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THREE-LAYER FLUID FLOW OVER A SMALL OBSTRUCTION ON THE BOTTOM OF A CHANNEL

Published online by Cambridge University Press:  20 January 2015

SRIKUMAR PANDA
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India email srikumarp@iitrpr.ac.in, scmartha@iitrpr.ac.in
S. C. MARTHA*
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India email srikumarp@iitrpr.ac.in, scmartha@iitrpr.ac.in
A. CHAKRABARTI
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India email aloknath.chakrabarti@gmail.com
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Abstract

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Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms.

Type
Research Article
Copyright
© 2015 Australian Mathematical Society 

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