1 Nu Nusselt number at the stationary plate Sh Sherwood number Sr Soret number (Thermal Diffusion) Dr Dufour number (Diffusion Thermo) r q Radiative heat flux K Chemical reaction of first order with rate constant y x  , Co-ordinate syste...

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