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I have these four PDE's that define the velocity $\vec u=u\hat i+v\hat j+w\hat k$ and pressure $p(x,y,z)$. I am less interested in calculating $p$, but I need to have it to complete the system. $$-f\cdot v(x,y,z)=-\frac{1}{\rho_0(z)}\frac{\partial p}{\partial x}$$ $$f\cdot u(x,y,z)=-\frac{1}{\rho_0(z)}\frac{\partial p}{\partial y}$$ $$0=-\frac{\partial p}{\partial z}-g\cdot\rho(z)$$ $$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$$ Here $\rho_0(z)\ne\rho(z)$ and $f$ and $g$ for my purposes can be treated as constant. The functions $\rho_0(z)$ and $\rho(z)$ I have tabulated values for.

These equations are a simplified model of the motion of a thin film over a sphere, assuming incompressibility, and using the Boussinesq approximation; and then applying some scaling arguments. Part of the simplification reduces the spherical coordinates to a local Cartesian system where the velocities would be solved within some local radius in the film.

I am thinking of using the following boundary conditions.

  1. $\vec u(x,y,H)=0$, where $0$ is at the top of the film and $H$ is at the bottom.
  2. $p(x,y,0)=P_0$

My instructor wants me to try and solve this problem as is, and then at some later point add a source term for $w$ using $w(X,Y,H)=\frac{Q}{A}$.

I have tried to solve this problem analytically by starting with the 3rd equation and getting $p(x,y,z)=P_0-g\int_0^H\rho_0(z)dz+A(x,y)$. Then I can get $u$ and $v$ by inserting this into the first two equations.$$u=-\frac{1}{f\rho_0(z)}\frac{\partial A}{\partial y}$$ $$v=\frac{1}{f\rho_0(z)}\frac{\partial A}{\partial x}$$ At this point I tried inserting them into the 4th equation only to find $$\frac{\partial w}{\partial z}=\frac{1}{f\rho_0(z)}\frac{\partial^2 A}{\partial y \partial x}-\frac{1}{f\rho_0(z)}\frac{\partial^2 A}{\partial x \partial y}\equiv0$$ which is trivial.

Am I missing something, or is there something wrong with the problem statement? This isn't a homework problem, but my instructor does want us to consider how we would go about solving this problem.

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    If you take the partial with respect to $y$ in the first equation, the partial with respect to $x$ in the second and equate, you get that $ \partial_{x} u = - \partial_{y} v$. Inserting into the divergence condition gives the same result you obtained. I think you are better off doing an analysis in normal modes. – 2017-02-01
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    @Mattos, did you delete your previous comment about normal modes? Could you please comment again with it because I don't see it here. – 2017-02-01
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    Google normal mode analysis. For example, note that $p = P_{0} \cos(\pi z)$ satisfies the second boundary condition for $H$ an integer. Inserting $p$ into the third PDE gives $\rho(z) = \sin(\pi z)$, $g = \pi P_{0}$. In turn, $p_{x}, p_{y} = 0$. So take $\rho_{0}(z)$ equal to any non-zero function of $z$. To satisfy the other BC, the stationary velocity $\vec{u} = 0$ satisfies the divergence condition and the first two PDEs. So take $f$ arbitrary. This set of solutions satisfies the problem. However, $p = P_{0} \cos(\pi z)$ is just one ansatz in a normal mode. You can choose others. – 2017-02-04

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