Vortex mode of 2-D quasi simple wave
Multi-dimensional quasi simple wave for weakly dissipation fluid is formulated by method of multi-scale or multi ordering analysis which is different from the standard perturbation. For sufficiently small amplitudes and small dissipation coefficients, solutions are possible which may be regarded as analogous to unidirectional simple waves. In order to solve the fundamental equation for the quasi-simple wave, the physical variables, such as pressure, density and velocity, are assumed to consist of two parts. The first part has the same functional form of the phase as for the corresponding exact simple wave and the second part is a function of (r,t) assumed to be in second order. The vortex simple wave which is studied in this article is one of physical solutions. It is shown that the vortex simple waves still propagate in an incompressible fluid with weakly dissipation the same as ideal incompressible fluid up to the second order. It is expressed that the pressure and density are constant in whole of fluid and variation of velocity with respect to phase is normal to normal vector. Generally, in dimension two new form of Burgers equation is obtained. The 2-D Burgers equation has not a unique form and our obtained equation was not extracted before at least in the fluid mechanics. Up to now, this equation has not been solved analytical. By using mathematical methods, new solutions are obtained for it. The methods used for the present study are quite powerful and systematic ones.