NDLI: Forward and Inverse Problems in Two-Phase Fluid Dynamics

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Zabarankin, Michael Grechuk, Bogdan
Copyright Year	2017
Abstract	Given an objective function, e.g., an error measure, which depends on an optimization variable and a parameter, a forward problem finds optimal value of the optimization variable as a function of the parameter, whereas an inverse problem finds optimal value of the parameter as a function of the optimization variable. In general, the two functions are not inverse one of the other. If the objective function is nonconstant on the solution set of either the forward or inverse problem, then it should necessarily be nonsmooth with respect to both the optimization variable and the parameter in order for the solutions of the forward and inverse problems to be inverse one of the other. In this case, a corresponding sufficient condition is suggested. It involves only single-variable functions, which make its verification computationally efficient. Of particular interest is the problem of finding a steady spheroidal drop freely suspended in an ambient fluid subjected to either an external linear flow or a uniform electric field---the ratios of the external flow stress to surface tension and of the electric stress to surface tension are defined by capillary numbers $Ca$ and $Ca_E$, respectively. The spheroid's axes ratio and $Ca$ (or $Ca_E$) are the optimization variable and parameter, respectively, whereas the objective function is a measure of the normal velocity on the interface chosen to be the sum of the absolute coefficients in the Legendre series expansion of a stream function in the prolate spheroidal coordinates. The solution of the inverse problem is found analytically. Remarkably, its inverse solves the forward problem for almost the same range of $Ca$ (or $Ca_E$), where it is defined. This justifies determining steady spheroids in [M. Zabarankin, SIAM J. Appl. Math., 4 (2016), pp. 1606--1632] as those for which the first coefficient in the stream function series expansion vanishes. As a result, the computational process and further analysis are simplified considerably.
Starting Page	3969
Ending Page	3989
Page Count	21
File Format	PDF
ISSN	03630129
DOI	10.1137/16M107726X
e-ISSN	10957138
Issue Number	6
Volume Number	55
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-12-14
Access Restriction	Subscribed
Subject Keyword	drop inverse problem Inverse problems Capillarity Magnetohydrodynamics and electrohydrodynamics forward problem electric field Stokes and related (Oseen, etc.) flows steady shape Stokes flow
Content Type	Text
Resource Type	Article
Subject	Control and Optimization Applied Mathematics

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