The constitutive law for a Newtonian fluid is
$$ \boldsymbol{\tau} = 2\mu \mathbf{D} + \lambda\left(\nabla\cdot\mathbf{v}\right)\mathbf{I}$$
where $\mu$ is the dynamic viscosity. Assuming we have a flow field that has a form
$$ \mathbf{v} = x\hat{\mathbf{i}} + 0\hat{\mathbf{j}} + 0\hat{\mathbf{k}}$$
then the normal stress in the $x$ direction is found to be
$$ \tau_{11} = 2\mu + \lambda$$
My question is: Why is dynamic viscosity affecting the normal stress at all? I am probably missing something really basic here but viscous forces come into picture only when there is movement between fluid layers and there is sliding. For the given velocity field, I cannot see any sliding of fluid layers that can cause viscosity to affect the stress. There is only stretching.
It may have something to do with the velocity field I chose and that it is not an admissible one. I am not sure if the validity of the constitutive law depends on whether the velocity field is admissible.
Nevertheless, how can viscosity affect normal components of stress?
Best Answer
The usual picture you see in wkipedia and other sources is indeed over-simplified. Viscosity resists more general velocity gradients, not just pure shear flows $\nabla_x v_y\neq 0$. For example, if you have a compressible fluid undergoing non-isotropic scaling expansion $$ \vec{v} = (\alpha x,\beta y,\gamma z) $$ then shear viscosity will try to equalize the expansion rates $\alpha\simeq \beta \simeq \gamma$. Bulk viscosity will resist the overall expansion of the fluid.
You can view you example $$ \vec{v} = (x,0,0) = \frac{1}{3} \left( (2x,-y,-z)+(x,y,z) \right) $$ as a linear combination of anisotropic shear flow, and pure expansion. Shear viscosity counteracts the first, and bulk viscosity the second term.