Formulation

We now consider the incompressible Navier-Stokes equations in a time-dependent domain $\dom(t)$

where $\nu$ is the viscosity and f_i is a body force. We assume for now homogeneous boundary conditions; specific boundary conditions will be presented in a later section. We present the variational form of the momentum equation following the derivation in [54]. To this end, we consider test functions v_i in the space where all functions and their derivatives are square-integrable, i.e. $H^1[\dom(t)]$. Multiplying equation (5.5) by test functions and integrating by parts we obtain  

$$\int_{\dom(t)} v_i (u_{i,t} + u_j u_{i,j}) d \bx = \int_{\dom(t)} v_{i,j} ( p \delta_{ij} - \nu u_{i,j} + v_if_i) d \bx.$$ (1)

The next step is to define the reference system on which time-differentiation takes place. This was accomplished in [54] by use of the Reynolds transport theorem and by using the fact that the test function v_i is following the material points; therefore, its time-derivative in that reference frame is zero, i.e.

$$\frac{d v_i}{d t}\vert _{X_p} = v_{i,t} +w_j v_{i,j} = 0$$

where w_j is a velocity that describes the motion of the time-dependent domain $\dom(t)$. Amd X_p denotes the material point. The final variational statement then becomes  

$$\frac{d}{d t} \int_{\dom(t)} v_i u_i d \bx + \int_{\dom(t)} ... ...} [v_{i,j} p \delta_{ij} - \nu v_{i,j} u_{i,u} + v_i f_i] d \bx$$ (2)

This is the ALE formulation of the momentum equation. It reduces to the familar Eulerian and Lagrangian form by setting w_j=0 and w_j = u_j, respectively. However, w_j can be chosen arbitrarily to minimize the mesh deformation.