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Numerical study of HeleShaw flow with suction Hector D. Ceniceros, Thomas Y. Hou, and Helen Si
 

Summary: Numerical study of Hele­Shaw flow with suction
Hector D. Ceniceros, Thomas Y. Hou, and Helen Si
Applied Mathematics, California Institute of Technology, Pasadena, California 91125
#Received 30 July 1998; accepted 27 May 1999#
We investigate numerically the effects of surface tension on the evolution of an initially circular
blob of viscous fluid in a Hele­Shaw cell. The blob is surrounded by less viscous fluid and is drawn
into an eccentric point sink. In the absence of surface tension, these flows are known to form cusp
singularities in finite time. Our study focuses on identifying how these cusped flows are regularized
by the presence of small surface tension, and what the limiting form of the regularization is as
surface tension tends to zero. The two­phase Hele­Shaw flow, known as the Muskat problem, is
considered. We find that, for nonzero surface tension, the motion continues beyond the
zero­surface­tension cusp time, and generically breaks down only when the interface touches the
sink. When the viscosity of the surrounding fluid is small or negligible, the interface develops a
finger that bulges and later evolves into a wedge as it approaches the sink. A neck is formed at the
top of the finger. Our computations reveal an asymptotic shape of the wedge in the limit as surface
tension tends to zero. Moreover, we find evidence that, for a fixed time past the zero­surface­tension
cusp time, the vanishing surface tension solution is singular at the finger neck. The
zero­surface­tension cusp splits into two corner singularities in the limiting solution. Larger
viscosity in the exterior fluid prevents the formation of the neck and leads to the development of
thinner fingers. It is observed that the asymptotic wedge angle of the fingers decreases as the

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara
Hou, Thomas Yizhao - Applied and Computational Mathematics Department, California Institute of Technology

 

Collections: Mathematics