We show that a viscous steady water flow of constant non-vanishing vorticity situated below a free surface (assumed to have the shape of a two-dimensional time dependent graph) and above a flat bottom has to be two-dimensional; that is, while satisfying the viscous three-dimensional water wave equations, it turns out that the free surface, the velocity field and the pressure present no variation in one of the horizontal directions. Moreover, the vorticity must have only one non-zero component which points in the horizontal direction orthogonal to the direction of the surface wave propagation. Compared with previous works, our study here has the advantage that it considers the Navier–Stokes equations and also takes into account the normal and tangential stress boundary conditions.

我们表明，位于自由表面下方（假设具有二维时间相关图的形状）和平底上方的具有恒定非零涡度的粘性稳定水流必须是二维的；即在满足粘性三维水波方程的同时，自由表面、速度场和压力在水平方向之一没有变化。此外，涡度必须只有一个非零分量，该分量指向与表面波传播方向正交的水平方向。与以前的工作相比，我们在这里的研究的优势在于它考虑了 Navier-Stokes 方程，还考虑了法向和切向应力边界条件。