Bubble dynamics research has many difficulties due to nonlinearity of the system of differential equations, so can we overcome on this difficulties by complex analysis techniques?
I wait for any contribution, paper, book, ... etc.
Best Regards.
Dear Khaled,
Please see this article
http://link.springer.com/article/10.1007/s12559-013-9214-3
Best regards,
Adnène
Maybe it hepls
https://www.google.pl/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwiy8_7-_q7LAhUiDZoKHcNHBiQQFggvMAE&url=http%3A%2F%2Fwwwf.imperial.ac.uk%2F~dgcrowdy%2FPubFiles%2FPaper-1.pdf&usg=AFQjCNHXYOTpX6uJfA0SKO7-4iBrm7igxg&bvm=bv.116274245,d.bGs
Anna recommendation is very useful.
In the paper
SIAM J. APPL. MATH. 2005, 65, No. 3, pp. 941–963
EXACT SOLUTIONS FOR THE EVOLUTION OF A BUBBLE IN STOKES FLOW: A CAUCHY TRANSFORM APPROACH
by DARREN CROWDY and MICHAEL SIEGEL,
the reader can find further information about the application of complex variable methods to
free boundary problems for 2-D Stokes bubbles.
The emphasis is placed on
consideration of the Cauchy transform of simple function \overline{z'}, which is defined as
1 /2πi\int_ ∂D(t) \{overline{z'}/(z'-z) dz',(2) where ∂D(t) denotes the boundary of the fluid domain D(t). When z is inside the bubble, the
line integral (2) defines an analytic function, say C(z,t).
The analytic continuation of C(z,t) outside the bubble
(and into the fluid region D(t)) contains a great deal of information about the bubble shape. It will be interesting find further application of of complex analysis in bubble dynamics research. My primary research interest revolves around complex analysis.
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