10  Lubricated flow

Whenever two solids surfaces should move relative to each other while at the same time being forced onto each other, it is beneficial to introduce a so-called lubricant in the gap to reduce friction. Such lubricant, often a fluid, then experiences lubricated flow. Lubrication constitutes a very common process with high relevance for many engineering, science and practical applications. Examples are

Similar to shallow flow models as discussed in Chapter 9, we are facing a situation, in which the characteristic horizontal extent \(L\) of the continuum at stake is much larger than its characteristic cross-flow extent \(H\) that typically constitutes the film thickness. As before, \(H/L = \epsilon\) is very small: \(\epsilon <<1\). In contrast to shallow flow, however, our flow has no free surface but is constrained by solid bounding surfaces. These mediate a load to the fluid in the gap.

10.1 Tower’s experiment

A series of experiments was conducted by Tower in 1883 that paved the way for a better understanding of bearing systems. He constructed a half-open journal bearing, in which the frictional resistance of the rotating interior journal could be investigated for a varying vertical loading, rotation speed and temperature regimes of the lubricant, see also Hori (2006). Tower observed that

  • the frictional resistance in the bearing is very small as soon as lubricants are introduced ( 1/1000)
  • the frictional resistance increases with sliding speed (number of revolutions per time unit)
  • the frictional resistance insignificantly changes as a result of the changing the bearing’s load
  • frictional resistance decreases with an increasing temerpature of the (oil-type) lubricant

Investigating these qualitative observations requires us to both derive a process model for the flow within the gap area, and consider the loading on the system. This means that our model has to provide information on the pressure \(p\), velocity \(\mathbf u\) and potentially temperature \(T\) within the gap, while additionally consider a force balance between pressure force developed in the gap and the external load \(F\) that the system is exposed to:

\[F = \int_{\Gamma} p d \sigma\]

10.2 Reynolds equations

We will first introduce the essential mathematical model for flow processes in the gap. The model had been first derived by Reynolds, who was inspired by Tower’s experiments, see also Reynolds (1886). It is therefore dubbed Reynold’s equation.

In what follows, all terminology relates to the sketch as provided in the lecture. We consider a Eulerian coordinate system \((x,y,z)\). We furthermore consider a fluid that is constrained by two surfaces denoted as \(\Gamma_1\) and \(\Gamma_2\). We assume a situation, in which the gap between both surfaces is (well-)defined as the gap width \(h\) in \(y\) direction. Both surfaces are boundaries of a rigid body \(\Omega_1\) and \(\Omega_2\).

The bodies are moving, which constitutes a motion of the respective surfaces given by \((U_i,V_i,W_i)^T\). Accordingly, the gap width \(h(t,x,z)\) is a function of spatial coordinates \((x,z)\) and time \(t\).

We are interested in the pressure field \(p(t,x,z)\) and velocity field \(\mathbf u(t,x,z) = (u(t,x,z),v(t,x,z),w(t,x,z))^T\) within the gap. In the following, we will restrict ourselves to a quasi 2d situation, in which \(W_i\) can be neglected effectively leading to \(p(t,x,z) = p(t,x)\) and \(\mathbf u(t,x,z) = \mathbf u(t,x)\)

In is original work, Reynolds formulated the following assumptions:

  1. The gap fluid is a Newtonian fluid of constant viscosity
  2. Compressibility of the fluid can be neglected
  3. No-slip conditions at the surfaces
  4. The gap fluid is in stationary laminar flow
  5. Gravity and inertia forces can be ignored compared to viscous forces
  6. The rate of change of \(u\) in \(x\) can be neglected in comparison to its change in \(y\)
  7. Fluid pressure does not change across the gap width

Note, that some of these assumptions would follow from the choice of the lubricant and the considered operating regime. Others can be justified via a scaling analysis as introduced earlier in this lecture. Assumptions 4, 5 and 6 jointly result in assuming Stokes flow in the lubricated gap, whereas condition 7 results from a small \(\epsilon\) and reminds of the shallowness assumption. Both have been discussed earlier in the lecture and will not be repeated here. We will simply assume these assumptions to be valid.

We now start (once again) from fundamental mass and momentum balance for an incompressible Newtonian fluid in the gap:

\[ \begin{align*} \nabla \cdot \mathbf v & = 0 \\ \partial_t \mathbf v + \mathbf v \cdot \nabla \mathbf v &= - \frac{1}{\rho} \nabla p + \nu \triangle \mathbf v + \mathbf g. \end{align*} \tag{10.1}\]

Under Reynold’s assumptions, the \(x-\) component of the momentum balance reduces significantly into

\[ \partial_x p = \mu \partial_y^2 u. \tag{10.2}\]

This states that the pressure distribution along the gap (in \(x\) direction) is determined by cross-gap (\(y\)) momentum diffusion. Reynolds equation now results from a series of analytical complexity reduction steps as follows:

First, we integrate Equation 10.2 twice and use as boundary conditions that velocities of both surfaces \(\Gamma_1\) and \(\Gamma_2\) are prescribed, namely

\[u=U_1 \quad \text{at} \quad y=0\]

and

\[u=U_2 \quad \text{at} \quad y=h.\]

This results in an explicit expression for the horizontal velocity component \(u\) as a function of the (yet unknown) pressure field:

\[ u = \underbrace{-\frac{1}{2 \mu} \partial_x p \, y (h-y)}_{\text{(I)}} + \underbrace{\left( (1-\frac{y}{h}) U_1 + \frac{y}{h} U_2 \right)}_{\text{(II)}}. \]

Remark

It is very interesting to observe that the velocity field is given by a superposition of a Poiseuille part (I) due to the pressure gradient \(\partial_x p\) (pressure induced flow), and a Couette part (II) due to the relative motion of the rigid surfaces (shear flow).

Integration of the horizontal velocity across the gap yields:

\[ \int_0^h u \, dy = -\frac{h^3}{12 \mu} \partial_x p + \frac{h}{2} (U_1 + U_2) \]

Note, that so far we still don’t know the pressure. If we knew the pressure, we could determine the velocity (and its integral) and vice versa!

In order to close the system, we integrate mass continuity \(\partial_x u + \partial_y v\) across the gap:

\[ 0 = \int_0^h \partial_x u \, dy + \int_0^h \partial_y v \, dy = \int_0^h \partial_x u \, dy + (V_2 - V_1), \]

in which we exploited boundary conditions

\[v=V_1 \quad \text{at} \quad y=0\]

and

\[v=V_2 \quad \text{at} \quad y=h.\]

Still the first integral needs to be evaluated. Similar to the shallow flow section, we utilize Leibniz’ integral rule to interchange differentian and integration. This time, however, we need to consider boundary velocities. Combination with the previously derived integrated horizontal velocity results in the famous Reynolds equation:

\[ \partial_x \left( h^3 \partial_x p \right) = 6 \mu \left( \underbrace{(U_1 - U_2) \partial_x h}_{\text{(I)}} + \underbrace{h \partial_x (U_1 + U_2) }_{\text{(II)}} + \underbrace{2 (V_2 - V_1)}_{\text{(III)}} \right) \]

Left hand side denotes the average curvature of the pressure distribution along the gap. The terms on the right hand side have th following interpretation

  1. stands for the pressure generation due to the fluid being driven from thick end of a wedge (large \(h\)) to a thin end (small \(h\)) and is referred to as the wedge effect.
  2. accounts for changes in the fluid’s boundary velocity along the gap and is referred to as the stretch effect.
  3. accounts for a pressure generation due to a change in gap width. It is called the squeeze effect.
Remark

The Reynold’s equation constitutes a process model that allows to determine the pressure field along a lubricant filled gap based on our knowledge on the motion of its bounding surfaces. Once the pressure field has been determined, the velocity can be evaluated in a post-processing step.