Steady flow and interfacial shapes of a highly viscous dispersed phase

Abstract

A perturbation theory for the steady flow of immiscible liquids is developed when the dispersed phase is much more viscous than the continuous phase, as is the case in emulsions of highly viscous bitumen in water and in water lubricated pipelines of heavy crude. The perturbation is nonsingular, but nonstandard; the partitioning of the boundary conditions at different orders is not conventional. At zero-th order the dispersed phase moves as a rigid solid with an as yet unknown, to-be-determined, pressure. The flow of the continuous phase at zero-th order is determined by a Dirichlet problem with prescribed velocities on a to-be-iterated interfacial boundary. The first order problem in the dispersed phase is determined from the solution of a Stokes flow problem driven by the previously determined shear strain on the as yet undetermined interfacial boundary. This Stokes problem determines the unknown, to-be-determined, lowest order pressure distribution. At this point we have enough information to test the balance of normal stresses at lowest order; by iterating the interface shapes we may now complete the description of the lowest order problems. The perturbation sequence in powers of the viscosity ratio has a similar structure at every order and all the problems may be solved sequentially with the caveat that interface shape must be determined iteratively in each perturbation loop. A perturbation solution for the wavy interfacial shapes on core-annular flows of very viscous oil is presented and the results are compared with experiments and a simpler approximation in which the core moves as rigid, but deformable body with no secondary motions. The perturbation theory gives rise to an accurate description of the bamboo waves observed in experiments when the holdup ratio measured in the experiments is assumed in the theoretical calculation. The perturbation solution and the rigid body approximation are in a relatively good agreement with errors of the order 10% in the flow curves and wave shapes; the error is associated with the neglect of the secondary motion in the rigid-deformable core approximation.