# Shear viscosity in an acoustic plane wave

In the absence of bulk viscosity, $$\eta_b = 0$$, in Landau and Lifshitz book and many other places, the viscous stress tensor is defined as:

$$$$\sigma'_{ik} = \eta_s\left(\frac{\partial v_i}{\partial x_k} + \frac{\partial v_k}{\partial x_i}-\frac{2}{3}\delta_{ik}\frac{\partial v_l}{\partial x_l}\right)$$$$

where $$\eta_s$$ is the shear viscosity and $$v_i$$ the fluid velocity. If we consider an acoustic plane wave propagating in the $$x_1$$ direction in a gas at rest, we have:

$$$$v_1 =v\cdot\mathrm{e}^{i(Kx_1-\Omega t)}, \quad v_2 = v_3 = 0$$$$

Then, we find the the non-zero viscous stress coefficients to be:

$$$$\sigma'_{11} = \frac{4}{3}i\eta_s K v_1,$$$$

$$$$\sigma'_{22} = \sigma'_{33} = -\frac{2}{3}i\eta_s K v_1$$$$

First, I thought that the shear viscosity would only apply forces parallel to a surface element (i.e. $$\sigma'_{ii} = 0$$), but it is not the case. Second, since we have a plane acoustic wave, the only velocity gradient is along the velocity direction itself, so I intuitively see this effect as a compression/relaxation of the fluid, for which only the bulk viscosity is concerned, as I thought.

My question is then: is the results I obtained correct? And if yes, how can we intuitively visualize what happens ?

You can think of uniaxial compression as a sum of bulk compression + volume preserving deformation. If it's hard to recognize a pure shear on the second picture, consider rotating coordinates 45°.

Finally, shear viscosity can apply forces perpendicular to element. The only forces that are always tangent are rotational (and the corresponding tensor is antisymmetric).

• When I see your decomposition, I agree there is a shear, and thus viscous forces in the diagram on the right. What I don't fully understand yet is why we perform this decomposition. If I simply imagine a square box getting compressed in one direction only, all the molecules just slide along that direction without generating any viscous forces (contrary to your decomposition, they are not first compressed and then sheared). But this image is probably inaccurate.
– Flav
Commented Oct 12, 2019 at 9:32
• You are thinking about gas/fluid as a solid body with atoms that generate stresses as they move and “slide” along each other. That's not how things work. Viscous forces appear in gases/fluids because of exchange of momentum between adjacent parts of gas/fluid due to the diffusion. Usually, in a classic textbook picture they draw exchange of horizontal momentum along vertical direction, because it is easy to imagine in constant flow. But if there is difference in vertical component, it is exchanged as well. Commented Oct 12, 2019 at 15:29
• Moreover, gas or classic fluid being is isotropic media and stays isotropic after the compression (apart from solids). So if you consider a picture of gas before and after the compression, you can see that gas became denser in both directions. So some molecules have to slide in between other molecules, creating your sideways motion. However, that is not the best way to think about it (see the comment above). Commented Oct 12, 2019 at 15:36