Numerical approach for bifurcation of nonlinear problem
Project/Area Number  09640295 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Nihon University 
Principal Investigator 
SHOJI Mayumi Nihon University, Department of General Educations, Associate Professor, 理工学部, 助教授 (10216161)

Project Period (FY) 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,200,000 (Direct Cost : ¥2,200,000)
Fiscal Year 1999 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥1,000,000 (Direct Cost : ¥1,000,000)

Keywords  bifurcation / gravity waves / vorticity / Biburcation / progriesive wave / corticity / progressive wave 
Research Abstract 
We have the following results on incompressible fluid. The first one is on the twodimensional stagnationpoint solution of the NavierStokes equations. On it Childress et al. ('89) investigated an unsteady example, namely they show some numerical examples of finite time blowup for large Reynolds number, They also gave the critical Reynolds number numerically. However we have obtained different results using modified formulation. We had no blowup and examined it by two methods, finite difference scheme and spectral mathod. ([1]) The second one is on bifurcation problem of gravity waves with constant vorticity. Our object is to see the global bifurcation structure, combining the results with our results for capillarygravity waves we have obtained before. The results on this work are as below : 1. It is found that bifurcation structures are unchanged qualitatively as vorticity varies. 2. As for symmetric waves, we conjectured numerically how many kinds of mode n bifurcation solutions exist. We checked it for n=1〜6 by simulations. 3. We gave the information about the flow beneath the free surface by plotting streamlines in the fluid region. It can be seen that eddy appears for positive vorticity and it expands as the vorticity becomes larger. 4. Zufiria ('87) gave nonsymmetric solutions for gravity waves of infinite depth numerically. We attempted to follow them by the nonsymmetric version of our algorithm, but we couldn't find any. We believe there might be no nonsymmetric solutions for the case of infinite depth, but for the case of finite depth. We will further study it.

Report
(4results)
Research Products
(11results)