astronomix._modules._viscosity._viscosity

Newtonian viscosity source terms.

Intuition of the 1D case

It might be intuitive that in a setting with viscosity, a velocity gradient partial_z v_x of v_x leads to a diffusive momentum flux -mu partial_z v_x = -tau_xz (the assumption of a linear relation here implies a Newtonian fluid) which gradient then enters the Euler equations, leading to a source term partial_z tau_xz = mu partial_z^2 v_x.

3D generalization

Given the velocity gradient tensor G_{ij} = ∂v_i/∂x_j, most generally, the viscous stress tensor (which encodes momentum fluxes in all directions) for a Newtonian fluid is given by

tau_{ij} = K_{ij}^{kl} G_{kl}

where K is a 4th order tensor with 3^4 = 81 components.

Our fluid is isotropic, which means that the components of K must be invariant under any rotation of the coordinate system R in SO(3). The only 2nd order isotropic tensor is delta_{ij}, so naturally,

K_{ij}^{kl} = lambda delta_{ij} delta^{kl} + mu delta_i^k delta_j^l + nu delta_i^l delta_j^k

so we are down to 3 parameters. Additionally, tau_{ij} must be symmetric (otherwise we would create a net torque and violate angular momentum conservation, imagine a small fluid element), implying K_{ij}^{kl} = K_{ji}^{kl}, which implies nu = mu (we are down to two parameters), such that

K_{ij}^{kl} = lambda delta_{ij} delta^{kl} + mu (δ_i^k δ_j^l + δ_i^l δ_j^k)

Inserting this into the definition of tau_{ij} gives

tau_{ij} = lambda delta_{ij} G_{kk} + mu (G_{ij} + G_{ji})

We can split the velocity gradient tensor into a symmetric and antisymmetric part

G_{ij} = ∂v_i/∂x_j = 1/2 * (∂v_i/∂x_j + ∂v_j/∂x_i) + 1/2 * (∂v_i/∂x_j - ∂v_j/∂x_i)

|-------- symmetric --------| |------ antisymmetric -------|

where the antisymmetric part corresponds to the vorticity and drops out by the symmetry of tau_{ij}, leaving only the symmetric part.

Therefore (also use G_{kk} = ∇·v)

tau_{ij} = lambda delta_{ij} ∇·v + mu * (∂v_i/∂x_j + ∂v_j/∂x_i)

As a next step, we split tau_{ij} into a (isotropic) hydrostatic part

h_{ij} = 1/3 * tau_{kk} delta_{ij} = (λ + 2/3 μ) ∇·v δ_{ij}

|- mean trace -|

acting like a pressure and a deviatoric part

s_{ij} = tau_{ij} - h_{ij} = mu * (∂v_i/∂x_j + ∂v_j/∂x_i - 2/3 δ_{ij} ∇·v)

and we impose Stoke’s hypothesis (λ + 2/3 μ = 0), eliminating the bulk isotropic viscosity, leaving only the deviatoric part, such that

tau_{ij} = s_{ij} = mu * (∂v_i/∂x_j + ∂v_j/∂x_i - 2/3 δ_{ij} ∇·v)

resulting in a momentum source term of ∇·τ and an energy source term of ∇·(v·τ).

Video explanation: https://www.youtube.com/watch?v=YPDaFQUqVE4

TODO: we might also support a variant without Stoke’s hypothesis.

Module Contents

Functions

fv_viscosity_update	Finite-volume viscosity update.
fd_viscosity_source	Finite-difference Newtonian viscosity source term.

API

astronomix._modules._viscosity._viscosity.fv_viscosity_update(primitive_state, params, config, registered_variables, dt)

Finite-volume viscosity update.

NOTE: not yet implemented. The finite-volume viscosity is intended to be folded into a unified all-source-term scheme in the future; until then it raises NotImplementedError.

astronomix._modules._viscosity._viscosity.fd_viscosity_source(primitive_state, params, config, registered_variables)

Finite-difference Newtonian viscosity source term.

Builds the deviatoric viscous stress tensor under Stokes’ hypothesis and returns its divergence as a momentum source plus the divergence of v·τ as an energy source (see the module docstring for the derivation).

Args:

primitive_state: The primitive state array. params: The simulation parameters (providing the viscosity). config: The simulation configuration. registered_variables: The registered variables.

Returns:

The viscous source term in the layout of the (primitive) state array.