When a viscous fluid occupies a gap whose thickness is much smaller than its length, the geometry creates a strong separation of scales. The lubrication approximation exploits that separation to simplify Navier-Stokes flow.

Let

• $H$ be a characteristic gap thickness;
• $L$ be the characteristic distance along the gap;
• $U$ be a characteristic tangential speed.

Define the small aspect ratio

$$\boxed{\varepsilon=\frac{H}{L}\ll1}.$$

The fluid is assumed Newtonian and incompressible.

Why transverse viscous gradients dominate

For a velocity component $u$ along the gap,

$$\frac{\partial^2u}{\partial y^2}\sim\frac{U}{H^2},$$

while

$$\frac{\partial^2u}{\partial x^2}\sim\frac{U}{L^2}.$$

Their ratio is

$$\frac{U/L^2}{U/H^2}=\left(\frac{H}{L}\right)^2=\varepsilon^2\ll1.$$

Thus viscous diffusion across the thin direction is much stronger than viscous diffusion along the gap.

When inertia is negligible

The axial inertial scale is approximately

$$\rho\frac{U^2}{L},$$

while transverse viscous stress contributes a momentum scale

$$\mu\frac{U}{H^2}.$$

Their ratio is

$$\frac{\rho U^2/L}{\mu U/H^2} =\frac{\rho UL}{\mu}\left(\frac{H}{L}\right)^2.$$

Defining

$$Re_L=\frac{\rho UL}{\mu},$$

inertia is small compared with transverse viscosity when

$$\boxed{Re_L\varepsilon^2\ll1}.$$

Lubrication flow can therefore be viscosity-dominated even when a Reynolds number based on the long length $L$ is not itself extremely small.

Leading-order equations

For a slowly varying gap $y=h(x,t)$ with $h/L\ll1$, the leading-order axial momentum balance becomes

$$\boxed{0=-\frac{\partial p}{\partial x} +\mu\frac{\partial^2u}{\partial y^2}}.$$

The transverse momentum balance implies, to leading order,

$$\boxed{\frac{\partial p}{\partial y}\approx0}.$$

So pressure is nearly uniform across the thin gap:

$$p\approx p(x,t).$$

The key idea is local fully developed flow. The gap can vary slowly with $x$ and time, but over each short axial slice the velocity profile is approximately the parallel-plate Couette-Poiseuille profile corresponding to the local gap thickness and local pressure gradient.

Local velocity profile

Take the lower wall at $y=0$ moving in the positive $x$ direction with speed $U$, while the upper wall at $y=h(x,t)$ has zero tangential speed. No slip gives

$$u(0)=U, \qquad u(h)=0.$$

Using the local Couette-Poiseuille solution,

$$\boxed{ u(x,y,t)=U\left(1-\frac{y}{h}\right) -\frac{1}{2\mu}\frac{\partial p}{\partial x},y(h-y)}.$$

The first term is wall-driven Couette flow. The second is pressure-driven Poiseuille flow.

Local flow rate

For a two-dimensional film, define the volumetric flow rate per unit width

$$q(x,t)=\int_0^{h(x,t)}u(x,y,t),dy.$$

Integrating the local profile gives

$$\boxed{q=rac{Uh}{2} -\frac{h^3}{12\mu}\frac{\partial p}{\partial x}}.$$

This compact relation is the central output of the lubrication approximation. It converts a two-dimensional velocity field into a one-dimensional relation among

• local gap thickness $h$;
• wall motion $U$;
• pressure gradient $\partial p/\partial x$;
• local flow rate $q$.

Worked local calculation

Suppose a film has

$$h=0.50,\mathrm{mm}=5.0\times10^{-4},\mathrm m,$$

$$U=1.0,\mathrm{m/s},$$

$$\mu=0.050,\mathrm{Pa,s},$$

and local pressure gradient

$$\frac{\partial p}{\partial x}=-2.0\times10^5,\mathrm{Pa/m}.$$

The wall-driven contribution is

$$q_C=\frac{Uh}{2} =\frac{(1.0)(5.0\times10^{-4})}{2} =2.50\times10^{-4},\mathrm{m^2/s}.$$

The pressure-driven contribution is

$$q_P=-\frac{h^3}{12\mu}\frac{\partial p}{\partial x}$$

$$=-\frac{(5.0\times10^{-4})^3}{12(0.050)}(-2.0\times10^5) \approx4.17\times10^{-5},\mathrm{m^2/s}.$$

Hence

$$\boxed{q\approx2.92\times10^{-4},\mathrm{m^2/s}}.$$

Both mechanisms drive flow in the positive $x$ direction in this example.

Scope of the approximation

Lubrication theory requires more than simply calling a gap "small." Its central assumptions are a slender geometry, slowly varying boundaries, dominant viscous transport across the gap, and sufficiently weak inertia. Abrupt geometry changes, strong separation, large slopes, turbulence, or inertia-dominated motion can invalidate the approximation.

When its assumptions hold, lubrication theory replaces a difficult multidimensional flow problem with a local Couette-Poiseuille profile plus a one-dimensional mass-conservation problem along the film.