### Two fluid dynamical problems of wave effect on stability of a planar interface of velocity discontinuity

** Time:** 10:00 to 11:30 Ngày 10/09/2018

**Venue/Location: ** C2-714, VIASM

**Speaker:**
Yasuhide Fukumoto (Institute of Mathematics for Industry, Kyushu University, Japan)

**Content:**

An interface across which velocity and/or thermodynamic quantities discontinuously vary serve as a universal model to mimic a wide range of phenomena of fluid flows, which is efficiently tractable by mathematical and numerical means. We address the following stability problems of an interface of discontinuity in tangential and normal velocities.

1) Drag induced instability of surface of velocity discontinuity of a shallow-water flow

2) Effect of compressibility on stability of a planar front of premixed flame

1) An interface of discontinuity in tangential velocity of an incompressible fluid is necessarily unstable and is well known as the Kelvin-Helmholtz instability (KHI). The compressibility drastically alters the situation. The interface is stabilized if the velocity jump $U$ is greater than $\sqrt{8}c$ with $c$ being the sound velocity (Landau 1944). This problem has a mathematical analogue with that of a shallow water. An interface of discontinuity in tangential velocity escapes the KHI for the velocity jump $U >\sqrt{8}c$, with $c$ being the propagating

speed of the gravity wave. We examine the frictional effect on this stability. The drag exerted by the bottom surface is incorporated in integrated form over the depth of the thin fluid layer as the resistance force on the layer. The drag has a destabilizing effect on the stabilized interface. This result is interpreted from the viewpoint of Hamiltonian mechanics.

2) A premixed flame front of combustion is very thin in the hydrodynamic scale and can be viewed as a density discontinuity surface across which the normal velocity varies discontinuously. A planar interface of discontinuity in normal velocity is necessarily unstable in the incompressible limit and is well known as the Darrieus-Landau instability (DLI). The effect of compressibility on the DLI is investigated in the form of the M^2 expansion for small Mach number M.

The method of matched asymptotic expansions is used to derive jump conditions for hydrodynamic variables across a flame front which is separated into the preheat zone and the reaction zone sandwiched by the former. With this jump conditions on the flame front, we obtain the correction to the growth rate of the DLI to O(M^2), showing that the compressibility effect can suppress the DLI. By taking account of compressibility, we can derive the heat-loss effect in a systematic manner without having to impose an ad hoc assumption.

1) And 2) are collaborations with Thi Thai Le and Keigo Wada respectively, of Graduate School of Mathematics, Kyushu University.

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**Yasuhide Fukumoto** is the Director and professor of the Institute of Mathematics for Industry, Kyushu University, Japan. He received his PhD from the University of Tokyo. He visited the University of Cambridge for one year, as a Visiting Fellow Commoner of Trinity College. His specialty is fluid mechanics, in particular, vortex dynamics and hydrodynamic stability, with interest in topological and Hamiltonian mechanical aspects. He is the first recipient of the Ryuumon Prize from the Japan Society of Fluid Mechanics. He is the leader of the Japanese government project for promoting collaboration of mathematics with other fields and industrial technologies. He is the Editor-in-Chief of Fluid Dynamics Research published by IOPP and an associate editor of several journals.