Journal of Fluid Mechanics

A singularity method for calculating time-dependent viscoelastic flows with integral constitutive equations

a1 Department of Chemical Engineering and Materials Science, University of California, Davis, Davis, CA 95616, USA


A new method is introduced for calculating time-dependent, non-Newtonian flows of fluids described by integral constitutive equations. The starting point for the method is the integral form of the solution to the equation sof motion, valid in the limit of low Reynolds number. Because of the non-Newtonian nature of the fluid, this solution includes an integral over the domain of the flow, which is not present in boundary integral methods. This integral over the fluid volume (in three dimensions) or area (in two dimensions) is converted to a Lagrangian reference frame, and discretized for numerical evaluation. Because points in the integrand move with the fluid velocity, values of the non-Newtonian portion of the stress can be found by integrating the deformation at those points in conjunction with a suitable integral constitutive equation. The contribution to the total velocity field of the non-Newtonian stress at each fluid element is that of a point dipole, and the method bears many similarities to the point-vortex method for calculating inviscid flows. Like the point-vortex method, it is necessary to introduce cutoff functions that remove the singular nature of the dipole–dipole interactions. In addition, to render the method computationally feasible, the interactions between the dipoles must be calculated by the fast-multipole method or some comparable approach. Methods for calculating cutoff functions and implementing the fast multipole method are discussed, and simulation results are presented for one- and two-phase time-dependent flows of viscoelastic fluids between eccentric and concentric rotating cylinders.

(Received September 21 2001)
(Revised October 18 2002)