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?

  • $\begingroup$ That's a weird flow. Your velocity equation says the speed in the $x$ direction increases as you move along the $x$ axis. Are you sure you didn't mean something like $\mathbf{v} = y\hat{\mathbf{i}} + 0\hat{\mathbf{j}} + 0\hat{\mathbf{k}}$ $\endgroup$ – John Rennie Jun 19 '16 at 18:35
  • $\begingroup$ I agree my velocity field is not divergence free and so it is definitely weird. I chose that particular velocity field because it has no shearing motion but yet gives some non-zero normal stress which is a function of dynamic viscosity. The velocity field that you have has shearing motion between fluid layers and does not give any non-zero normal stresses. $\endgroup$ – shk92 Jun 19 '16 at 18:47

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.

  • $\begingroup$ Thank you! So it is indeed true that dynamic viscosity can generate normal stresses, even if there is no "sliding" of fluid layers. I have been studying fluid mechanics for 3 years now and never thought about normal viscous stresses (for incompressible flows) and how dynamic viscosity can affect them. $\endgroup$ – shk92 Jun 19 '16 at 19:39

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