Formulas are derived that make it possible to construct new exact solutions for three-dimensional stationary and nonstationary Navier-Stokes and Euler equations using simpler solutions to the respective two-dimensional equations. The formulas contain from two to five additional free parameters, which are not present in the initial solutions to the two-dimensional equations. It is important that these formulas do not contain quadratures in the steady-state case. We consider some examples of constructing new three-dimensional exact solutions to the Navier-Stokes equations using the derived formulas. The results are used to solve some problems of the hydrodynamics of a viscous incompressible liquid. Examples of thenonuniqueness of solutions to steady-state problems are given. Some three-dimensional solutions to the Navier-Stokes equations are constructed using nonviscous solutions to the Euler equations. Two classes of new exact solutions to the Grad-Shafranov equation that contain functional arbitrariness are specified. Note that, in this study, a new method for constructing exact solutions is applied that can be useful for analyzing other nonlinear physical models and phenomena. Several physical and physicochemical systems where this method can be used are considered.
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Polyanin et al. (2011) studied this question.