Cross Bush

08/21/2024 · High School

Use the methods of symmetry analysis to solve the system of equations: \[ \begin{array}{l}u_{t}+u u_{x}+v u_{y}+w u_{z}=-p_{x}+\nabla^{2} u \\ v_{t}+u v_{x}+v v_{y}+w v_{z}=-p_{y}+\nabla^{2} v, \\ w_{t}+u w_{x}+v w_{y}+w w_{z}=-p_{z}+\nabla^{2} w \\ \text { and } \\ \qquad u_{x}+v_{y}+w_{z}=0,\end{array} \] where \( t, x, y \) and \( z \) are the independent variables of \( u, v, w \) and \( p \), the dependent variables.