Part VII: Rheology

As we saw during our discussion on transport theory, the principal equation of fluid dynamics really boils down to a transport equation for the momentum in a fluid, or for the transport of momentum density through fluid points:

\displaystyle \frac{\partial\left(\rho\vec{v}\right)}{\partial t}\ + \nabla \cdot \left(\rho \vec{v} \otimes \vec{v}\right) = \vec{f} + \vec{r}_{mol}

where the \vec{r}_{mol} term represents forces stemming from molecular effects. It is this molecular term which exclusively “identifies” a fluid, as everything else in the governing equation above was generated through a very general mathematical framework describing the movement of “things” in space rather than through specific physical insight associated with fluids. Consequently, understanding this connection between continuum-scale momentum and molecular effects is a humongously important task for physicists and engineers alike, and is the principal endeavor of rheology.

It’s often extremely difficult to generate refined models for \vec{r}_{mol} from first principles, so often the process of determining an \vec{r}_{mol} for a given fluid consists of making some basic assumptions about the fluid and then testing the fluid experimentally to see if the behavior of the fluid is consistent with those assumptions. Rheology, as a result, is largely an experimental science.

This doesn’t mean we’re simply left to fiddle around with rheometers from now on, however—in fact, we’ll rapidly find that a couple of simple assumptions and some not-so-simple tensor calculus will lead us to an equation that applies to nearly all fluids.

To that end, the first thing that we can assume is that molecular forces from a fluid point affect only the fluid points immediately around that point. This assumption is referred to by physicists as the principle of locality, but is largely taken for granted by fluid mechanicists; this is because it’s almost always true, since the range of molecular interactions tends to be extremely small, and the strength of those interactions is only notable when molecules are really close together. In other words, this assumption implies that molecules don’t warp momentum around in a fluid from one location to another; they transport it. Mathematically, this means that molecular forces always manifest as momentum density fluxes, and as a result we can rewrite the equation above as:

\displaystyle \frac{\partial\left(\rho\vec{v}\right)}{\partial  t}\ + \nabla \cdot \left(\rho \vec{v} \otimes \vec{v}\right) = \vec{f} +  \nabla \cdot \boldsymbol\sigma

That molecular momentum density flux \boldsymbol\sigma, which is a tensor just like the other molecular momentum density flux term \rho \vec{v} \otimes \vec{v}, is commonly called stress.

A diagram of a series of fluid points, each represented as a cube, with yellow arrows representing the possible interactions between them. The locality assumption indicates that interactions between non-neighboring points of fluid are either negligible or non-existent, meaning forces between two non-neighboring points of fluid have to be transported through points in between them. As a result, intermolecular forces can be described as the divergence of a tensor called the stress.

The stress tensor \boldsymbol\sigma is just a second-order tensor that represents, for a given fluid point, the traction/force per unit area \vec{t} acting on that element thanks to its neighboring fluid point in the \hat{n} direction. If you’ve studied solid mechanics, you’ve likely heard the term stress thrown around before; it means the same thing there as it does here. In fact, the only difference between a solid and a fluid is that the stress is a function of different continuum parameters in each type of substance. Notice we never did anything fluid-specific when we derived the momentum transport equation above!

Luckily, we already know one piece of the puzzle of \boldsymbol\sigma; it has to contain the forces coming from pressure gradients, since we spent two Parts discussing how pressure (and the forces it induces) is an entirely molecular effect. We can make our lives easier and pull that pressure gradient term out, making sure we make it negative so that everything is consistent with the hydrostatic equation we derived in Part II:

\displaystyle \frac{\partial\left(\rho\vec{v}\right)}{\partial   t}\ + \nabla \cdot \left(\rho \vec{v} \otimes \vec{v}\right) = \vec{f} - \nabla p +  \nabla \cdot \boldsymbol\tau

That new tensor \boldsymbol\tau is usually called the deviatoric stress tensor, which I think is a silly name.

The contents of the deviatoric stress tensor are in some sense what we’re really looking for, since the notion of pressure and pressure gradients is universal in solids and liquids. And here, we can classify fluids rheologically into two important & distinct branches based on a key characteristic of their behavior when standing still; fluids whose static behavior is fully characterized by pressure gradients (commonly called simple fluids), and fluids for which it is not (commonly called non-Newtonian fluids). For the latter, the molecules in the fluid will usually have some kind of extra molecular force that “causes” them to try to remain in some specific macroscopic shape or configuration—just like a solid. Some examples of this are ketchup (it doesn’t like to flow out unless you smack the bottle or pump it out), mucus, pancake mix, melted rubber, etcetra; there’s too many to count. A simple experimental way to figure out whether or not a fluid is simple or non-Newtonian is to pour it into onto a surface and then tilt the surface very, very slightly; if the fluid flows at even a slight angle, then it’s safe to say the fluid behaves like a simple fluid. (Professionals use rheometers). Modeling non-Newtonian fluids is very tricky, and usually done on a case-by-case basis; we won’t concern ourselves here with understanding how they’re modeled, other than that their additional static molecular forces usually depend on the same thing that they depend on for solids, which is some displacement off of an “equilibrium” configuration like in a spring.

The “tilt test”; a puddle of simple fluid in blue and a puddle of non-Newtonian fluid in yellow are placed on a surface. When the surface is tilted slightly, the simple fluid loses its form and flows while the non-Newtonian fluid deforms but doesn’t flow/preserves form.

Looking back at simple fluids, modeling them is far easier than modeling non-Newtonian fluids since the assumption on the behavior of simple fluids induces quite a big constraint on that deviatoric stress tensor \boldsymbol\tau; if we take the assumption that every molecular effect that induces forces in an unmoving fluid is by definition captured through pressure gradients, then we find that \boldsymbol\tau can’t possibly contain any terms involving the physics of the fluid when it’s standing still. It is entirely a function of the properties of the fluid’s motion, which is why I like to call \boldsymbol\tau the hydrodynamic stress tensor when modeling simple fluids.

With that in hand, let’s continue with some extra assumptions. The next assumption we can make is that \boldsymbol\tau, which we’ve said depends on the motion of the fluid, is independent of the absolute velocity of the fluid. This is pretty obvious when you think about it; if the stress of a fluid directly depended on the velocity, we would be able to observe a fluid deform differently depending on whether or not we were walking forwards or backwards relative to the fluid, since velocity is always relative to the observer. Instead, we can assume that \boldsymbol\tau is only a function of velocity gradients, which is almost always true unless you’re moving extremely fast. Mathematically, this means:

\displaystyle \frac{\partial\left(\rho\vec{v}\right)}{\partial t}\ + \nabla \cdot \left(\rho \vec{v} \otimes \vec{v}\right) = \vec{f} - \nabla p +  \nabla \cdot \left(\boldsymbol\tau\!\left[\nabla \vec{v}\right]\right)

Here the brackets just indicate that \boldsymbol\tau is a function of \nabla \vec{v}. But remember; when talking about velocity gradients, we have to consider the gradients in each direction of the velocity components in each direction. So instead of representing three pieces of independent information like \vec{v} does, velocity gradients represents nine, meaning that it’s—you guessed it—a tensor.

As a matter of fact, the assumption above puts a further restriction on \boldsymbol\tau; if I decided to somehow spin myself around a static fluid I’m observing in the lab, I would observe a nonzero velocity gradient even though the fluid is standing perfectly still. As a result, a simple fluid’s deviatoric stress tensor can’t depend on every part of the velocity gradient tensor, just some parts. Some behind-the-scenes tensor manipulations show that in a simple fluid, the deviatoric stress tensor can only depend on the symmetric part of the velocity gradient tensor. This property is commonly referred to as objectivity, and the symmetric part of the velocity gradient tensor is usually called the strain-rate tensor \mathbf{E}, calculated from the velocity gradients by:

\displaystyle \mathbf{E} = \frac{\nabla\vec{v} + \nabla\vec{v}^\intercal}{2}

This still doesn’t really get us anywhere; the nine components of \boldsymbol\tau are so far completely arbitrary functions of the nine components of \mathbf{E}. But we can section simple fluids out into two other categories; fluids for which \boldsymbol\tau is strictly proportional to the strain-rate tensor, which are commonly called Newtonian fluids, and fluids for which it is not, which are called generalized Newtonian fluids. (For what it’s worth, I think the naming conventions for Newtonian/non-Newtonian/generalized Newtonian fluids is confusing and outright ridiculous, but there’s not much I can do about it.) Examples of generalized Newtonian fluids include blood, nail polish, paint, syrup, etcetra. For generalized Newtonian fluids, the behavior of the fluid is properly “liquid”, but the magnitude of the molecular forces induced by the flow of the fluid are complicated functions of the components of the gradients in the velocity. Models for these functions are more of an art than a science, based largely on experimental results; here are some examples.

For a Newtonian liquid, however, we now find that there is a linear relationship between the second-order strain-rate tensor \mathbf{E}, with nine independent components, and the second-order deviatoric stress tensor \boldsymbol\tau, also with nine independent components. This means that we can represent the linear relationship by a fourth-order tensor with 81 different components (yikes!) which I’ll call the viscosity tensor \mathbf{M}:

\displaystyle \boldsymbol\tau = \mathbf{M}:\mathbf{E}

For better or worse, there’s an extra assumption that nearly always applies to fluids of this type; the molecular interactions are independent of direction, and as a result the stress tensor is independent of the orientation of the velocity gradients. Fluids for which this holds true are called isotropic, and fluids for which it’s not are called anisotropic. I’ve never heard of an anisotropic simple Newtonian fluid, but maybe you’ll be the first to find one!

For a locally-interacting simple Newtonian isotropic fluid, doing some behind-the-scenes tensor calculus reveals that our updated conservation of momentum equation has to take the following form once we all these assumptions:

\displaystyle \frac{\partial\left(\rho\vec{v}\right)}{\partial  t}\ + \nabla \cdot \left(\rho \vec{v} \otimes \vec{v}\right) = \vec{f} -  \nabla p\ +\  \mu \nabla^2 \vec{v}\ +\ \left(\lambda + \mu\right)\nabla\left(\nabla\cdot\vec{v}\right)

This absolute mess of an equation is called the compressible Navier-Stokes equation, and is the equation nearly everyone uses when dealing with flow of compressible fluids. The two symbols \mu and \lambda are undetermined proportionality factors that are the only surviving components of that viscosity tensor \mathbf{M}, and are usually determined through experiment; the first is universally referred to as the dynamic viscosity or just viscosity, and the second is so inconsistently defined in the literature that the only name we can give this thing that rheologists wouldn’t argue over is the profoundly unhelpful moniker of second Lamé parameter.

If we further assume that the fluid is incompressible and that the viscosity is the same everywhere, we can polish the equation up a lot more and pull the density out of a lot of expressions, get rid of that last term that depends on \lambda, and wind up with:

\boxed{\displaystyle  \frac{\partial\left(\rho\vec{v}\right)}{\partial  t}\ + \nabla \cdot \left(\rho \vec{v} \otimes \vec{v}\right)  = \vec{f} -   \nabla p\ +\  \mu \nabla^2 \vec{v}}

This is the incompressible Navier-Stokes equation, also known as just the Navier-Stokes equation, and it is the most famous equation in fluid dynamics; understanding it and analyzing its components will be the sole subject of the next Part. Given how involved this sequence of assumptions is for an introduction to fluid mechanics, I’ve put a table below describing how each assumption we made brought us from the Cauchy momentum equation at the beginning of this Part to the Navier-Stokes equation above.

AssumptionMathematical ConsequenceWhen It’s Wrong
Locality\vec{r}_{mol} = \nabla \cdot \boldsymbol\sigmaLong-range molecular interactions
Simplicity \vec{r}_{mol} = -\nabla p + \nabla \cdot \boldsymbol\tau Material has solid-like properties, pressure defined differently
Objectivity\boldsymbol\tau =  \boldsymbol\tau[\mathbf{E}]Velocities/rotations are extremely fast
Linearity\boldsymbol\tau = \mathbf{M}:\mathbf{E}Molecular forces have a complicated relationship with velocity gradients
Isotropy\boldsymbol\tau = 2 \mu \mathbf{E} + \lambda \text{Tr}(\mathbf{E})\mathbf{I}I don’t know (molecules would likely have to be very asymmetric)
Incompressibility\boldsymbol\tau = \mu \nabla \vec{v}Density is very low, fluid speeds are comparable to molecular speed (see Part II)

Things to Think About

1. What physical effects do you think the viscosity and second Lamé parameter quantify? Can you describe flows where only one of the terms is nonzero?

2. Can you think of other ways to test if a fluid is simple or not? How about to test if a fluid is Newtonian or not?

3. If you’re brave enough to handle some tensors, can you find why the form of the deviatoric stress tensor goes from 2 \mu \mathbf{E} + \lambda Tr(\mathbf{E})\mathbf{I} to \mu \nabla \vec{v} when we take the incompressibility assumption? Remember the definition of the strain-rate tensor!

4. How would the momentum transport equation change if you considered non-local molecular forces? Can you come up with some kind of parameter to describe if the locality assumption is valid? (This is something I think about often.)

5. What would the Navier-Stokes equation look like if the fluid was a locally-interacting simple Newtonian isotropic fluid but with a non-uniform viscosity?

6. Let’s say that you could describe the forces per unit area in the definition of the stress as coming from an energy density gradient. How would you define the stress in that case, and what would be the associated directions?

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