Numerical approach for bifurcation of nonlinear problem

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We have the following results on incompressible fluid.The first one is on the two-dimensional stagnation-point solution of the Navier-Stokes equations. On it Childress et al. ('89) investigated an unsteady example, namely they show some numerical examples of finite time blow-up for large Reynolds number, They also gave the critical Reynolds number numerically. However we have obtained different results using modified formulation. We had no blow-up 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 capillary-gravity 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 stream-lines 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 non-symmetric 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 non-symmetric solutions for the case of infinite depth, but for the case of finite depth. We will further study it.