Steady Navier–Stokes in a channel (parabolic inflow from the left, no-slip walls, free outflow). Drag the object in the flow; reshape it in the editor on the left; set the viscosity. All demos
$$u\cdot\nabla\omega = \nu\,\Delta\omega - \tfrac{1}{\eta}\,\nabla\times\big(\mathbf 1_{D}\,u\big),\qquad -\Delta\psi = \omega,\qquad u = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi)\ \text{ in } (0,1)^2,$$
$$u = \big(4x_2(1-x_2),\,0\big) \text{ at } x_1 = 0,\qquad u = 0 \text{ at } x_2 \in \{0,1\},\qquad \partial_{x_1}\psi = \partial_{x_1}\omega = 0 \text{ at } x_1 = 1,\qquad \eta = 10^{-3}.$$
The object \(D = \{x : |x - x_c| < r(\theta(x))\}\) is impermeable (a Brinkman region of permeability \(\eta\)); \(r(\theta)\) interpolates 8 control radii \(0.1\,(1+0.5\rho_m)\). Parameters: centre \(x_c \in [0.2,0.6]\times[0.3,0.7]\), \(\rho_m \in [-1,1]\), viscosity \(\nu \in [0.001, 0.04]\) (object Reynolds number \(\approx 0.2/\nu\)).