Date of Award

Spring 2002

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Demetrius T. Papageorgiou

Second Advisor

Michael Siegel

Third Advisor

Charles M. Maldarelli

Fourth Advisor

John Kenneth Bechtold

Fifth Advisor

Lou Kondic

Sixth Advisor

Daniel Goldman

Abstract

This work is an analytical and computational study of the nonlinear interfacial instabilities found in core-annular flows in the presence of surfactants. Core-annular flows arise when two immiscible fluids (for example water and oil) are caused to flow in a pipe under the action of ail axial pressure gradient. In one typical type of flow regime, the fluids arrange themselves so that the less viscous (e.g. water) lies in the region of high shear near the pipe wall, with the more viscous fluid occupying the core region. Tech nologically, this arrangement provides an advantage since the highly viscous fluid is lubricated by the less viscous annulus and for a given pressure gradient the core-fluid flux can be greatly increased.

The stability of these flows is of fundamental scientific and practical importance. The sharp interface between the two phases can become unstable by several physical meclianisins and one such mechanism of practical importance is surface tension. In this work we incorporate into our model the effects of insoluble surfactants on the instability. The full problem is derived with particular emphasis paid to the surfactant transport equation which is novel. NVe then carry out an asymptotic solution of the problem when the annular layer is thin compared to the core-fluid radius and for waves which are of the order of the pipe radius (that is long compared to the annular layer thickness); these scales are in accord with both linear theory as well as experimental observations. The result of the matched asymptotic analysis is a system of coupled nonlinear partial differential equations for the interfacial aniplitude and the surfactant concentration on the interface. In the absence of surfactarits, the system reduces to the Kuramoto-Sivashinsky equation which has been extensively studied as a paradigm for one-dimensional turbulence in dissipative systems. The surfactant modifies the flow by inducing Marangoni forces along the interface which in turn modify both the velocities and interfacial amplitudes. There are two parameters present in the nonlinear system, the length of the system and a surface Peclet number which measures the diffusion of surfactant on the interface.

In order to gain an understanding of the dynamics, we carry out extensive computations using accurate and stable numerical methods capable of following the solution for long times. We map out the dynamics by numerically solving initial value problems on spatially periodic domains where the length of the system is the bifurcation parameter, keeping the Peclet number fixed and equal to one. We find that surfactant acts to suppress chaotic behavior found in its absence for extensive ranges of the bifurcation parameter. The new flow consists of successive windows (in parameter space) of steady-state traveling waves separated by time-periodic attractors. As the length of the system increases a self-similar structure has been found to govern the shapes of the traveling waves as we move from a given window to a lower one. This is elucidated analytically and numerically.

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