STOKES_2D_EXACT
Exact solutions to the
2D Incompressible Steady Stokes Equations

STOKES_2D_EXACT, a Python library which evaluates exact solutions to the incompressible steady Stokes equations in 2D.

The incompressible steady 2D Stokes equations seek functions u(x,y), v(x,y), (the velocity vector) and p(x,y) (the pressure) such that

        - uxx - uyy + dpdx = f
        - vxx - vyy + dpdy = g
          ux  + vy         = h
      

Three exact solution fields are provided here.

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages:

STOKES_2D_EXACT is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

NAVIER_STOKES_2D_EXACT, a Python library which evaluates an exact solution to the incompressible time-dependent Navier-Stokes equations over an arbitrary domain in 2D.

NAVIER_STOKES_MESH2D, MATLAB data files which define triangular meshes for several 2D test problems involving the Navier Stokes equations for fluid flow, provided by Leo Rebholz.

SPIRAL_DATA, a Python library which computes a velocity vector field that satisfies the continuity equation, writing the data to a file that can be plotted by gnuplot.

Reference:

1. Junping Wang, Yanqiu Wang, Xiu Ye,
   A robust numerical method for Stokes equations based on divergence-free H(div) finite element methods,
   SIAM Journal on Scientific Computing,
   Volume 31, Number 4, 2009, pages 2784-2802.
2. Howard Elman, Alison Ramage, David Silvester,
   Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics,
   Oxford, 2005,
   ISBN: 978-0198528678,
   LC: QA911.E39.

Source Code:

• grid_2d.py, returns a regular 2D grid.
• r8vec_amax.py, returns the maximum absolute value entry in an R8VEC.
• r8vec_amin.py, returns the minimum absolute value entry in an R8VEC.
• r8vec_max.py, returns the maximum entry in an R8VEC.
• r8vec_min.py, returns the minimum entry in an R8VEC.
• r8vec_norm_l2.py, returns the L2 norm of an R8VEC.
• r8vec_print.py, prints an R8VEC.
• r8vec_uniform_ab.py, returns a scaled pseudorandom R8VEC.
• resid_stokes1.py, evaluates the residual at any point (x,y) for flow #1.
• resid_stokes2.py, evaluates the residual at any point (x,y) for flow #2.
• resid_stokes3.py, evaluates the residual at any point (x,y) for flow #3.
• rhs_stokes1.py, evaluates the residual at any point (x,y) for flow #1.
• rhs_stokes2.py, evaluates the residual at any point (x,y) for flow #2.
• rhs_stokes3.py, evaluates the residual at any point (x,y) for flow #3.
• stokes_gnuplot.py, writes files to create a GNUPLOT image of the flow.
• stokes_matplotlib.py, writes files to create a MATPLOTLIB image of the flow.
• timestamp.py prints the YMDHMS date as a timestamp.
• uvp_stokes1.py, evaluates the velocity and pressure field at any point (x,y) for flow #1.
• uvp_stokes2.py, evaluates the velocity and pressure field at any point (x,y) for flow #2.
• uvp_stokes3.py, evaluates the velocity and pressure field at any point (x,y) for flow #3.

Examples and Tests:

• stokes_2d_exact_test.py, calls all the tests.
• stokes_2d_exact_test.sh, runs all the tests.
• stokes_2d_exact_test.txt, the output file.

Graphics files for flow #1:

• stokes1_commands.txt, commands to draw the velocity field.
• stokes1_data.txt, velocity field data.
• stokes1.png, a PNG plot of the velocity field.
• stokes1_matplotlib.png, a PNG plot of the velocity field created by MATPLOTLIB.

Graphics files for flow #2:

• stokes2_commands.txt, commands to draw the velocity field.
• stokes2_data.txt, velocity field data.
• stokes2.png, a PNG plot of the velocity field.
• stokes2_matplotlib.png, a PNG plot of the velocity field created by MATPLOTLIB.

Graphics files for flow #3:

• stokes3_commands.txt, commands to draw the velocity field.
• stokes3_data.txt, velocity field data.
• stokes3.png, a PNG plot of the velocity field.
• stokes3_matplotlib.png, a PNG plot of the velocity field created by MATPLOTLIB.

You can go up one level to the MATLAB source codes.