Exercise 7

Weak Form of Stokes

Consider the steady Stokes equations for slow viscous flow in a domain \(\Omega\): \[ \begin{aligned} \mu \, \Delta \mathbf{u} - \nabla p + \mathbf{f} &= 0 \quad \text{in } \Omega, \\ \nabla \cdot \mathbf{u} &= 0 \quad \text{in } \Omega, \end{aligned} \] with boundary conditions \[ \begin{aligned} \mathbf{u} &= \mathbf{u}_D \quad & \text{on } \Gamma_D, \\ \mu\,\nabla \mathbf{u} \cdot \mathbf{n} - p\,\mathbf{n} &= \mathbf{g} \quad & \text{on } \Gamma_N. \end{aligned} \]

  1. Multiply the momentum equation by a test function \(\mathbf{v}\) and integrate over \(\Omega\). Apply integration by parts to the viscous term \(\mu\,\Delta\mathbf{u}\) and to the pressure term \(\nabla p\). State which boundary terms arise and how the Neumann boundary condition enters the formulation.
  2. Multiply the continuity equation by a test function \(q\) and integrate over \(\Omega\). What condition on the test space \(\mathbb{Q}\) is required?
  3. Combine the results of 1. and 2. to state the complete weak form: find \((\mathbf{u}, p) \in \mathbb{V} \times \mathbb{Q}\) such that \[ \begin{aligned} a(\mathbf{u}, \mathbf{v}) + b(\mathbf{v}, p) &= L(\mathbf{v}) \quad \forall\,\mathbf{v} \in \mathbb{V},\\ b(\mathbf{u}, q) &= 0 \quad \forall\,q \in \mathbb{Q}. \end{aligned} \] Give explicit expressions for the bilinear forms \(a(\mathbf{u},\mathbf{v})\), \(b(\mathbf{v},p)\), and the linear functional \(L(\mathbf{v})\).
  4. The sign of \(b(\mathbf{u},q)\) in the second equation is chosen deliberately. What structural property of the resulting block matrix system does this sign choice preserve?

Newton Shear Flow Experiment

Recall the Newton shear flow experiment setup given in the figure below:

Assume that

  • the plates are infinite
  • the flow is steady
  • the flow is fully developed
  • \(\frac{\partial p_0}{\partial x} =\) constant
  • gravity is acting in negative z-direction

and that you are given the boundary conditions

\[ \begin{aligned} u(z=0) & = U_b \\ u(z=h) & = U_t \\ p(z=h) & = p_0(x) \end{aligned} \]

  1. Compute the pressure.

  2. Compute the velocity \(\mathbf{u} = (u, w)^T\).

  3. Sketch the velocity profile for the cases

\[ \begin{aligned} a) \, & \frac{\partial}{\partial x} p_0 > 0 \quad \text{ and } U_b = U_t = 0, \\ b) \, & \frac{\partial}{\partial x} p_0 = U_b = 0 \quad \text{ and } U_t > 0, \\ c) \, & \frac{\partial}{\partial x} p_0 = U_t = 0 \quad \text{ and } U_b > 0, \\ d) \, & \frac{\partial}{\partial x} p_0 = 0 \quad \text{ and } U_b = 2 \, U_t > 0, \\ e) \, & \frac{\partial}{\partial x} p_0 > 0 \quad \text{ and } U_b = 2 \, U_t > 0. \end{aligned} \]

  1. Name the different cases sketched in 3.