Abstract
The motion of bubbles dispersed in a liquid when a small-amplitude oscillatory motion is imposed on the mixture is examined in the limit of small frequency and viscosity. Under these conditions, for bubbles with a stress-free surface, the motion can be described in terms of added mass and viscous force coefficients. For bubbles contaminated with surface-active impurities, the introduction of a further coefficient to parametrize the Basset force is necessary. These coefficients are calculated numerically for random configurations of bubbles by solving the appropriate multibubble interaction problem exactly using a method of multipole expansion. Results obtained by averaging over several configurations are presented. Comparison of the results with those for periodic arrays of bubbles shows that these coefficients are, in general, relatively insensitive to the detailed spatial arrangement of the bubbles. On the basis of this observation, it is possible to estimate them via simple formulas derived analytically for dilute periodic arrays. The effect of surface tension and density of bubbles (or rigid particles in the case where the no-slip boundary condition is applicable) is also examined and found to be rather small.
Highlights
Flows involving bubblesdispersedin a liquid are important becausethey occur in a variety of processesT. he rigorous analysisof such flows is, in general,quite complicated as the overall propertiesof the flow dependon the detailsof the microstructure of the medium which, in turn, depend on the nature of flow
It is hoped that by studying a number of different physical situations in a rigorous manner, it may be possible to develop a framework and a qualitative understanding that could be usedfurther for modeling more complex flows
In view of the effect of the velocity distribution on the computed value of C, one can expect that, for a periodic arrangement of particles, the two different approaches will give the sameresult. This hasindeed beenfound by Biesheuvel and Spoelstra.It may benoted that, the numerical results for the C, of nondilute periodic arrays presented by these authors on the basis of their expression (35) are correct, the subsequentexpression (36) that purports to give an approximate formula for the C, of nondilute random arrays is incorrect as it suggeststhat this quantity will diverge asp approachesits maximum packing value
Summary
Flows involving bubblesdispersedin a liquid are important becausethey occur in a variety of processesT. he rigorous analysisof such flows is, in general,quite complicated as the overall propertiesof the flow dependon the detailsof the microstructure of the medium (i.e., the size,shape,spatial, and velocity distribution of the bubbles) which, in turn, depend on the nature of flow. In this paper we confine ourselvesto the caseof small-amplitude oscillatory motion Under these conditions, one may write the following expression for the total force actmg on a single bubble immersed in a unidirectional liquid flow at high Reynolds number, IT- i&xJb(Ii, -i? In view of the effect of the velocity distribution on the computed value of C,, one can expect that, for a periodic arrangement of particles, the two different approaches will give the sameresult This hasindeed beenfound by Biesheuvel and Spoelstra.It may benoted that, the numerical results for the C, of nondilute periodic arrays presented by these authors on the basis of their expression (35) are correct, the subsequentexpression (36) that purports to give an approximate formula for the C, of nondilute random arrays is incorrect as it suggeststhat this quantity will diverge asp approachesits maximum packing value.
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