The cooling and solidification of a non-Newtonian material in the presence of reciprocating flow is analysed. Finite difference solutions are obtained for a transient 1-space-dimensional model of a shear thinning material with temperature dependent properties, contained between plane parallel surfaces to which it looses heat by conduction in the presence of periodically reversing flow. This situation is relevant to live feed injection moulding - an advanced technology for thermoplastics processing - in which alternating direction shear is imposed on melt in the mould cavity during the holding and packing stages. Interest centres on the effects of flow on cooling and solidification and the development of frozen-in strains, which have important implications for the control of material alignment and mechanical properties of mouldings. The governing dimensionless parameters are identified and their influence investigated in a series of numerical simulations, where the parameters range over the following values: Pearson number 0.5-5.0; Brinkman number 0.001-1.0; Stefan number 0.1-1.0; Power Law Index 0.2-1.0; dimensionless flow reversal lime 0.01-1.0. Two operating modes are considered: (1) fixed applied pressure gradient, with a falling flow rate, (2) flow rate initially fixed, followed by a falling flow rate period at a fixed limiting pressure gradient. In mode 1 operation viscous heating does not significantly influence cooling rates, but in mode 2, for high Brinkman numbers, a dynamic equilibrium temperature field can be attained. Complex profiles of frozen-in material strain through the thickness of the moulding are predicted. In mode 1 operation the magnitude or the principle stretch rises to a peak close to the mould walls, and falls in a series of oscillations to unity (zero deformation) on the centre plane. High values of the Pearson number or low values of the power Law Index damp these oscillations, In mode 2 operation sharply alternating regions of high and low strain are obtained indicating a structure with alternating layers of high and low material orientation.