Spherical Couette flow of Oldroyd 8-constant model. Part II. Third-order approximation and the stream function Ψ(3)

Abstract

In the previous work, Part I, Abu-El Hassan (Can. J. Phys. 84, 345 (2006)), the steady flow of an incompressible Oldroyd 8-constant fluid in the annular region between two concentric spheres is investigated up to the second-order approximation. Hence, the normalized second-order velocity field, V is seen to be V = W(0)[Formula: see text]+λU[Formula: see text](1)+λ(2)W(2)[Formula: see text]+O(λ3) with U[Formula: see text](1) = –[Formula: see text][Formula: see text](Ψ(1)/r sin θ) =  [Formula: see text]U(1) + [Formula: see text]V(1). The leading velocity term represents the Newtonian flow in the ϕ-direction, while the first-order term denoted by the stream function Ψ(1)(r,θ), produces a secondary flow field that divides the flow region into four parts symmetric about the z-axis, which is the axis of rotation. The second-order approximation gives a viscoelastic contribution W(2)(r,θ) in the ϕ-direction. λ is the retardation time parameter. The present work is devoted to the solution of the third-order approximation of the same problem treated in Abu-El Hassan (Can. J. Phys. 84, 345 (2006)). The solution produces a stream function Ψ(3)(r, θ), which being a secondary flow field divides the domain of flow into two similar regions symmetric about an axis perpendicular to the axis of rotation. The streamlines Ψ(3)(r,θ) = const. are sketched for Maxwell, Oldroyd-B, and Oldroyd-8 constant model fluids, respectively. The results show that the distribution of flow for these fields is mainly affected by the values of the field's elastic parameters.PACS No.: 47.15.Rq