Abstract: In the present paper an unsteady thermal flow of non-Newtonian fluid is investigated which is of the flow into axisymmetric mould cavity.In the second part an unsteady thermal flow of upper-convected Maxwell fluid is studied.For the flow into mould cavity the constitutive equation of power-law fluid is used as a Theological model of polymer fluid.The apparent viscosity is considered as a function of shear rate and temperature.A characteristic viscosity is introduced in order to avoid the nonlinearity due to the temperature dependence of the apparent viscosity.As the viscosity of the fluid is relatively high the flow of the thermal fluid can be considered as a flow of fully developed velocity field.However, the temperature field of the fluid flow is considered as an unsteady one.The governing equations are constitutive equation, momentum equation of steady flow and energy conservation equation of non-steady form.The present system of equations has been solved numerically by the splitting difference method.The numerical results show that the splitting difference method is suitable for the 2D problem of non-Newtonian fluid.The present application of the splitting diffference method is at first developed by us for non-Newtonian case.For the unsteady flow in the tube the finite difference scheme is given which leads to a tridiagonal system of equations.

Key words: internal thermal flow, power law fluid, Maxwell fluid, splitting difference scheme, unsteady flow