Oscillatory Carreau flows in straight channels

The present paper studies the oscillatory flow of Carreau fluid in a channel at different Womersley and Carreau numbers. At high and low Womersley numbers, asymptotic expansions in small parameters, connected with the Womersley number, are developed. For the intermediate Womersley numbers, theoretical bounds for the velocity solution and its gradient, depending on the problem parameters, are proven and explicitly given. It is shown that the Carreau number changes the type of the flow velocity to be closer to the Newtonian velocity corresponding to low or high shear or to have a transitional character between both Newtonian velocities. Some numerical examples for the velocity at different Carreau and Womersley numbers are presented for illustration with respect to the similar Newtonian flow velocity.