SELF-SIMILAR SPIRAL INSTABILITIES IN ELASTIC FLOWS BETWEEN A CONE ANDA PLATE

Experimental observations and linear stability analysis are used to qu antitatively describe a purely elastic flow instability in the inertia less motion of a viscoelastic fluid confined between a rotating cone a nd a stationary circular disk. Beyond a critical value of the dimensio nless rotation rate, or Deborah number, the spatially homogeneous azim uthal base flow that is stable in the limit of small Reynolds numbers and small cone angles becomes unstable with respect to non-axisymmetri c disturbances in the form of spiral vortices that extend throughout t he fluid sample. Digital video-imaging measurements of the spatial and temporal dynamics of the instability in a highly elastic, constant-vi scosity fluid show that the resulting secondary how is composed of log arithmically spaced spiral roll cells that extend across the disk in t he self-similar form of a Bernoulli Spiral. Linear stability analyses are reported for the quasi-linear Oldroyd-B constitutive equation and the nonlinear dumbbell model proposed by Chilcott & Rallison. Introduc tion of a radial coordinate transformation yields an accurate descript ion of the logarithmic spiral instabilities observed experimentally, a nd substitution into the linearized disturbance equations leads to a s eparable eigenvalue problem. Experiments and calculations for two diff erent elastic fluids and for a range of cone angles and Deborah number s are presented to systematically explore the effects of geometric and theological variations on the spiral instability. Excellent quantitat ive agreement is obtained between the predicted and measured wavenumbe r, wave speed and spiral mode of the elastic instability. The Oldroyd- B model correctly predicts the nonaxisymmetric form of the spiral inst ability; however, incorporation of a shear-rate-dependent first normal stress difference via the nonlinear Chilcott-Rallison model is shown to be essential in describing the variation of the stability boundarie s with increasing shear rate.