What is the physical interpretation of the second term in the viscous stress tensor in the Navier-Stokes equations?

Question

I have been searching for this answer for awhile. I've read numerous texts and even watched some lectures online, but often times this is never explained and just given. The viscous stress term in the Navier-Stokes equations looks like

\begin{equation} \nabla \cdot \tau = \nabla \cdot \mu \left(\nabla\vec{u} + (\nabla\vec{u})^T\right) \end{equation}

Now the term $\nabla \cdot \mu \nabla\vec{u}$ is easy enough to understand as it is just velocity diffusion, but I have a hard time coming up with a physical interpretation of the term $\nabla \cdot \mu (\nabla\vec{u})^T$. After I expanded this term I ended up with

\begin{equation} \nabla \cdot \mu (\nabla\vec{u})^T = \begin{pmatrix} \frac{\partial}{\partial x} \nabla \cdot \vec{u} \\ \frac{\partial}{\partial y} \nabla \cdot \vec{u} \\ \frac{\partial}{\partial z} \nabla \cdot \vec{u} \end{pmatrix} \end{equation}

which seems to imply that this effect is not present in a divergence-free velocity field, but I still can't come up with or find any physical intuition about what this term actually means. Does anyone understand what this term physically represents?

Answer 1

You should not separate those two terms in search for physical interpretation. The term $\nabla\vec{u}+\left(\nabla\vec{u}\right)^T$ is the strain rate tensor $\dot{\gamma}$. The momentum flux (or stress) due to the fact we have a flowing fluid is accounted for by the whole term $\mu\left(\nabla\vec{u}+\left(\nabla\vec{u}\right)^T\right)$. In the NS equation both terms can be thought of as force densities (force per unit volume). You are right, that the second term is zero for incompressible flows (see here).

UPDATE: The complete derivation of the strain rate tensor is complex and it might be out of scope here. If you are interested I have found that a good resource is Introduction to Fluid Mechanics by Whitaker. Briefly, lets accept that the tensor $\nabla \vec{u}$ represents the strain rate and solid like rotational motion. Any tensor can be decomposed in the following way: $$\nabla \vec{u} = \frac{1}{2}\left(\nabla\vec{u}+\left(\nabla\vec{u}\right)^T\right)+\frac{1}{2}\left(\nabla\vec{u}-\left(\nabla\vec{u}\right)^T\right)$$ The first term is typically called the strain rate tensor, is symmetric, and it can be shown that it includes no rigid rotational motion. The second term is typically called the vorticity tensor, its is skew symmetric, and it can be shown that it does not contribute to the rate of strain and that it represents rigid like rotational motion.

Answer 2

I agree with @sturgman one should not look at individual parts but try to understand it in ints context.

Looking at the very basic version of the Navier-Stokes-Equation (using Einstein-Notation):

$$\rho\frac{\mathrm{D}u_i}{\mathrm{D}t} = \rho k_i + \frac{\partial}{\partial x_i} \left( -p + \lambda^*\frac{\partial u_k}{\partial x_k}\right) + \underbrace{\frac{\partial}{\partial x_j} \left( \eta \left[ \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right] \right)}_{\nabla \,\cdot\, (\eta \, \left[ (\nabla\vec{u})+(\nabla\vec{u})^\mathrm{T}\right])} $$

The underbraced part in its original can be rewritten.

$$ \frac{\partial}{\partial x_j} \left( \eta \left[ \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right] \right) = \eta \left( \frac{\partial^2u_i}{\partial x_j \partial x_j} + \frac{\partial}{\partial x_i}\left[\frac{\partial u_k}{\partial{x_k}} \right]\right)$$

Which leads to:

$$\rho\frac{\mathrm{D}u_i}{\mathrm{D}t} = \underbrace{\rho k_i}_{\text{I}} - \underbrace{\frac{\partial p}{\partial x_i}}_{\text{II}} +\underbrace{(\lambda^* + \eta)\frac{\partial}{\partial x_i}\left[\frac{\partial u_k}{\partial x_k}\right]}_{\text{III}} + \underbrace{\eta \left[ \frac{\partial^2u_i}{\partial x_j \partial x_j}\right]}_{\text{IV}}$$

In symbolic notation this should look like:

$$\rho\frac{\mathrm{D}\vec{u}}{\mathrm{D}t} = \rho\vec{k} - \nabla p + (\lambda^* + \eta)\nabla(\nabla\cdot\vec{u}) + \eta\nabla\cdot\nabla\vec{u}$$

Part $\text{III}$ is not always shown like this depending on the way the Newtonian stress tensor was introduced. Since $\lambda^*$ is a fluid property which is very hard to measure but varies only little, Stokes Hypothesis sets it to $-2/3 \eta$ (Which is technically only true for monoatomic gases).

Part $\text{III}$ describes a feature of a fluid where the atomic structure of the fluid-molecule can absorb energy, it is sometimes referred to as pressure-viscosity. Whereas Part $\text{IV}$ describes the resistance of the flow when sheared, part $\text{III}$ describes the resistance of a fluid-volume when it is "isobarically" expanded or compressed.