Browsing by Author "Chan, Chi Hin"

Now showing 1 - 9 of 9

An ODE for boundary layer separation on a sphere and a hyperbolic space
(2014-07-15)
Chan, Chi Hin
;
Czubak, Magdalena
;
Yoneda, Tsuyoshi
;
應用數學系
;
Department of Applied Mathematics
Ma and Wang derived an equation linking the separation location and times for the boundary layer separation of incompressible fluid flows. The equation gave a necessary condition for the separation (bifurcation) point. The purpose of this paper is to generalize the equation to other geometries, and to phrase it as a simple ODE. Moreover we consider the Navier-Stokes equation with the Coriolis effect, which is related to the presence of trade winds on Earth. (C) 2014 Elsevier B.V. All rights reserved.
An ODE for boundary layer separation on a sphere and a hyperbolic space
(2014-07-15)
Chan, Chi Hin
;
Czubak, Magdalena
;
Yoneda, Tsuyoshi
;
應用數學系
;
Department of Applied Mathematics
Ma and Wang derived an equation linking the separation location and times for the boundary layer separation of incompressible fluid flows. The equation gave a necessary condition for the separation (bifurcation) point. The purpose of this paper is to generalize the equation to other geometries, and to phrase it as a simple ODE. Moreover we consider the Navier-Stokes equation with the Coriolis effect, which is related to the presence of trade winds on Earth. (C) 2014 Elsevier B.V. All rights reserved.
Antithesis of the Stokes Paradox on the Hyperbolic Plane
(1970-01-01)
Chan, Chi Hin
;
Czubak, Magdalena
;
應用數學系
;
Department of Applied Mathematics
We show there exists a nontrivial H-0(1) solution to the steady Stokes equation on the 2D exterior domain in the hyperbolic plane. Hence we show there is no Stokes paradox in the hyperbolic setting. In fact, the solution we construct satisfies both the no-slip boundary condition and vanishing at infinity. This means that the solution is in some sense actually a paradoxical solution since the fluid is moving without having any physical cause to move. We also show the existence of a nontrivial solution to the steady Navier-Stokes equation in the same setting, whereas the analogous problem is open in the Euclidean case.
Asymptotic behavior of the steady Navier-Stokes equation on the hyperbolic plane
(2017-01-01)
Chan, Chi Hin
;
Chen, Che-Kai
;
Czubak, Magdalena
;
應用數學系
;
Department of Applied Mathematics
We develop the asymptotic behavior for the solutions to the stationary Navier-Stokes equation in the exterior domain of the 2D hyperbolic space. More precisely, given the finite Dirichlet norm of the velocity, we show the velocity decays to 0 at infinity. We also address the decay rate for the vorticity and the behavior of the pressure.
Liouville theorems for the stationary Navier-Stokes equation on a hyperbolic space
(2018-04-01)
Chan, Chi Hin
;
Czubak, Magdalena
;
應用數學系
;
Department of Applied Mathematics
The problem for the stationary Navier-Stokes equation in 3D under finite Dirichlet norm is open. In this, paper we answer the analogous question on the 3D hyperbolic space. We also address other dimensions and more general manifolds. (C) 2017 Elsevier Inc. All rights reserved.
Non-uniqueness of the Leray-Hopf solutions in the hyperbolic setting
(2013-03-01)
Chan, Chi Hin
;
Czubak, Magdalena
;
應用數學系
;
Department of Applied Mathematics
The Leray-Hopf solutions to the Navier-Stokes equation are known to be unique on R-2. We show the uniqueness of the Leray-Hopf solutions breaks down on H-2(-a(2)), the two dimensional hyperbolic space with constant sectional curvature -a(2). We also obtain a corresponding result on a more general negatively curved manifold for a modified geometric version of the Navier-Stokes equation. Finally, as a corollary we also show a lack of the Liouville theorem in the hyperbolic setting both in two and three dimensions.
On the stationary Navier-Stokes flow with isotropic streamlines in all latitudes on a sphere or a 2D hyperbolic space
(2013-09-01)
Chan, Chi Hin
;
Yoneda, Tsuyoshi
;
應用數學系
;
Department of Applied Mathematics
In this paper, we show the existence of real-analytic stationary Navier-Stokes flows with isotropic streamlines in all latitudes in some simply-connected flow region on a rotating round sphere. We also exclude the possibility of having a Poiseuille's flow profile to be one of these stationary Navier-Stokes flows with isotropic streamlines. When the sphere is replaced by a 2-dimensional hyperbolic space, we also give the analog existence result for stationary parallel laminar Navier-Stokes flows along a circular-arc boundary portion of some compact obstacle in the 2-D hyperbolic space. The existence of stationary parallel laminar Navier-Stokes flows along a straight boundary of some obstacle in the 2-D hyperbolic space is also studied. In any one of these cases, we show that a parallel laminar flow with a Poiseuille's flow profile ceases to be a stationary Navier-Stokes flow, due to the curvature of the background manifold.
Remarks on the weak formulation of the Navier-Stokes equations on the 2D hyperbolic space
(2016-05-06)
Chan, Chi Hin
;
Czubak, Magdalena
;
應用數學系
;
Department of Applied Mathematics
The Leray-Hopf solutions to the Navier Stokes equation are known to be unique on R-2. In our previous work, we showed the breakdown of uniqueness in a hyperbolic setting. In this article, we show how to formulate the problem in order so the uniqueness can be restored. (C) 2015 Elsevier Masson SAS. All rights reserved.
The formulation of the Navier-Stokes equations on Riemannian manifolds
(2017-11-01)
Chan, Chi Hin
;
Czubak, Magdalena
;
Disconzi, Marcelo M.
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應用數學系
;
Department of Applied Mathematics
We consider the generalization of the Navier Stokes equation from R-n to the Riemannian manifolds. There are inequivalent formulations of the Navier Stokes equation on manifolds due to the different possibilities for the Laplacian operator acting on vector fields on a Riemannian manifold. We present several distinct arguments that indicate that the form of the equations proposed by Ebin and Marsden in 1970 should be adopted as the correct generalization of the Navier-Stokes to the Riemannian manifolds. (C) 2017 Elsevier B.V. All rights reserved.