today a new problem has appeared.
Usually, the shear stress tensor Pi is calculated in the form
Pi_ij = eta * (dv_i / dx_j + dv_j / dx_i)
Obviously, this expression is symmetric. It can be calculated in the LB code in a local way,
Pi_ij = (1 - 1 / (2 * tau)) * sum_k f^neq_k * (c_ik * c_jk - delta_ij / 3)
in three dimensions. However, this quantity is not physical. Why is that so? If you have an area element in the xy-plane and you want to measure (in an experiment) the force due to shear on that area element PARALLEL to the xy-plane, then you measure
Pi'_xz = eta * dv_z / dx
Pi'_yz = eta * dv_z / dy
Those two quantities are not the same as Pi_xz and Pi_yz using the definition at the top. The shear tensor Pi_ij has five independent components, since it is symmetric and traceless. The shear tensor Pi’_ij has EIGHT independent components (if the flow is incompressible). The missing three components are the asymmetric parts, which have been removed.
Now, since I want to simulate the physical shear stress tensor, which can be measured in an experiment, I want to calculate the quantity Pi’ and not Pi. It is not possible to extract Pi’ from Pi, since the asymmetric information is gone.
But is it possible to find a local expression usable in a LB code, which directly produces Pi’?
My understanding is that this is not possible, since one cannot find a second moment of the velocities which is asymmetric.
My conclusion is that, if one is interested in Pi’, one has to use a finite difference scheme, which of course is not local.
Does anybody have an idea regarding this issue?
Thanks a lot,