A complex variable integration technique for the two-dimensional Navier-Stokes equations
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Starting from a complex variable formulation for the two-dimensional steady flow equations describing the motion of a viscous incompressible liquid, a method is developed which carries out three integrations of the fourth order system in parametric form containing three arbitrary real functions. Introduction. It is a feature of nonlinear differential equations that even when an exact solution is available it is not always possible to express the dependent variables as explicit functions of the independent variables. Clearly this can be a disadvantage when the dependent variables represent unknown physical quantities and the independent variables may be space and time. However, in some cases the solution can be parametrized in terms of derivatives, as in the elementary example x = pep , p = dy/dx . Differentiation with respect to y , followed by integration with respect to p, leads to a second equation y = {p2 p + \)ep + c from which the net gain is a parametrization of x and y in terms of the first derivative p containing an arbitrary constant c. In this example it is possible to eliminate p to determine a relation between x, y, and c, although this is not the case with the more general equation x f(p), which can be treated by the same technique. In general parametric representation represents a powerful method for displaying solutions of nonlinear differential equations especially when the solutions bifurcate as in the case of the Navier-Stokes equations. The present paper attempts to extend this solution method to a certain class of partial differential equations and in particular the two-dimensional steady flow NavierStokes equations. Starting from a concise complex variable formulation for the flow equations first given in [1], and subsequently rediscovered by others, an integration technique is developed which exhibits solutions in implicit parametric form. The dependent variables in the complex system are the stream function y and an auxiliary function 0 associated with the Bernoulli function, or total head of pressure. One major advantage in connection with the present analysis is that the complex flow equation is quasi-linear, autonomous, and contains only z = x iy as independent variable. Received June 6, 1990. ©1991 Brown University 555
[1] R. Courant,et al. Methods of Mathematical Physics, Vol. I , 1954 .