Categories: Fluid dynamics, Fluid mechanics, Physics.

Viscosity

The viscosity of a fluid describes how “sticky” its constituent molecules are; when one part of the fluid moves, it “drags” neighbouring parts by an amount proportional to the viscosity.

Imagine a liquid in a canal, flowing in the xx-direction at a velocity v(z)v(z) as a function of depth zz. Due to the liquid’s viscosity, its molecules are “stuck” to the bottom of the canal z=0z = 0, such that it is stationary there v(0)=0v(0) = 0. However, at the surface z=zsz = z_s, there is a flow at v(zs)=vsv(z_s) = v_s.

This difference in vv means that there is a velocity gradient across zz. Each infinitesimal layer of the liquid is dragging on the layers above and below it, meaning there is a nonzero shear stress σxz\sigma_{xz} (see Cauchy stress tensor). Formally, the dynamic viscosity η\eta is defined as follows:

σxz=ηdvdz\begin{aligned} \boxed{ \sigma_{xz} = \eta \dv{v}{z} } \end{aligned}

This is Newton’s law of viscosity, and fluids obeying it are known as Newtonian. In a Newtonian fluid at rest, there are no such shear stresses, and the Cauchy stress tensor σ^\hat{\sigma} is diagonal:

σij=−pδij\begin{aligned} \sigma_{ij} = - p \delta_{ij} \end{aligned}

Where pp is the pressure, and δij\delta_{ij} is the Kronecker delta. If the fluid flows according to a velocity field v⃗\va{v}, then a more general definition of η\eta is as follows, in index notation with ∇i ⁣= ⁣∂/∂xi\nabla_i \!=\! \ipdv{}{x_i}:

σij=−pδij+η(∇ivj+∇jvi)\begin{aligned} \boxed{ \sigma_{ij} = - p \delta_{ij} + \eta (\nabla_i v_j + \nabla_j v_i) } \end{aligned}

The double term ∇ivj+∇jvi\nabla_i v_j + \nabla_j v_i comes from the fact that the stress tensor of a Newtonian fluid is always symmetric; this definition of σij\sigma_{ij} enforces that.

Another quantity is the kinematic viscosity ν\nu, which is simply η\eta divided by the density ρ\rho:

ν≡ηρ\begin{aligned} \boxed{ \nu \equiv \frac{\eta}{\rho} } \end{aligned}

With this, Newton’s law of viscosity is written using the momentum density P=ρvP = \rho v:

σxz=νdPdz\begin{aligned} \sigma_{xz} = \nu \dv{P}{z} \end{aligned}

Because momentum is “more fundamental” than velocity, is ν\nu often more useful than η\eta. However, this comes at the cost of our intuition: for example, as you would expect, ηwater>ηair\eta_\mathrm{water} > \eta_\mathrm{air}, but you may be surprised that νwater<νair\nu_\mathrm{water} < \nu_\mathrm{air}. Since air is less dense, it is easier to set in motion, hence we expect it to be less viscous than water, but in fact air’s molecules are stickier than water’s.

References

  1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.