in fluid mechanics, the taylorâproudman theoremthe taylor-proudman theorem was first derived by sydney samuel hough (1870-1923), a mathematician at cambridge university. see:  (after g. i. taylor and joseph proudman) states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity \omega, the fluid velocity will be uniform along any line parallel to the axis of rotation. \omega must be large compared to the movement of the solid body in order to make the coriolis force large compared to the acceleration terms. that this is so may be seen by considering the navierâstokes equations for steady flow, with zero viscosity and a body force corresponding to the coriolis force, which are: : \rho({\mathbf u}\cdot\nabla){\mathbf u}={\mathbf f}-\nabla p, where {\mathbf u} is the fluid velocity, \rho is the fluid density, and p the pressure. if we now make the assumption that f=\nabla\phi is scalar potential and the advective term may be neglected (reasonable if the rossby number is much less than unity) and that the flow is incompressible (density is constant) then the equations become: : 2\rho\omega\times{\mathbf u}=\nabla \phi -\nabla p, where \omega is the angular velocity vector. if the curl of this equation is taken, the result is the taylorâproudman theorem: : ({\mathbf\omega}\cdot\nabla){\mathbf u}={\mathbf 0}. to derive this, one needs the vector identities :\nabla\times(a\times b)=a(\nabla\cdot b)-(a\cdot\nabla)b+(b\cdot\nabla)a-b(\nabla\cdot a) and :\nabla\times(\nabla p)=0\ and :\nabla\times(\nabla \phi)=0\ (because the curl of the gradient is always equal to zero). note that \nabla\cdot{\mathbf\omega}=0 is also needed (angular velocity is divergence-free). the vector form of the taylorâproudman theorem is perhaps better understood by expanding the dot product: : \omega_x\frac{\partial {\mathbf u}}{\partial x} + \omega_y\frac{\partial {\mathbf u}}{\partial y} + \omega_z\frac{\partial {\mathbf u}}{\partial z}=0. now choose coordinates in which \o
Fluid_dynamics Physics_theorems