Spherical Couette flow of Oldroyd 8-constant model Part I. Solution up to the second-order approximation

Abstract

The steady flow of an incompressible Oldroyd 8-constant fluid in the annular region between two spheres, or so-called spherical Couette flow, is investigated. The inner sphere rotates with anangular velocity Ω about the z-axis, which passes through the center of the spheres, while the outer sphere is kept at rest. The viscoelasticity of the fluid is assumed to dominate the inertia such that the latter can be neglected in the momentum equation. An analytical solution is obtained through the expansion of the dynamical variables in a power series of the dimensionless retardation time. The leading velocity term denotes the Newtonian rotation about the z-axis. The first-order term results in a secondary flow represented by the stream function that divides the flow region into four symmetric parts. The second-order term is the viscoelastic contribution to the primary viscous flow. The first-order approximation depends on the viscosity and four of the material time-constants of the fluid. The second-order approximation depends on the eight viscometric parameters of the fluid. The torque acting on the outer sphere has an additional term due to viscoelasticity that depends on all the material parameters of the fluid under consideration. For an Oldroyd-B fluid this contributed term enhances the primary torque but in the case of fluids with higher elasticity the torque components may be enhanced or diminished depending on the values of the viscometric parameters.PACS Nos.: 47.15.Rq