I have been given a flow with Lagrange path trajectories: $$x(\alpha,t)=(\alpha_1\cos(t)+\alpha_2\sin(t),\alpha_2\cos(t)-\alpha_1\sin(t),\alpha_3)$$ and I have to determine whether it is an incompressible flow or not. I know that incompressible flow means its mass density is always constant along a particle path, which leads to the statement that the material derivative of the density equals zero, and hence the divergence of the velocity function equals zero, or: $$\nabla \cdotp v=0$$ and so I worked out the Lagrangian velocity of the fluid: $$v(\alpha,t)=(-\alpha_1\sin(t)+\alpha_2\cos(t),-\alpha_2\sin(t)-\alpha_1\cos(t),0)$$ and then calculated the divergence of $v$, which gave: $$\nabla \cdotp v=-2sin(t)$$ which is not equal to zero, and hence the fluid is incompressible. Is this correct? I get a sense from the question that the fluid IS incompressible and I could have done this completely wrong, so I'm not sure. Any help would be greatly appreciated!


Hint: you have to find an expression $v(x)$ instead of $v(t)$: the divergence operator is applied to spacial variables.

details:$$ v(x) = (y,-x,0)\\ \nabla\cdot v = \partial_x y + \partial_y(-x) + \partial_z(0)= 0+0+0=0 $$

| cite | improve this answer | |
  • $\begingroup$ makes sense... thank you! $\endgroup$ – Lucy Mar 25 '14 at 18:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.