1D Shear-Thinning Flow: A Numerical Challenge for Physical Reliability

A 1D shear-thinning model is considered as a first approximation of the synovial fluid flow which occurs in an infinitely long channel bounded by moving solid boundaries. The viscosity is assumed to obey a power law expression, namely Modified Cross Model. The concentration of the hyaluronic acid influences the shear-thinning behavior. Consequently, the viscosity depends on both the concentration and the strain rate. The problem contains two length scales whose ratio can reach extremely large values, 10^{200}. The failure is guaranteed to double-precision solvers. Their results can open the way to fake behaviors and erroneous interpretations. A physically reliable solution can be obtained provided the machine accuracy is fixed up to 200 accurate digits. Using the Mathematica software is a way to manage this control.