#### Date of Award

Spring 2009

#### Document Type

Dissertation

#### Degree Name

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

#### Department

Mathematical Sciences

#### First Advisor

Demetrius T. Papageorgiou

#### Second Advisor

Linda Jane Cummings

#### Third Advisor

Peter G. Petropoulos

#### Fourth Advisor

David Sheldon Rumschitzki

#### Fifth Advisor

Michael Siegel

#### Abstract

This study is concerned with several problems related to self-similar flows in pulsating channels. Exact or similarity solutions of the Navier-Stokes equations are of practical and theoretical importance in fluid mechanics. The assumption of self-similarity of the solutions is a very attractive one from both a theoretical and a practical point of view. It allows us to greatly simplify the Navier-Stokes equations into a single nonlinear one-dimensional partial differential equation (or ordinary differential equation in the case of steady flow) whose solutions are also exact solutions of the Navier-Stokes equations in the sense that no approximations are required in order to calculate them. One common characteristic to all applications of self-similar flows in real problems is that they involve fluid domains with large aspect ratios. Self-similar flows are admissible solutions of the Navier-Stokes equations in unbounded domains, and in applications it is assumed that the effects of the boundary conditions at the edge of the domain will have only a local effect and that a self- similar solution will be valid in most of the fluid domain. However, it has been shown that some similarity flows exist only under a very restricted set of conditions which need to be inferred from numerical simulations. Our main interest is to study several self-similar solutions related to flows in oscillating channels and to investigate the hypothesis that these solutions are reasonable approximations to Navier-Stokes flows in long, slender but finite domains.

#### Recommended Citation

Espin Estevez, Leonardo Xavier, "Self similar flows in finite or infinite two dimensional geometries" (2009). *Dissertations*. 903.

https://digitalcommons.njit.edu/dissertations/903