PHYSICAL MATHEMATICS SEMINAR


TOPIC:		A Numerical Approach for Simulating Two-Dimensional Viscous
       		Incompressible Flows with Interfaces

SPEAKER: 	Jin Wang
         	Department of Mathematics
		The Ohio State University

Abstract:

Incompressible flows with interfaces occur in a wide variety of physical
phenomena and the numerical study of such flows has been a very active
area over the past years.  Mathematically, the motion of flows is governed
by the incompressible Navier-Stokes equations together with the
interfacial conditions.

In this talk, we present a new approach to simulate the two-dimensional
viscous incompressible flows with interfaces. First we introduce some new
coordinates so that the interface is mapped into a coordinate line which
enables us to work on a rectangular domain instead of a deformed geometry.
Then an iterative approach combined with the backward difference formula
is applied to do the time marching.  Meanwhile the Fourier transform and
the pseudo-spectral technique are applied in the horizontal direction,
$X$, under the assumption that the solutions are periodic in $X$.  Then,
we write the semi-discretized equations as a 1st-order ODE system with
respect to the vertical coordinate, $Z$, and an efficient ODE solver is
developed to construct the solutions.

As an application of our numerical approach, we study the problem of
steady progressive interfacial waves (Stokes' waves).  In contrast to all
the previous works which were concerned with inviscid fluids, we study the
Stokes' waves in the presence of viscosity.  Our numerical results show
that the effect of viscosities is somehow equivalent to decreasing the
value of the expansion parameter in the series expansion of the inviscid
Stokes' waves. Our work suggests a new expansion form for Stokes' waves in
viscous fluids.

Finally, we mention some new results obtained in our numerical simulations
to a more practical problem where a wind is blowing over a progressive
water wave and where complicated interactions occur between air and water.

DATE:		Tuesday, March 16, 2004

TIME:		2:30 PM

LOCATION:	Building 2, Room 338


Refreshments at 3:30 PM in Building 2, Room 349.


Massachusetts Institute of Technology
Department of Mathematics
Cambridge, MA  02139