Poiseuille flow in a pipe and plane Couette flow of a viscous incompressible fluid are stable in the linear approximation at all Reynolds numbers, yet these flows certainly do not remain laminar as the Reynolds number is increased indefinitely. In this paper a study is made of some nonlinear differential equations, containing a parameter, which model this aspect of the fluid-mechanical problems. In the model equations there is no finite value of the parameter at which bifurcation occurs, but there are solutions whose norm tends to zero as the parameter tends to infinity. It is shown that these small-norm bifurcating solutions are characterized by strongly-nonlinear balances. In the fluid-mechanical context the viscous and inertial terms would be comparable.
ASJC Scopus subject areas
- Applied Mathematics