Date of Award

Spring 4-1-2019

Document Type


Degree Name

Doctor of Philosophy (PhD)


Applied Mathematics

First Advisor

Mark A. Hoefer

Second Advisor

Gennady El

Third Advisor

Keith Julien

Fourth Advisor

John Crimaldi

Fifth Advisor

Daniel Appelo

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.


Viscous fluid conduits provide an ideal system for the study of dissipationless, dispersive hydrodynamics. A dense, viscous fluid serves as the background medium through which a lighter, less viscous fluid buoyantly rises. If the interior fluid is continuously injected, a deformable pipe forms. The long wave interfacial dynamics are well-described by a dispersive nonlinear partial differential equation called the conduit equation.

Experiments, numerics, and asymptotics of the viscous fluid conduit system will be presented. Structures at multiple length scales are characterized, including solitary waves, periodic waves, and dispersive shock waves. A more generic class of large-scale disturbances is also studied and found to emit solitary waves whose number and amplitudes can be obtained. Of particular interest is the interaction of structures of different scales, such as solitary waves and dispersive shock waves. In the development of these theories for the conduit equation, we have uncovered asymptotic methods that are applicable to a wide range of dispersive hydrodynamic systems.

The conduit equation is nonintegrable, so exact methods such as the inverse scattering transform cannot be implemented. Instead, approximations of the conduit equation are studied, including the Whitham modulation equations, which can be derived for any dispersive hydrodynamic system with a periodic wave solution family and at least two conservation laws. The combination of the conduit equation's tractability and the relative ease of the associated experiments make this a model system for studying a wide range of dispersive hydrodynamic phenomena.