# Accession Number:

## ADA276472

# Title:

## Oscillations in Piston-Driven Shear Flow of a Non-Newtonian Fluid

# Descriptive Note:

## Summary rept.

# Corporate Author:

## WISCONSIN UNIV-MADISON CENTER FOR MATHEMATICAL SCIENCES

# Personal Author(s):

# Report Date:

## 1994-02-01

# Pagination or Media Count:

## 17.0

# Abstract:

In recent experiments on piston-driven shear flow of a highly elastic and very viscous non-Newtonian fluid. Lim and Schowalter observed nearly periodic oscillations in the particle velocity at the channel wall for particular values of the constant volumetric flow rate. Such periodicity has been characterized as a stick-slip phenomenon caused by the failure of the fluid to adhere to the wall. We suggest an alternative explanation for these oscillations using a dynamic mathematical model based on the Johnson-Segalman- Oldroyd constitutive relation, the key feature of which is a non-monotonic relationship between steady shear stress and strain rate. The resulting three- dimensional equations of motion and stress are reduced to one space dimension, consistent with experimental results. In the inertialess approximation, the equations governing the flow can be viewed as a continuous family of quadratic ordinary differential equations coupled by the non-local constraint that fixes the volumetric flow rate. Varying the flow rate, the numerical simulation of solutions to this system exhibits transitions to and from a regime with persistent oscillations that compare favorably with the Lim-Schowalter observations. When the time-asymptotic behavior is cyclic, large shear strain rates are observed in a thin but macroscopic layer near the wall. The transitions are explained using spectral analysis of the linear infinite dimensional operator resulting from linearization of the quadratic system about a discontinuous steady state with a jump discontinuity in the stress components. The persistent oscillations arise as a Hopf bifurcation to periodic orbits as the volumetric flow rate is increased beyond a critical value.

# Descriptors:

- *MATHEMATICAL MODELS
- *FLOW RATE
- *OSCILLATION
- *NONNEWTONIAN FLUIDS
- VELOCITY
- SIMULATION
- STEADY STATE
- ORBITS
- PISTONS
- EQUATIONS OF MOTION
- DYNAMICS
- LAYERS
- OBSERVATION
- ELASTIC PROPERTIES
- FAILURE
- MOTION
- RATES
- STRAIN RATE
- TIME
- THREE DIMENSIONAL
- HIGH VISCOSITY
- PERIODIC VARIATIONS
- CHANNELS
- DISCONTINUITIES
- VALUE
- FLOW
- BEHAVIOR
- TRANSITIONS
- WALLS
- DIFFERENTIAL EQUATIONS
- CONSTANTS
- PARTICLES

# Subject Categories:

- Operations Research
- Fluid Mechanics