A One-Dimensional Model of the Navier-Stokes

A one-dimensional nonlinear dynamical system is examined as a simplified model of the dynamics ensued by the Navier-Stokes equations. The model has a richer dynamical behaviour than the Burgers equation and shows several features similar to the ones that are associated with the threedimensional Navier-Stokes. Although the spatial dimension is only one, there are still three velocity components and three “directions.” The gradients along the transverse (virtual) directions are given by the product of the gradient along the real dimension and two arbitrary parameters, α and β, which can be either constant or variable. In general, differentiation with respect to the yand z-axis is replaced by differentiation in the x-axis and multiplication by α and β, respectively. This model, for various values of the two parameters, is solved numerically with a pseudo-spectral method and the results are analyzed. The dynamics of the proposed model differs from the well studied dynamics of the Burgers equation. For example, in the case of variable coefficients, the shock formation which characterizes Burger-like solutions is not present in the proposed model.