STOKES_2D_EXACT
Exact solutions to the
2D Incompressible Steady Stokes Equations

STOKES_2D_EXACT, a FORTRAN90 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

while satisfying boundary conditions for u and v, and a normalization condition for p, such as specifying its value at one point, or requiring that the integral of p over the region be 0.

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 FORTRAN90 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 FORTRAN90 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:

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.
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:

s2de.f90, the source code.

Examples and Tests:

s2de_test.f90, a sample calling program.
s2de_test.txt, the output file.

Graphics files:

stokes1_commands.txt, commands to draw the velocity field.
stokes1_data.txt, velocity field data.
stokes1.png, a PNG plot of the velocity field.
stokes2_commands.txt, commands to draw the velocity field.
stokes2_data.txt, velocity field data.
stokes2.png, a PNG plot of the velocity field.
stokes3_commands.txt, commands to draw the velocity field.
stokes3_data.txt, velocity field data.
stokes3.png, a PNG plot of the velocity field.

List of Routines:

GET_UNIT returns a free FORTRAN unit number.
GRID_2D returns a regular 2D grid.
R8VEC_NORM_L2 returns the L2 norm of an R8VEC.
R8VEC_UNIFORM_AB returns a scaled pseudorandom R8VEC.
RESID_STOKES1 returns residuals of the exact Stokes solution #1.
RESID_STOKES2 returns residuals of the exact Stokes solution #2.
RESID_STOKES3 returns residuals of the exact Stokes solution #3.
RHS_STOKES1 returns the right hand sides of the exact Stokes solution #1.
RHS_STOKES2 returns the right hand sides of the exact Stokes solution #2.
RHS_STOKES3 returns the right hand sides of the exact Stokes solution #3.
STOKES_GNUPLOT writes the Stokes vector field to files for GNUPLOT.
TIMESTAMP prints the current YMDHMS date as a time stamp.
UVP_STOKES1 evaluates the exact Stokes solution #1.
UVP_STOKES2 evaluates the exact Stokes solution #2.
UVP_STOKES3 evaluates the exact Stokes solution #3.

You can go up one level to the FORTRAN90 source codes.

Last revised on 23 January 2015.

STOKES_2D_EXACT Exact solutions to the 2D Incompressible Steady Stokes Equations