Bifurcation of viscoelastic fluid counterflows

Abstract

UDC 532.5.013.4:532.135 One of the phenomena of structural instability observed earlier in viscoelastic fluid flows in cross-channels, namely, the asymmetry of flows in the region of the central singular point is considered. It is shown analyti- cally, by demonstrating symmetric and asymmetric asymptotic solutions in the vicinity of this singular point, that the latter exhibits bifurcation. The fields of velocities and distributions of stresses and pressure provided by asymptotic solutions are compared with their analogs observed experimentally and in numerical simula- tion. Close association of the existence of asymmetric flows with the viscoelastic nature of fluid is shown. Counterflows of a viscoelastic fluid considered in the present work relate to the class of small-scale flows with a singular point which are of current interest in connection with the development of nanotechnologies (1). Considerable stresses and deformations frequently originating in the region of the singular point in such kind of flows are capable of leading to various phenomena of the structural rearrangement of flow such as the origination of vortex-like structures and local zones of large stresses, as well as to a change in the direction of flows (2, 3) often related to elastic instability (4). In recent years, of special interest is one of such kind of phenomena: spontaneous development of the asym- metry of flow under symmetric conditions of the process; it is frequently considered as the most significant manifes- tation of the viscoelastic nature of fluid in the process of its flow. In the publications describing this phenomenon on the basis of experiment or numerical simulation (e.g., (4-6)) an analysis of its reasons is given which is of pheno- menological character and which suggests some factors as the probable reason for this phenomenon (e.g., an abnor- mally high pressure in the region of central point which causes a noticeable compressibility of fluid (6)). In the present work, an asymptotic solution is obtained analytically for a system of equations which determine the motion of a viscoelastic fluid in the vicinity of the central singular point of vector field where the fluid velocity vanishes (7). The solution has symmetric and asymmetric modes suggesting the presence of bifurcation at this point which follows directly from the rheological properties of the fluid. This result may point to the fact that the viscoe- lasticity as such, apart from other factors of flow and assumptions, is the reason for its asymmetry. This assumption is discussed in more detail in the last section of this article. 1. Statement of the Problem. We consider a two-dimensional flow of a viscoelastic fluid in symmetric cross- channels. One of the variants of a symmetric flow (identical in the four quadrants of the domain) which was investi- gated earlier in (2, 3, 8) is shown in Fig. 1. It was assumed in those works that the fluid was set in stationary motion by a smooth increase in the pressure at the inlet into horizontal channels from zero to a stationary value. We will determine the non-Newtonian properties of fluid using the rheological equations of state of the upper convected Maxwell model (UCM) which very accurately describe the nonstationary manifestations of viscoelasticity and which are frequently used in investigations of such a class of phenomena (2, 3, 6, 8):