# deriveShearStress Subroutine

## public recursive subroutine deriveShearStress(fun, varSys, elempos, time, tree, nElems, nDofs, res)

Calculate the deviatoric shear stress for Newtonian fluid (exclude pressure) (no mixtures).\n Shear Stress depends on variable: nonEquilibirium

The formula is where $\tau_{\alpha \beta}$ is the stress in the $\beta$-direction on a face normal to the $\alpha$-axis,\n $f^{neq}_i = f_i - f^{eq}_i$ is the non-equilibirium density.\n For more information, please refer to:\n Krueger T, Varnik F, Raabe D. Shear stress in lattice Boltzmann simulations. Physical Review E. 2009;79(4):1-14.\n Thus this variable is dependent on another variable: nonEquilibirium.

For multi-level mesh, Omega on finer level needs to be adjusted in order to get the correct shearstress calculation.\n First, we defines c as the dx ratio between finer and coarse level.\n $c={ \Delta dx }_{ c }/{ \Delta dx }_{ f }$ Then the viscosity on the different levels must satisfy:\n $\frac { { \nu }_{ f } }{ { \nu }_{ c } } =c$ This constrain leads to a relationship of omega on different levels:\n ${\omega}_f = \frac {1}{ {\lambda}(\frac{1}{{\omega}_c}-0.5)+0.5 }$ For more information, please refer to:\n Manuel H, Harald K, Joerg B, Sabine R. Aeroacoustic validation of the lattice boltzmann method on non-uniform grids. ECCOMAS 2012

### Arguments

TypeIntentOptionalAttributesName
class(tem_varSys_op_type), intent(in) :: fun

Description of the method to obtain the variables, here some preset values might be stored, like the space time function to use or the required variables.

type(tem_varSys_type), intent(in) :: varSys

The variable system to obtain the variable from.

integer, intent(in) :: elempos(:)

Position of the TreeID of the element to get the variable for in the global treeID list.

type(tem_time_type), intent(in) :: time

Point in time at which to evaluate the variable.

type(treelmesh_type), intent(in) :: tree

global treelm mesh info

integer, intent(in) :: nElems

Number of values to obtain for this variable (vectorized access).

integer, intent(in) :: nDofs

Number of degrees of freedom within an element.

real(kind=rk), intent(out) :: res(:)

Resulting values for the requested variable.

Linearized array dimension: (n requested entries) x (nComponents of this variable) x (nDegrees of freedom) Access: (iElem-1)fun%nComponentsnDofs + (iDof-1)*fun%nComponents + iComp

## Variables

TypeVisibilityAttributesNameInitial
type(mus_varSys_data_type), private, pointer:: fPtr
type(mus_scheme_type), private, pointer:: scheme
integer, private :: posNonEq
integer, private :: nCompsNonEq
integer, private :: iLevel
integer, private :: iElem
real(kind=rk), private :: omega
real(kind=rk), private, allocatable:: nonEq(:)
real(kind=rk), private, allocatable:: tau(:)