# Unsteady Stokes equations in ALE framework

I'm trying to solve Unsteady Stokes equations on a moving domain, using an ALE formulation, that is $$\frac{\partial \mathbf{u}}{\partial t} - \mathbf{w}\cdot \nabla\mathbf{u} = \nu\Delta\mathbf{u} - \nabla p$$

where $$\mathbf{w}$$ is the velocity of the mesh. I'm starting from the simple case of a rigid sphere translating with constant velocity, $$\mathbf{W}$$, in an unbounded viscous fluid. What I'm testing is that, if I solve the problem in a fixed control volume (the standard way, let's say), simply having an imposed velocity on the sphere in the middle of a large domain, I have that the force acting on it is given by the Stokes formula, that is $$\mathbf{F} = 6\pi R \nu \mathbf{W}$$ (normalized by density), once a stationary is reached.

Now, if I solve for the same thing on a control volume that moves following the sphere, I should have the same result, that is the same force acting on the sphere. I just plug the constant velocity $$\mathbf{W}$$ in place of $$\mathbf{w}$$ in the equation above:

$$\frac{\partial \mathbf{u}}{\partial t} - \mathbf{W}\cdot \nabla\mathbf{u} = \nu\Delta\mathbf{u} - \nabla p$$

and use the same boundary conditions: $$\mathbf{u} = \mathbf{W}$$ on the surface of the sphere, and $$\mathbf{u} = \mathbf{0}$$ on a boundary sufficiently far, or zero stress, should make no difference I think.

I solve the Unsteady Stokes equations problem with P2/P1 finite elements and a standard splitting technique of the Chorin kind: for reference I use the pressure correction projection method reported by Guermond and Shen

Guermond, J. L.; Minev, P.; Shen, Jie, An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng. 195, No. 44-47, 6011-6045 (2006). ZBL1122.76072,

with BDF1 method for time stepping.

The problem is that the force results to be wrong: while in the fixed domain case it approaches the theoretical result, even though with some approximation, in the ALE case the error is much larger, of the order of 35%.

Changing the mesh or the time step seems to have no effect whatsoever on the final value, which makes me think that there must be something wrong in the setup, but I don't see what since I'm simply moving the points of the mesh, translating with a constant velocity (so in theory I don't even have to actually move them), and this shouldn't modify the physics.

Am I wrong? Are there any special precautions to be taken when the problem is that of Unsteady Stokes??

• What are your external boundary conditions and how big is the domain? – EMP Dec 15 '20 at 23:15
• The domain is a cylinder with diameter and height 40 and 50 times the diameter of the sphere, respectively. I set a Dirichlet condition of zero velocity on the outer boundary. – gc11 Dec 16 '20 at 7:36