Load the package:

Solve a Poisson equation on a unit Disk:

Visualize the solution:

Compute the total flux through the boundary of the region through a second-order approximation of the boundary region:

Compute the total flux through the boundary of a subregion:

Solve a Poisson equation on a unit Disk:

Compute the total flux through the outer boundary:

Compute the total flux through the boundary of the region through a second-order approximation of the boundary region:

Specify a differential equation and a region:

On the left, set up a NeumannValue with a boundary unit normal and solve the equation:

Visualize the solution:

For this domain, the equivalent of the BoundaryUnitNormal at the left is :

Show that the solutions are equal:

Create a tangential for a NeumannValue:

Make use of a Indexed component of a BoundaryUnitNormal to compute a NeumannValue:

Solve a Poisson equation on a geometry:

Plot the solution:

Compute the total flux through the boundary:

Set up a NeumannValue with a boundary normal:

Use a Neumann 0 boundary condition and solve the equation again:

Inspect how the solutions start to differ over time:

For this domain, the equivalent of the BoundaryUnitNormal at the left is :

Check that the solutions are the same:

The following example considers a Stokes flow with prescribed traction boundary conditions in an annulus region. On the outer edge, at , there are no-slip boundary conditions. These set the fluid velocity in the and directions to 0, . On the inner boundary, at , a traction is prescribed. The traction with the stress vector is given as:

    

Here, is the radial unit vector given as and is the tangent unit vector given as . and are the Cartesian unit vectors in the and directions, respectively. For this example, the traction is set in normal direction to and the tangential traction is set to . In other words, at the inner boundary the velocity is not specified, only the traction is specified.

The fluid stress tensor is given by:

    

Stokes equation is given by:

    

and the continuity equation:

    .

Set up parameters, geometry and refined mesh:

Set up a Stokes flow operator:

Set up a no-slip condition on the outer boundary:

On the inner surface, you can compute the tangent unit normal from the outward-pointing unit normal with a cross product. Use a cross product to compute the unit tangent from the unit normal vector :

To extract the first and second components of the cross product for the and directions, respectively, Indexed is used.

Set up the traction boundary conditions:

Equivalently, note that the tangent unit normal can also be given as because the unit normal vector on the inner surface is and the unit tangent then is :

Without specifying a boundary condition for the pressure, the pressure value will be floating and NDSolve will give a warning that not enough boundary conditions are specified:

Visualize the pressure solution and the velocity field:

To verify that the traction boundary condition works, you can compute radial and tangential components of the velocities and plot them at for all . For the radial velocity, a solution proportional to is expected, and for the tangential velocity, a solution proportional to is expected.

Define a function to compute the radial and tangential components of the velocities:

Plot the radial component of the velocities and a function proportional to :

Plot the tangential component of the velocities and a function proportional to :

Next, compute the normal stress and the shear at the inner boundary and check that they match the prescribed traction boundary conditions. The stress tensor is given by:

    .

The normal and tangential components of the traction vector can be computed by

    

and

    .

Create a function to compute the stress tensor :

Compute the unit normal vector:

Define a function for the tangent vector:

Define a function to compute the normal stress:

Define a function to compute the shear stress:

Visualize the computed normal and shear stress at :

Visualize the difference of the normal and shear stress at from the boundary conditions: