Date

2017

Document Type

Thesis

Degree

Master of Science

Department

Mechanical Engineering

First Adviser

Kazakia, Jacob

Abstract

Newtonian viscous flows under laminar regime across tubes of arbitrary varying cross sections were analyzed. The simple form of this problem for constant circular cross section is a well-known result from the early years of Fluid Mechanics. [1] Nowadays, many modern applications such as nanotubes manufacturing, MEMS, Biomechanics and medical procedures, require determining pressure losses for Newtonian and non-Newtonian fluids in tubes with varying geometries. These applications have renewed the interest in this classical problem and the search for accurate solutions based on numerical, analytic and experimental methods. [2] In the present work, several methods were explored as means to determine pressure gradients for arbitrary shapes and varying cross sections. In all cases, non-dimensional forms of momentum equation were employed. The motivation to work with non-dimensional expressions was clearly justified by the fact that in laminar regime, velocity distributions are basically the same for any cross section as long as its shape remains constant. Thus, once the numerical solution is found for a typical section, solution for any other section can be obtained by a re scaling process of by an appropriate characteristic length. This approach enabled to determine not only pressure gradients, but also Fanning's friction factor for any shape.Lubrication approximation and perturbation of axial convective acceleration term were first addressed. Non-dimensional numerical solutions by finite element method were implemented in a computer code developed for this purpose as part of the work. Results were compared to analytic solutions [2,3,4] for elliptical cross sections exhibiting good correlation for small aspect ratios (characteristic length / axial length) along the tube. Furthermore, an alternative regular perturbation approach applied to viscous and axial convective acceleration was tested. A computer code was also developed to implement finite element solution for this approach. Results were in good correlation with analytic results based on lubrication and perturbations methods previously detailed. Finally, a different approach identified as convective acceleration correction was defined. In this case, the effect that transverse acceleration components have on axial velocity throughout the tube were considered. Once again, a non-dimensional form of the momentum equation was developed and solved by means of finite element procedure. In all cases, convective acceleration correction exhibited less deviation in pressure gradient estimation than any other method however of which, further tests must be conducted and compared to physical experimental data in future research. Detailed analysis of uncertainties for all methods were developed for two elliptical cross sections with different aspect ratios. CFD solutions obtained from commercial software were accepted as experimental values for comparison purposes.

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