fluid dynamics- is this flow incompressible?

Question

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!

Answer 1

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

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