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Equations problem Navier-Stokes

The Navier-Stokes equations are named after Claude-Louis Navier and George Gabriel Stokes. This is a set of partial differential equations describing the nonlinearmotion of a fluid. These equations govern the Earth's atmosphere, ocean currents and the flow around vehicles or projectiles and, in general, any phenomenon in which Newtonian fluids involved.
Theseequations are obtained by applying the conservation principles of mechanics and thermodynamics to fluid volume. Doing this yields the formulation called integral equations. To reach theirdifferential formulation considerations apply handled mainly one in which the tangential forces bear a linear relationship with the velocity gradient (Newton law of viscosity), thereby obtaining the differentialformulation is generally more useful for solving problems arising in fluid mechanics.

1.- Between two concentric cylinders rotating at different speeds has a Newtonian fluid in the whichestablishes the following velocity field:


a).- What should be the field of pressure to satisfy the Navier-Stokes?
Much is the term for the viscous forces?
b).- Calculate the powerdissipated in the region defined by:



Respuesta:
a) La ecuación de continuidad en coordenadas cilíndricas es:





Todos los términos se anulan porque no es funciónde
La ecuación de Navier-Stokes en la dirección radial es:



Reemplazando el campo de velocidades dato resulta:


Laecuación de Navier-Stokes en la dirección angular es:


Reemplazando el campo de velocidades dato resulta:









La ecuación de Navier-Stokes en la dirección z es:Reemplazando el campo de velocidades dato se obtiene:

Finalmente, integrando:

b).- La potencia disipada...