I. On a New Constitutive Equation for Non-Newtonian Fluids. II. Brownian Motion with Fluid-Fluid Interfaces

Citation

El-Kareh, Ardith W. (1989) I. On a New Constitutive Equation for Non-Newtonian Fluids. II. Brownian Motion with Fluid-Fluid Interfaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/xd3m-5r94. https://resolver.caltech.edu/CaltechETD:etd-02092007-131907

Abstract

This thesis is a small contribution to the ambitious goal of understanding some of the more complex flows that are found in nature, namely, flows of fluids with a microstructure. There is a great diversity of such flows: suspensions, emulsions, polymeric solutions, . . ., each exhibiting phenomena not found in the flow of homogeneous Newtonian fluids. A bit of this diversity has been incorporated in this thesis: The first part of it is on some aspects of non-Newtonian fluid flow, and in the second part Brownian motion involving interfaces between Newtonian fluids is studied.

Given the large amount of effort devoted recently to the numerical simulation of non-Newtonian fluid flow, the absence of mathematical proofs that any of the standard computational methods for solving the equations will converge except for nearly Newtonian flows seems somewhat disturbing. While there is evidence that investigators may have overcome the so-called "high Weissenberg-number problem," at least in specific cases, confidence in the numerical solutions would undoubtedly be increased by a rigorous mathematical foundation for the numerical algorithm. The first, and in many cases nontrivial, step towards this is to prove that a solution actually exists. In the first part of this thesis, a proof of existence without restriction on the parameters is given for a particular modified finitely extendible nonlinear elastic dumbbell model. A physical basis for the modifications is given.

For numerical computation, the issue of stability of a flow is also an important one, as the small errors introduced by discretization are essentially perturbations in the flow, which, if they grow too fast, can make convergence impossible. An energy method calculation is given here for the same FENE dumbbell model considered in the existence proof (except for the modifications) to show that for any flow in a bounded domain, at small enough Reynolds number and high enough Deborah number, all disturbances will remain bounded. While the estimate found for the highest Reynolds number and lowest Deborah number for guaranteed stability may be very conservative, the result is nevertheless useful in that it shows that if there is an instability, it must occur at a critical Reynolds or Deborah number.

While the Brownian motion of a rigid particle has received much attention in the literature, and the Stokes-Einstein diffusivity of a rigid particle is a result almost as well-known as the Stokes drag law, the Brownian motion of systems that are more complex hydrodynamically has only recently begun to be investigated. Most recent work on such systems has been for systems with rigid boundaries, e.g., suspensions of rigid spheres. In this thesis, the case of deformable fluid-fluid interfaces is considered. Since the understanding of the behavior of clusters or suspensions of particles can only follow an understanding of the behavior of a single particle, the two cases considered here are a drop in an infinite fluid, and an isolated particle in the presence of an approximately planar interface. Expressions for statistical quantities, such as the velocity autocorrelation, of the particle and drop motion are derived. In the case of the interface, the nature of its effect on the particle's behavior, beyond the obvious fact that it changes the particle's mobility, is explored. Similarly, the surface-tension dependence of the drop's motion is investigated.

Finally, in a slight digression, the problem of high-frequency oscillatory Stokes flow around two spheres, with specified velocity at their surfaces, is reduced to an infinite system of algebraic equations for the (frequency-dependent) coefficients in a spherical harmonic expansion of the solution. This is expected to be useful for computations of such flows where better accuracy than an approximate solution obtained by the method of reflections is desired.

Thesis Files