Global existence and numerical approximation in adiabatic shearing of incompressible fluids

We first study the existence of global solutions to an adiabatic shearing model of incompressible fluids between two parallel plates when 1⩽α⩽3, where the viscosity is μ(θ)=θ−α. Then we propose two finite difference schemes for μ(θ)⩾μ0>0 (0<θ<∞) and for μ(θ)↓0 as θ→∞, respectively. Numerical experiments are presented which demonstrate the high accuracy of the schemes.