OverviewΒΆ
Assess is a python package that implements a number of analytical solutions, in cylindrical and spherical shell domains, to the Stokes equations
with velocity \(\mathbf{u}\) and pressure \(p\). The domain is assumed to be a spherical shell consisting of points with radius \(R_-\leq r \leq R_+\). \(\hat{\mathbf{r}}\) denotes the radial, outward unit-vector.
The gravitational acceleration \(g\) and viscosity \(\nu\) are two user specified constants and the perturbation density \(\rho'\) is assumed to either have the following smooth form
where in 2D, we use cylindrical coordinates with radius \(r\) and azimuthal angle \(\varphi\),
and in 3D, spherical coordinates with radius \(r\), co-latitude \(\theta\),
and longitude \(\varphi\). The radial dependency is a simple polynomial (monomial)
of order \(k\). In 2D, \(n\) is the wave number and in 3D \(l\) and \(m\) are the
degree and order of the spherical harmonic function \(Y_{lm}\) (see assess.Y()
for definition).
Or, \(\rho'\) is a perturbation at a specified radius \(r'\)
where \(\delta(r-r')\) is the Dirac delta function. Combined with two types of boundary conditions
this leads to eight analytical solutions, which are implemented in the following classes
assess.CylindricalStokesSolutionSmoothFreeSlip
assess.CylindricalStokesSolutionSmoothZeroSlip
assess.CylindricalStokesSolutionDeltaFreeSlip
assess.CylindricalStokesSolutionDeltaZeroSlip
assess.SphericalStokesSolutionSmoothFreeSlip
assess.SphericalStokesSolutionSmoothZeroSlip
assess.SphericalStokesSolutionDeltaFreeSlip
assess.SphericalStokesSolutionDeltaZeroSlip