Unsteady MHD boundary-layer flow and heat transfer of nanofluid over a permeable shrinking sheet in the presence of thermal radiation

Forced convection in unsteady boundary layer flow of nanofluid over a permeable shrinking sheet in the presence of thermal radiation is studied. A variable magnetic field is applied normal to the sheet. The nanofluid model includes Brownian motion and thermophoresis effects. The governing momentum, energy and nanofluid solid volume fraction equations are solved numerically using fourth order Runge–Kutta method with shooting technique. The effects of various physical parameters on dimensionless velocity, temperature, nanoparticle volume fraction, as well as the skin friction, local Nusselt and local Sherwood numbers are analyzed. The numerical results indicate that dual solutions exist for certain values of the magnetic parameter ðMÞ, wall mass suction (s) and unsteadiness parameter (A). It is found that both magnetic field and wall mass suction widen the range of unsteadiness parameter for which the solution exists. The skin friction coefficient, local Nusselt and Sherwood numbers increase for the first solution and decrease for the second solution with the increase in M.