Publisher Summary

This chapter discusses the translation of bubbles and solid spheres in a viscoelastic fluid. The motion is assumed to be caused by a gravitational field or an imposed force. The chapter presents some experimental observations of bubble shapes, velocity fields, and friction coefficients. One of the most striking features of translating air bubbles in viscoelastic fluids is that they develop a cusp at the rear pole. Often this effect may be observed in an almost full shampoo bottle by rapidly turning it upside down and watching an air bubble rise. More controlled experiments by Astarita and Apuzzo with a series of bubbles of increasing volume show a transition from a spherical bubble shape to an elongated ellipsoidal shape and the development of the cusp at the rear pole. They also measured the rise velocity of the bubbles as a function of the volume, and showed that for some viscoelastic fluids there is a critical volume at which the rise velocity appears to have a discontinuous increase when plotted as function of the volume.

INTRODUCTION

This paper is concerned with the translation of bubbles and solid spheres in a viscoelastic fluid. The motion is assumed to be caused by a gravitational field or an imposed force, and we will consider the situations in which the motion takes place in an unbounded fluid, quiescent far from the object as well as the situation in which the motion takes place in a finite container (a cylinder). First we will review some experimental observations of bubble shapes, velocity fields and friction coefficients. In the next section we then consider perturbation solutions for motions that are so slow that the viscoelastic fluid behaves almost as a Newtonian fluid. In this so-called low Deborah number limit analytical solutions may be obtained for bubble shapes, friction factors and velocity fields by perturbation methods. In the last section we consider some numerical simulations that apply also for intermediate Deborah numbers.

2 EXPERIMENTAL EVIDENCE

One of the most striking features of translating air bubbles in viscoelastic fluids is that they develop a cusp at the rear pole. Often this effect may be observed in an almost full shampoo bottle by rapidly turning it upside down and watching an air bubble rise. More controlled experiments by Astarita and Apuzzo (1965) with a series of bubbles of increasing volume show a transition from a spherical bubble shape to an elongated ellipsoidal shape and the development of the cusp at the rear pole. Astarita and Apuzzo also measured the rise velocity of the bubbles as a function of the volume, and showed that for some viscoelastic fluids there is a critical volume at which the rise velocity appears to have a discontinuous increase when plotted as function of the volume. Similar measurements have been performed by Calderbank, Johnson and Loudon (1970) and Leal, Skoog and Acrivos (1971) who also documented discontinuous jumps in the rise velocity. The actual value of the jump depends on the particular polymer/solvent system as well as the temperature. The above investigators have reported jumps by factors in the range of 2–10, and the effect is in fact quite remarkable. It is possible that the velocity discontinuity is related to the development of the cusp at the rear pole. It is proposed that the “cusped” bubbles are really not closed surfaces, but rather open surfaces in which the “cusps” continue into thin gas filaments that eventually dissolve in the liquid. In some fluids (Hassager (1979)) the “cusps” lose rotational symmetry and take the form of a knife edge. In these circumstances the knife edge appears to continue into a thin sheet of air that supposedly eventually dissolves in the liquid. Both in the situation where the bubbles extend into filaments or into sheets there must be a critical total volume at which the boundary condition at the rear pole changes from one involving a stagnation point into another condition with no stagnation point. This could certainly give a discontinuous change in rise velocity and one may argue qualitatively that at least two mechanisms that would retard the bubble when it has a rear stagnation point will not be present when the bubble does not close at the rear pole. First as long as the bubble has a rear stagnation point there may be an accumulation of surface active impurities near that point that would cause immobilization of the surface. Second with the stagnation point present there will be an elongational flow near the rear pole that could account for much of the drag on the bubble.

More detailed information on the flow around bubbles and spheres may be obtained by laser Doppler measurements of the fluid velocity fields. The technique, unfortunately, is limited to fluid velocities well above those at which the velocity discontinuity takes place. We will refer to this region as the high Deborah number region. In this region the following two phenomena have been found and documented by laser Doppler anemometry:

First, in the wake region behind the bubble the fluid velocity as seen by an observer stationary with respect to the fluid far from the bubble is in the opposite direction to that in which the bubble is moving (Hassager (1979)). This wake flow is strikingly different from wake flow in Newtonian fluids where fluid is always pulled with the bubble, and has been termed “negative wake”. It has been demonstrated also by flow visualization by Contanceau and Hajjam (1982).

Second, in the wake flow region the fluid velocity field (referred to an observer on the bubble) is not steady even at Reynolds numbers much less than unity (Bisgaard 1983). This observation however is currently limited to one particular polymer solution.

The above described two phenomena in the wake behind bubbles in viscoelastic fluids have been observed also in wakes behind solid spheres, whereas the velocity discontinuity at low Deborah numbers does not occur for spheres.

3 LOW DEBORAH NUMBER FLOW

For intermediate and high Deborah numbers there is considerable ambiguity as to the correct constitutive equation to be used for incompressible viscoelastic fluids. However in the limit as the Deborah number tends to zero the correct expression for the stress tensor is the retarded motion expansion, which through terms of third order may be written:

__=−b1[γ__(1)+B2γ__(2)+B11γ__(1)·γ__(1)+B3γ__(3)+B12(γ__(1)·γ__(2)+γ__(2)·γ__(1))+B1:11γ__(1)(γ__(1):γ__(1))+…] (3.1)

where

__(1)=▽v_+(▽v_)†  andγ__(n+1)=DDtγ__(n)−{(▽v_)†·γ__(n)+γ__(n)·(▽v_)} (3.2)

Here b1 is the zero shear-rate viscosity, B2 and B11 are constants with dimension of time and B3, B12 and B1:11 are constants with dimension of time squared. The B2 and B11 are related to the zero-shear-rate first and second normal stress coefficients Ψ1,0 and Ψ2,0 by Ψ1,0 = −2b1 B2 and Ψ2,0 = b1 B11. Values of the parameters for various molecular models may be obtained from Table 1 of Bird (1984).