I am not an expert in cavitation problem, I immagine you work on a spherical bubble and use the 1D assumption for the radial flow, r=0 being the center of the bubble. Therefore, v = ir u, the unit vector being in the direction of the increasing radius. The boundary of the bubble is therefore coincident to ir for positive r and -ir for negarive r.

This way you could write the vector form of the momentum equation, that is totally independent from the system and projecting along ir to deduce the proper sign.

Vikash,

I follow your logic in the attachment, and I agree that your final momentum conservation equation seems appropriate given your intention to follow Case 1 convention. I haven't worked specifically with bubble cavitation, but I'm wondering if it would be simpler if you used the "standard" convention, and wouldn't the sign problem work itself out as it is solved? In other words: is it necessary to alter convention and use a negative sign for ρ(Du/Dt) to indicate that fluid motion is directed radially inwards, or can that be implicit from your application of the standard form to the cavitation dynamics problem specifically? In that case you could dispense with the need to use Rayleigh's +∇p and can instead stick with the standard −∇p on the right hand side of the equation.

@Sarah, I have given this quite a thought. I have concluded that for the physics to remain correct the viscous-Laplacian term must have opposite sign than the grad P term. So the final equation considering all sign convention is somewhat similar to the Navier-Stokes equation. rho Du/Dt = - grad p + mu Laplacian u + f_ext

I will agree with your last statement in that the momentum equation can stay in its standard form and the solution would provide the right signs for various quantities.

I too recommend that −∇p could be on the RHS of momentum equation.