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##### IMI(Institute of Mathematics for Industry)

# Publications

## MI Preprints

2017-8

Title：Frictional effect on stability of discontinuity surface of tangential velocity in shallow water

Author : Liangbing Jin, Thi Thai Le & Yasuhide Fukumoto

Author : Liangbing Jin, Thi Thai Le & Yasuhide Fukumoto

Abstract: In the absence of waves, a surface of tangential discontinuity in velocity is necessary unstable, regardless of strength of discontinuity, being well known as the Kelvin-Helmholtz instability (KHI). This problem has intensively been investigated from the linear stability to the turbulence stage. Landau (1944) demonstrated a surprising result that, for a compressible fluid, KHI is suppressed when the discontinuity velocity U is greater than √8 times sound speed c. There is an analogy between gravity waves in shallow-water flow of an incompressible fluid and sound waves in a compressible gas flow. Exploiting this analogy, Bazdenkov and Pogutse (1983) showed that a discontinuous surface of tangential velocity is stabilized when the discontinuous velocity is greater than √8 times propagating speed c of gravity wave. Compared with the classical KHI, the stability of the KHI in the presence of waves is less studied. This paper is concerned with the influence of a bottom drag, which dissipates the kinetic energy of the flow, on the KHI in the shallow-water flow. This paper gives a surprise that the drag force destabilized otherwise the stable flow. This is true even when the drag force is very strong. Ref [11] gives a result that for a similar shallow-water system with bottom drag, stability is recovered as the drag force becomes strong.

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2017-8

2017-7

Title：$M^2$ expansion for effect of compressibility on Darrieus-Landau instability of a premixed flame

Author : Keigo Wada & Yasuhide Fukumoto

Author : Keigo Wada & Yasuhide Fukumoto

Abstract: The effect of compressibility on the stability of a plane flame front of the premixed flame is investigated in the form of the $M^2$ expansion. The method of the matched asymptotic expansions employed, by dividing the region of the flame front into preheat and reaction zones. The small parameter of the asymptotic expansion for the former is the thickness of the diffusion zone relative to the hydrodynamic scale and for the parameter for the latter is the inverse of activation energy. We manipulate the jump conditions on the flame front, whereby we obtain the correction in the growth rate for the Darrieus-Landau instability to order $M^2$. We reveal the sensitive dependence, on the heat release, of the weak compressibility effect.

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2017-7

2017-6

Title：Finite-thickness effect on speed of a counter-rotating vortex pair at high Reynolds numbers

Author : Ummu Habibah, Hironori Nakagawa & Yasuhide Fukumoto

Author : Ummu Habibah, Hironori Nakagawa & Yasuhide Fukumoto

Abstract: We establish a general formula for the traveling speed of a counter-rotating vortex pair, being valid for thick cores, moving in an incompressible fluid with and without viscosity. We extend, to a higher order, the method of matched asymptotic expansions developed by Ting and Tung (1965 Phys. Fluids Vol. 8 pp. 1039-1051). The solution of the Euler or the Navier-Stokes equations is constructed in the form of a power series in a small parameter, the ratio of the core radius to the distance between the core centers. For a viscous vortex pair, the small parameter should be $\sqrt{\nu/\Gamma}$ where $\nu$ is the kinematic viscosity of the fluid and $\Gamma$ is the circulation of each vortex. A correction due to the effect of finite thickness of the vortices to the traveling speed makes its appearance at fifth order. A drastic simplification is achieved of expressing it solely in terms of the strength of the second-order quadrupole field associated with the elliptical deformation of the core. For a viscous vortex pair, we exploit the conservation law of the hydrodynamic impulse to derive the growth of the distance, cubically in time, between the vortices.

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2017-6

2017-5

Title：Stability of bifurcating stationary solutions of the artificial compressible system

Author : Yuka Teramoto

Author : Yuka Teramoto

Abstract: The artificial compressible system gives a compressible approximation of the incompressible Navier-Stokes system. The latter system is obtained from the former one in the zero limit of the artificial Mach number $\epsilon$ which is a singular limit. The sets of stationary solutions of both systems coincide with each other. It is known that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion, then it is also stable as a solution of the artificial compressible one for sufficiently small $\epsilon$. In general, the range of $\epsilon$ shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of $\epsilon$ can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.

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2017-5

2017-4

Title：Submodular Function Minimization with Submodular Set Covering Constraints and Precedence Constraints

Author : Naoyuki Kamiyama

Author : Naoyuki Kamiyama

Abstract: In this paper, we consider the submodular function minimization problem with submodular set covering constraints and precedence constraints, and we prove that the algorithm of McCormick, Peis, Verschae, and Wierz for the precedence constrained covering problem can be generalized to our setting.

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2017-4

2017-3

Title：Pareto Stable Matchings under One-Sided Matroid Constraints

Author : Naoyuki Kamiyama

Author : Naoyuki Kamiyama

Abstract: The Pareto stability is one of solution concepts in two-sided matching markets with ties. It is known that there always exists a Pareto stable matching in the many-to-many setting. In this paper, we consider the following generalization of the Pareto stable matching problem in the many-to-many setting. Each agent v of one side has a matroid defined on the set of edges incident to v, and the set of agents assigned to v must be an independent set of this matroid. By extending the algorithm of Kamiyama for the many-to-many setting, we prove that there always exists a Pareto stable matching in this setting, and a Pareto stable matching can be found in polynomial time.

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2017-3

2017-2

Title：On the spectrum for the artificial compressible system

Author : Yoshiyuki Kagei, Takaaki Nishida & Yuka Teramoto

Author : Yoshiyuki Kagei, Takaaki Nishida & Yuka Teramoto

Abstract: Stability of stationary solutions of the incompressible Navier-Stokes system and the corresponding artificial compressible system is considered. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero limit of the artificial Mach number $\ep$ which is a singular limit. It is proved that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small $\ep$. The result is applied to the Taylor problem.

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2017-2

2017-1

Title：Gyroscopic Analogy of Coriolis Effect of Rotating Stratified Flows Confined in a Spheroid

Author : Yuki Miyachi & Yasuhide Fukumoto

Author : Yuki Miyachi & Yasuhide Fukumoto

Abstract: An insight is gained into the mechanism of system rotation for suppressing the Rayleigh-Taylor instability (RTI) by drawing analogy with the gyroscopic effect. A rotating flow of a stratified fluid confined in a spheroid, subject to gravity force, whose velocity field is linear in coordinates, is equivalent, in the Boussinesq approximation, to the motion of the Lagrange top, a heavy symmetrical rigid body. The sleeping top corresponds to the state in which a heavy fluid lies on top of a lighter fluid. Specifically, we investigate the incompressible two-layer RTI confined in the lower-half of a spheroid rotating about the axis of symmetry oriented parallel to the vertical direction. We derive the dispersion relation and the critical rotation rate for suppressing the axisymmetric mode of RTI. The gyroscopic analogy accounts for decrease of the critical rotation rate with oblateness of the spheroid. The stabilizing effect of rotation is enhanced for the half spheroid as compared with a circular cylinder of finite length.

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2017-1