Optimal error estimates for a semi-implicit Euler scheme for incompressible fluids with shear dependent viscosities

Certain rheological behaviors of fluids in engineering sciences are modeled by power law ansatz with p is an element of (1,2]. In the present paper a semi-implicit time discretization scheme for such fluids is proposed. The main result is the optimal O(k) error estimate, where k is the time step size. Our results hold in the range p is an element of (3/2, 2] ( in the three-dimensional setting) for strong solutions of the continuous problem, whose existence is guaranteed under appropriate assumptions on the data. The estimates are uniform with respect to the degeneracy parameter delta is an element of [0, delta_0] of the extra stress tensor. Additional regularity properties of the solution of the discrete problem are proved.