The turbulent flow behaviors of viscoelastic liquids has aroused considerable interest in recent years. It has been observed that one of the characteristics of the turbulent flow of slightly viscoelastic fluids is a very pronounced suppression of turbulence accompanied by major reduction in drag coefficient.
No consistent theoretical interpretation of the turbulent flow behaviors has yet appeared in the literature. It is thought that a study of the stability of the flow of viscoelastic liquids would be a useful first step. In this paper the problem of the stability of parallel flow of viscoelastic fluids is studied. The analog of the Orr-Sommerfeld equation is solved by the method used by Michael.
It is found that the presence of elasticity destabilizes the flow and this is apparently contrary to recent experimental observations. Possible reasons for this discrepancy are discussed.
In recent years there has been a growing interest in the study of the turbulent flow, behaviors of viscoelastic fluids. Several investigators have observed that the addition of very small amounts of certain polymeric solutes to a Newtonian fluid can greatly reduce the turbulent drag coefficient.
A consistent theoretical interpretation of the turbulent flow behaviors of viscoelastic fluids has not yet appeared in the literature. As the problem of turbulence is related to the stability problem, it is thought that a study of the stability of parallel flow of a viscoelastic liquid would be a useful first step. Furthermore, Seyer observed that the transition from laminar to turbulent flow is delayed by the addition of polymeric solutes. This observation suggests that the flow of viscoelastic liquids is more stable. Thus, a study of the stability of parallel flow of viscoelastic liquids would be of great practical importance.
The linearized theory of stability of a viscoelastic liquid has been formulated by Walters. By considering a parallel flow with velocity components [U(y), 0, 0] referred to a Cartesian coordinate axis [x, y, z], with the x-axis along the direction of flow, and by superposing a small two-dimensional disturbance, the analog of the Orr-Sommerfeld equation for Liquids A and B has been shown to be:
subject to one boundary conditions v = Dv = 0 at y = 1. It is worth noting that the analog of the Orr-Sommerfeld equation for a second-order fluid is identical with Eq. 1 above.
The method used is similar to that employed by Michael and the reader is referred to that paper for the details.
Suppose (, RO) be a point on the neutral stability curve for a Newtonian fluid, with vo and co the corresponding wave function and wave velocity. We consider a perturbation about that point and write: