The lubrication approximation gives the local flow rate in a thin viscous gap. Conservation of mass then determines how that flow rate must vary along a gap whose thickness changes in space or time.

For a two-dimensional incompressible film of thickness $h(x,t)$, let

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

be the volumetric flow rate per unit width.

If the lower wall moves tangentially with constant speed $U$ while the upper boundary has zero tangential speed, lubrication theory gives

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

Mass conservation in a thin slice

Consider a thin slice between $x$ and $x+dx$. If its local thickness changes, the difference between inflow and outflow must supply that change in fluid volume.

For unit width,

$$\boxed{\frac{\partial h}{\partial t}+\frac{\partial q}{\partial x}=0}.$$

This relation can also be obtained by integrating incompressible continuity across the gap and applying the kinematic boundary condition at the moving boundary.

Substituting the lubrication flow rate gives

$$\frac{\partial h}{\partial t} +\frac{U}{2}\frac{\partial h}{\partial x} -\frac{1}{12\mu} \frac{\partial}{\partial x} \left(h^3\frac{\partial p}{\partial x}\right)=0.$$

Therefore

$$\boxed{ \frac{\partial}{\partial x} \left(h^3\frac{\partial p}{\partial x}\right) =6\mu U\frac{\partial h}{\partial x} +12\mu\frac{\partial h}{\partial t} }.$$

This is a one-dimensional form of the Reynolds lubrication equation for an incompressible Newtonian fluid with constant viscosity.

What creates pressure in a lubricating film?

The equation shows two basic pressure-generating mechanisms.

Wedge action

If the gap varies along $x$,

$$\frac{\partial h}{\partial x}\ne0,$$

and a wall drags fluid through the converging or diverging geometry, the term

$$6\mu U\frac{\partial h}{\partial x}$$

can generate a pressure field. This is the basic mechanism of hydrodynamic journal and slider bearings.

Squeeze action

If the gap changes with time,

$$\frac{\partial h}{\partial t}\ne0,$$

then fluid must be expelled from or drawn into the gap. The term

$$12\mu\frac{\partial h}{\partial t}$$

produces squeeze-film pressure.

A parallel gap can therefore generate pressure even with no tangential wall motion if the two surfaces approach one another.

Steady fixed geometry

If the geometry is steady,

$$\frac{\partial h}{\partial t}=0,$$

then

$$\boxed{ \frac{d}{dx} \left(h^3\frac{dp}{dx}\right) =6\mu U\frac{dh}{dx}}.$$

For a constant gap $h=$ constant, the right side vanishes and

$$\frac{d}{dx}\left(h^3\frac{dp}{dx}\right)=0,$$

so $dp/dx$ is constant. This recovers ordinary plane Couette-Poiseuille flow.

Worked example: squeeze film between parallel plates

Two parallel plates form a fluid film of spatially uniform thickness $h(t)$ over

$$-\frac L2\le x\le\frac L2.$$

There is no tangential sliding,

$$U=0,$$

and the upper plate approaches the lower one, so

$$\dot h<0.$$

Because $h$ is uniform in $x$, Reynolds' equation becomes

$$h^3\frac{d^2p}{dx^2}=12\mu\dot h.$$

Assume symmetry about $x=0$:

$$\left.\frac{dp}{dx}\right|_{x=0}=0,$$

and let pressure at both edges equal the ambient gauge pressure:

$$p\left(\pm\frac L2\right)=0.$$

Integrating twice gives

$$\boxed{ p(x)=\frac{6\mu\dot h}{h^3} \left(x^2-\frac{L^2}{4}\right)}.$$

Because $\dot h<0$ and

$$x^2-\frac{L^2}{4}\le0$$

inside the film, the gauge pressure is positive.

The maximum pressure occurs at the center:

$$\boxed{ p_{\max}=p(0) =-\frac{3\mu\dot hL^2}{2h^3}}.$$

For example, take

$$\mu=0.20,\mathrm{Pa,s},$$

$$L=0.10,\mathrm m,$$

$$h=1.0,\mathrm{mm}=1.0\times10^{-3},\mathrm m,$$

and

$$\dot h=-1.0\times10^{-4},\mathrm{m/s}.$$

Then

$$p_{\max} =-\frac{3(0.20)(-1.0\times10^{-4})(0.10)^2} {2(1.0\times10^{-3})^3} =300,\mathrm{Pa}.$$

Thus

$$\boxed{p_{\max}=300,\mathrm{Pa}}.$$

The strong $h^{-3}$ dependence explains why very thin viscous films can resist rapid squeezing surprisingly strongly.

Scope and extensions

This form of Reynolds' equation assumes a thin film, incompressible Newtonian fluid, constant viscosity, weak inertia, and the usual lubrication scaling. More general forms can include

• two-dimensional variation in the film plane;
• both walls moving;
• compressible fluids;
• variable viscosity;
• porous boundaries;
• curved geometries.

The core structure remains the same: a locally Couette-Poiseuille velocity profile is integrated across the thin direction, and mass conservation determines the pressure field along the film.