Categories: Fluid dynamics, Fluid mechanics, Physics.

Lubrication theory

Lubricants are widely used to reduce friction between two moving surfaces. In fluid mechanics, lubrication theory is the study of fluids that are tightly constrained in one dimension, especially those in small gaps between moving surfaces.

For simplicity, we limit ourselves to 2D by assuming that everything is constant along the zz-axis. Consider a gap of width dd (along yy) and length LL (along xx), where dLd \ll L, containing the fluid. Outside the gap, the lubricant has a Reynolds number ReUL/ν\mathrm{Re} \approx U L / \nu.

Inside the gap, the Reynolds number Regap\mathrm{Re}_\mathrm{gap} is different. This is because advection will dominate along the xx-axis (gap length), and viscosity along the yy-axis (gap width). Therefore:

Regap(v)vν2vU2/LνU/d2d2L2Re\begin{aligned} \mathrm{Re}_\mathrm{gap} \approx \frac{|(\va{v} \cdot \nabla) \va{v}|}{|\nu \nabla^2 \va{v}|} \approx \frac{U^2 / L}{\nu U / d^2} \approx \frac{d^2}{L^2} \mathrm{Re} \end{aligned}

If dd is small enough compared to LL, then Regap1\mathrm{Re}_\mathrm{gap} \ll 1. More formally, we need dL/Red \ll L / \sqrt{\mathrm{Re}}, so we are inside the boundary layer, in the realm of the Prandtl equations.

Let Regap1\mathrm{Re}_\mathrm{gap} \ll 1. We are thus dealing with Stokes flow, in which case the Navier-Stokes equations can be reduced to the following Stokes equations:

px=η(2vxx2+2vxy2)py=η(2vyx2+2vyy2)\begin{aligned} \pdv{p}{x} = \eta \: \Big( \pdvn{2}{v_x}{x} + \pdvn{2}{v_x}{y} \Big) \qquad \quad \pdv{p}{y} = \eta \: \Big( \pdvn{2}{v_y}{x} + \pdvn{2}{v_y}{y} \Big) \end{aligned}

Let the y=0y = 0 plane be an infinite flat surface, sliding in the positive xx-direction at a constant velocity UU. On the other side of the gap, an arbitrary surface is described by h(x)h(x).

Since the gap is so narrow, and the surfaces’ movements cause large shear stresses inside, vyv_y is negligible compared to vxv_x. Furthermore, because the gap is so long, we assume that vx/x\ipdv{v_x}{x} is negligible compared to vx/y\ipdv{v_x}{y}. This reduces the Stokes equations to:

px=η2vxy2py=0\begin{aligned} \pdv{p}{x} = \eta \pdvn{2}{v_x}{y} \qquad \quad \pdv{p}{y} = 0 \end{aligned}

This result could also be derived from the Prandtl equations. In any case, it tells us that pp only depends on xx, allowing us to integrate the former equation:

vx=p2ηy2+C1y+C2\begin{aligned} v_x = \frac{p'}{2 \eta} y^2 + C_1 y + C_2 \end{aligned}

Where C1C_1 and C2C_2 are integration constants. At y=0y = 0, the viscous no-slip condition demands that vx=Uv_x = U, so C2=UC_2 = U. Likewise, at y=h(x)y = h(x), we need vx=0v_x = 0, leading us to:

vx=p2ηy2(p2ηh+Uh)y+U\begin{aligned} v_x = \frac{p'}{2 \eta} y^2 - \Big( \frac{p'}{2 \eta} h + \frac{U}{h} \Big) y + U \end{aligned}

The moving bottom surface drags fluid in the xx-direction at a volumetric rate QQ, given by:

Q=0h(x)vx(x,y)dy=[p6ηy3p4ηhy2U2hy2+Uy]0h=p12ηh3+U2h\begin{aligned} Q = \int_0^{h(x)} v_x(x, y) \dd{y} = \bigg[ \frac{p'}{6 \eta} y^3 - \frac{p'}{4 \eta} h y^2 - \frac{U}{2 h} y^2 + U y \bigg]_0^{h} = - \frac{p'}{12 \eta} h^3 + \frac{U}{2} h \end{aligned}

Assuming that the lubricant is incompressible, meaning that the same volume of fluid must be leaving a point as is entering it. In other words, QQ is independent of xx, which allows us to write p(x)p'(x) in terms of measurable constants and the known function h(x)h(x):

p=6η(Uh22Qh3)\begin{aligned} \boxed{ p' = 6 \eta \: \Big( \frac{U}{h^2} - \frac{2 Q}{h^3} \Big) } \end{aligned}

Then we insert this into our earlier expression for vxv_x, yielding:

vx=3y(yh)(Uh22Qh3)Uhh2y+Uh2h2\begin{aligned} v_x &= 3 y (y - h) \Big( \frac{U}{h^2} - \frac{2 Q}{h^3} \Big) - \frac{U h}{h^2} y + \frac{U h^2}{h^2} \end{aligned}

Which, after some rearranging, can be written in the following form:

vx=U(3yh)(yh)h2Q6y(yh)h3\begin{aligned} \boxed{ v_x = U \frac{(3 y - h) (y - h)}{h^2} - Q \frac{6 y (y - h)}{h^3} } \end{aligned}

With this, we can find vyv_y by exploiting incompressibility, i.e. the continuity equation states:

vyy=vxx=2hUh3Qh4(2hy3y2)\begin{aligned} \pdv{v_y}{y} = - \pdv{v_x}{x} = - 2 h' \frac{U h - 3 Q}{h^4} \big( 2 h y - 3 y^2 \big) \end{aligned}

Integrating with respect to yy thus leads to the following transverse velocity vyv_y:

vy=2hUh3Qh4y2(hy)\begin{aligned} \boxed{ v_y = - 2 h' \frac{U h - 3 Q}{h^4} y^2 (h - y) } \end{aligned}

Typically, the lubricant is not in a preexisting pressure differential, i.e it is not getting pumped through the system. Although the pressure gradient pp' need not be zero, we therefore expect that its integral vanishes:

0=Lp(x)dx=6ηUL1h(x)2dx12ηQL1h(x)3dx\begin{aligned} 0 = \int_L p'(x) \dd{x} = 6 \eta U \int_L \frac{1}{h(x)^2} \dd{x} - 12 \eta Q \int_L \frac{1}{h(x)^3} \dd{x} \end{aligned}

Isolating this for QQ, and defining qq as below, yields a simple equation:

Q=12UqqLh2dxLh3dx\begin{aligned} Q = \frac{1}{2} U q \qquad \quad q \equiv \frac{\int_L h^{-2} \dd{x}}{\int_L h^{-3} \dd{x}} \end{aligned}

We substitute this into vxv_x and rearrange to get an interesting expression:

vx=U3y2hy3hy+h2h2Uq3y23hyh3=U(1yh)(13y(hq)h2)\begin{aligned} v_x &= U \frac{3 y^2 - h y - 3 h y + h^2}{h^2} - U q \frac{3 y^2 - 3 h y}{h^3} \\ &= U \Big( 1 - \frac{y}{h} \Big) \Big( 1 - \frac{3 y (h - q)}{h^2} \Big) \end{aligned}

The first factor is always positive, but the second can be negative, if for some yy-values:

h2<3y(hq)    y>h23(hq)\begin{aligned} h^2 < 3 y (h - q) \quad \implies \quad y > \frac{h^2}{3 (h - q)} \end{aligned}

Since h>yh > y, such yy-values will only exist if hh is larger than some threshold:

3(hq)>h    h>32q\begin{aligned} 3 (h - q) > h \quad \implies \quad h > \frac{3}{2} q \end{aligned}

If this condition is satisfied, there will be some flow reversal: rather than just getting dragged by the shearing motion, the lubricant instead “rolls” inside the gap. This is confirmed by vyv_y:

vy=Uh2h3qh4y2(hy)\begin{aligned} v_y = - U h' \frac{2 h - 3 q}{h^4} y^2 (h - y) \end{aligned}


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