Flow of a Stokes fluid in a pressure-driven channel

\[\begin{split} \mathbb{S}_{\mathbf{u}, p} \begin{cases} \Omega = [0, L_x] \times [0, L_y] \\ \textbf{u}_{\text{E}}(x,y=L_y)=\textbf{0} & \text{no-slip and no-penetration on upper and lower boundaries} \\ \textbf{u}_{\text{E}}(x,y=0)=\textbf{0} \\ \boldsymbol{\tau}_{\text{N}}(x=0,y)=p_{\text{in}}\textbf{e}_x & \text{equivalent to } p(x=0,y)=p_{\text{in}}\\ \boldsymbol{\tau}_{\text{N}}(x=L_x,y)=\textbf{0} & \text{equivalent to } p(x=L_x,y)=0\\ \end{cases} \end{split}\]

from ufl import grad
from lucifex.mesh import rectangle_mesh, mesh_boundary
from lucifex.fem import Function, Constant, cross_section_grid
from lucifex.solver import bvp, BoundaryConditions, OptionsPETSc
from lucifex.plt import plot_colormap, plot_line, save_figure
from lucifex.utils.fenicsx_utils import extract_component_functions
from lucifex.pde.stokes import stokes_incompressible


Lx = 2.0
Ly = 1.0
mesh = rectangle_mesh(Lx, Ly, 200, 200, 'right')
boundary = mesh_boundary(
    mesh, 
    {
        "left": lambda x: x[0],
        "right": lambda x: x[0] - Lx,
        "lower": lambda x: x[1],
        "upper": lambda x: x[1] - Ly,
    },
)
mu = Constant(mesh, 1.0, 'mu')
stress = lambda u: mu * grad(u)

u_elem = ('P', 2, 2)
p_elem = ('P', 1)
up_elem = [u_elem, p_elem]
up = Function((mesh, up_elem), name="up")

bcs_e = BoundaryConditions(
    ('essential', boundary['upper', 'lower'], (0.0, 0.0), 0),
)
p_in = 1.0
bcs_n = BoundaryConditions(
    ('natural', boundary['left'], (p_in, 0.0), 1),
    ('natural', boundary['right'], (0.0, 0.0), 1),
)

petsc = OptionsPETSc(pc_type='lu', pc_factor_mat_solver_type='mumps')
up_solver = bvp(stokes_incompressible, bcs_e, petsc)(up, stress, bcs=bcs_n)
up_solver.solve()

u, p = up.split(('u', 'p'), collapse=True)
ux, uy = extract_component_functions(('P', 1), u, names=('ux', 'uy'))

fig, ax = plot_colormap(p, title=f'${p.name}$')
save_figure('p(x,y)')(fig)

fig, ax = plot_colormap(ux, title=f'${u.name}_x$')
save_figure('ux(x,y)', thumbnail=True)(fig)

ux_x, y_value = cross_section_grid(ux, 'x', 0.5)
fig, ax = plot_line(ux_x, x_label='$y$', y_label=f'${u.name}_x(y={y_value:.2f})$')
save_figure(f'ux(x,y={y_value}')(fig)