10  Lubricated flow

10.1 Motivation

When two solid surfaces

  1. are in relative motion, and
  2. are being forced onto each other,

it is beneficial to introduce a lubricant into the gap between them to reduce friction. Such lubricant then experiences lubricated flow. Lubrication constitutes a very common process with high relevance for engineering, science and practical applications.

Examples:

  • a journal bearing, in which the interior journal is separated from the bearing’s bush via a lubricant,
  • a joint in our body, in which synovial fluid mediates forces and reduces friction between bones,
  • an air-floating slider in a magnetic disk memory device,
  • contact phase-change as seen in hot wire cutting or melting cryobots.

The content of this chapter uses Hori (2006) as a starting point.

Lubricated flow does not imply a specific constitutive relation per se. Here, however, we will assume to lubricant to by given as a Newtonian fluid.

  • The approach to derive a process model for lubricated flow follows ideas introduced in the section on shallow flow in that we exploit the characteristic horizontal scale \(L\) to be much larger than its characteristic cross-flow extent \(H\) and hence \(\epsilon:= H/L << 1.\)

  • In contrast to shallow flow, however, our flow has no free surface but is constrained by solid bounding surfaces. These surfaces mediate the load into the fluid in the gap.

Figure 10.1: Shallow flow versus lubricated flow.

10.2 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 figure Figure 10.2.

Figure 10.2: Tower’s experiment.

Tower observed that

  • Frictional resistance almost constant regardless the bearing load \(P\)
  • Frictional resistance increases with sliding speed
  • Frictional resistance decreases with temperature
  • The friction coefficient is very small, around 1/1000

Investigating these qualitative observations requires us to derive a process model for flow within the gap area that takes into consideration the load exerted to the fluid via its solid bounding surfaces.

This means that our model has to provide information on the pressure \(p\), velocity \(\mathbf u\) and potentially temperature \(T\) within the lubricated flow in the gap, while additionally it has to consider a force balance between hydrodynamic pressure force \(F\) developed in the gap and the external load \(P\) that the system is exposed to:

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

10.3 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.

Figure 10.3: Problem setting

The problem setting is sketched in figure Figure 10.3. We consider a 2d Eulerian coordinate system \((x,y)\) and a fluid that is constrained by two surfaces denoted as \(\Gamma_1\) and \(\Gamma_2\). Both surfaces are boundaries of a rigid body \(\Omega_1\) and \(\Omega_2\). The gap between both surfaces has width \(h\).

The bodies \(\Omega_1\) and \(\Omega_2\) are moving, which constitutes a motion of the respective surfaces given by \(\mathbf U_1 = (U_i,V_i)^T\) for \(i \in \{1,2\}\). Accordingly, the gap width \(h(t,x)\) is a function of spatial coordinate \(x\) and time \(t\).

We are interested in the pressure field \(p(t,x)\) and velocity field \(\mathbf u(t,x) = (u(t,x),v(t,x))^T\) within the gap. Note that the complete section assumes a 2d situation. An generalization in three space dimensions is straight forward.

In his original work, Reynolds formulated the following assumptions:

  1. The gap fluid is a Newtonian fluid of constant viscosity \(\mu\)

  2. Compressibility of the fluid can be neglected, hence \(\rho \approx\) const

  3. No-slip conditions at the surfaces, hence \(\mathbf u(\Gamma_i) = \mathbf U_i\)

  4. The gap fluid is in stationary laminar flow

  5. Gravity and inertia forces can be ignored compared to viscous forces

  6. The variation of \(u\) in \(x\) (in-flow direction) can be neglected in comparison to its change in \(y\) (cross-flow direction)

  7. Fluid pressure does not change across the gap width, hence \(\partial_y p \approx 0\)

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.

In order to derive Reynold’s equations, we start from the fundamental mass and momentum balance for an incompressible Newtonian fluid in the gap:

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

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 in flow direction along the gap, hence in \(x\) direction, is determined by cross-gap \(y\)-direction momentum diffusion.

Now, we integrate Equation 10.2 twice and use as boundary conditions the velocities prescibed at surfaces \(\Gamma_1\) and \(\Gamma_2\):

\[ \mathbf u (\Gamma_i) = \mathbf U_i \; \text{for} \; i \in \{1,2\} \quad \text{hence} \; u (\Gamma_i) = U_i \; \text{for} \; i \in \{1,2\} \]

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

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

The velocity field hence constitutes a superposition of a pressure induced Poiseuille part (I), and a shear flow Couette part (II), see also exercises.

Integration of the horizontal velocity across the gap in \(y\) direction 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 would be able to determine the velocity (and its integral)!

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

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

in which we exploited boundary conditions

\[ v (\Gamma_i) = V_i \; \text{for} \; i \in \{1,2\} \]

Leibniz’ integral rule is used to interchange differentiation and integration. Combination with the previously derived integrated horizontal velocity results in the famous Reynold’s equation:

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

LHS denotes the pressure distribution along the gap. RHS has the following interpretation

  1. wedge effect: pressure generation due to the fluid being driven from large \(h\) to small \(h\)
  2. stretch effect: accounts for changes in the fluid’s boundary velocities
  3. squeeze effect accounts for pressure generation due to a change in gap width.

The Reynold’s equation constitutes a process model that allows to determine the pressure field along lubricated flow when the motion of its bounding surfaces is known. The velocity can be evaluated in a post-processing step.

10.4 Lubrication in a journal bearing

We will now apply Reynold’s equation to a concrete and historically important configuration: the journal bearing that already motivated Tower’s experiment in (lubexperiment?). A journal bearing supports a rotating shaft, the journal, inside a slightly larger cylindrical bush, the bearing. The narrow gap between both is filled with a lubricant. As the journal rotates, it drags fluid into the converging part of the gap and thereby builds up the hydrodynamic pressure that carries the external load.

10.4.1 Problem setting

The geometry of a journal bearing is sketched in figure Figure 10.4. It is characterized by

  • the geometry, given by the radius of the bearing \(R_b\) and the radius of the journal \(R_j\),
  • the operating conditions, given by the number of revolutions \(N\) of the journal and the load \(P\) acting on it,
  • the material properties of the lubricant, here represented by its viscosity \(\mu\).
Figure 10.4: Schematic of a journal bearing.

For given geometry, operating conditions and material properties, we aim to determine

  • the thickness of the lubricant film \(h(\phi)\), and
  • the pressure distribution \(p(\phi)\)

within the gap, both expressed as functions of the polar angle \(\phi\) that parametrizes the circumference of the bearing.

10.4.2 Geometry and notation

If journal and bearing were perfectly concentric, the gap width would be constant. By Reynold’s equation, no pressure could then build up to support the load. A load-carrying film therefore requires the journal centre to be displaced with respect to the bearing centre. To describe this configuration, we introduce, see figure Figure 10.5,

  • the radial clearance \(c := R_b - R_j\), i.e. the gap width in the concentric configuration,
  • the eccentricity \(e\), i.e. the distance between the centres of bearing and journal,
  • the relative eccentricity \(k := e/c \in [0,1)\),
  • the polar coordinate \(\phi\) measured along the circumference, and
  • the offset angle \(\theta\) by which the coordinate system is rotated so as to comply with the location of minimum gap width.
Figure 10.5: Extended schematic of a journal bearing accounting for the eccentricity.

For a small clearance, \(c \ll R_b\), the gap width follows from elementary geometry as

\[ h(\phi) \approx c + e \cos \phi = c \left( 1 + k \cos \phi \right). \tag{10.3}\]

The film thickness hence varies between a maximum \(c(1+k)\) and a minimum \(c(1-k)\), which constitutes the converging-diverging wedge that is required for pressure generation.

10.4.3 Reynold’s equation for the bearing

In steady operation, only the journal surface moves, with tangential surface speed \(U\), while the bush is at rest. Unwrapping the circumference via the arclength coordinate \(x\), the general Reynold’s equation reduces to its wedge contribution only,

\[ \partial_x \left( h^3 \partial_x p \right) = 6 \mu U \, \partial_x h, \tag{10.4}\]

since the bearing exhibits neither a stretch effect, as \(U\) is constant, nor a squeeze effect, as the gap is stationary. We complement Equation 10.4 with Sommerfeld boundary conditions

\[ p(0) = p(2\pi) = 0, \]

which express that the pressure is single-valued and periodic around the bearing.

10.4.4 Pressure distribution

In the limit of a small clearance, the two radii nearly coincide and we introduce a representative radius \(R := R_b \approx R_j\). Integrating Equation 10.4 with the film ansatz Equation 10.3 subject to the Sommerfeld boundary conditions yields the dimensionless pressure

\[ \bar p(k,\phi) = \frac{k \, ( 2 + k \cos \phi) \sin \phi}{(2+k^2)(1+k \cos \phi)^2}, \tag{10.5}\]

from which the physical pressure distribution, the so-called Sommerfeld pressure, follows as

\[ p(k,\phi) = \frac{6 \mu U R}{c^2} \, \bar p(k,\phi). \tag{10.6}\]

Note that \(\bar p\) is positive in the converging half of the gap, where \(\phi \in (0,\pi)\), and negative in the diverging half. The latter foreshadows a limitation of the model that will be discussed below.

10.4.5 Force balance and eccentricity

The hydrodynamic pressure in the film generates a resultant force \(F\) that has to balance the external load \(P\) acting on the journal, see figure Figure 10.6. Decomposing the balance into the component along the line of centres and the one perpendicular to it reads

\[ \begin{aligned} \underbrace{L R \int_0^{2 \pi} p(k,\phi) \cos \phi \, d \phi}_{=F_1} + P \cos \theta &= 0, \\ \underbrace{L R \int_0^{2 \pi} p(k,\phi) \sin \phi \, d \phi}_{=-F_2} - P \sin \theta &= 0, \end{aligned} \tag{10.7}\]

in which \(L\) denotes the axial length of the bearing.

Figure 10.6: Force balance in a journal bearing.

By the symmetry of the dimensionless pressure, the first integral vanishes, hence \(F_1 = 0\). This implies \(P \cos \theta = 0\) and therefore \(\theta = \pi/2\), so that the minimum gap width is located at \(90^\circ\) with respect to the load direction. Evaluating the remaining integral provides a relation between the load and the relative eccentricity,

\[ P = \mu U L \left( \frac{R}{c} \right)^2 \frac{12 \pi k}{(2 + k^2)(1 - k^2)^{1/2}}. \tag{10.8}\]

10.4.6 The Sommerfeld number

The result can be cast in dimensionless form. To this end, we introduce the mean bearing pressure \(p_m := P / (2 R L)\) and the number of revolutions \(N := U / (2 \pi R)\), and define the Sommerfeld number

\[ S := \frac{\mu N}{p_m} \left( \frac{R}{c} \right)^2. \tag{10.9}\]

With these, the load relation Equation 10.8 takes the compact form

\[ \frac{1}{S} = \frac{12 \pi^2 k}{(2 + k^2)(1 - k^2)^{1/2}}. \tag{10.10}\]

The Sommerfeld number collects geometry via \(R/c\), operating conditions via \(\mu\) and \(N\), and the load via \(p_m\) into a single dimensionless group. Two journal bearings that share the same Sommerfeld number \(S\) therefore operate under similar conditions; in particular they exhibit the same relative eccentricity \(k\).

10.4.7 Workflow and limitations

In summary, the Sommerfeld number reflects geometry, operating conditions and material properties of the bearing. Once it is known, the model allows us to compute, in turn,

  • the relative eccentricity \(k\), and thereby the eccentricity \(e\), from the Sommerfeld relation Equation 10.10,
  • the lubricant film thickness \(h(\phi)\) from the film ansatz Equation 10.3, and
  • the pressure distribution \(p(\phi)\) from Reynold’s equation in form of Equation 10.6.

The model is subject to limitations. The negative pressures predicted in the diverging part of the gap can in practice not be sustained by the lubricant and lead to cavitation. Furthermore, the model neither accounts for thermal effects nor for a mechanical coupling between the lubricant film and the deformation of the bounding solids.

10.5 List of Symbols

Symbol Type Description SI Unit
\(H,L\) scalar characteristic height and length scake \([\mathrm{m}]\)
\(P\) vector load / force \([\mathrm{kg m} \ \mathrm{s}^{-2}]\)
\(\Omega_1, \Omega_2\) domain rigid bodies constraining the flow \([-]\)
\(\Gamma_1, \Gamma_2\) curve surfaces constraining the flow \([-]\)
\(p\) scalar pressure field \([\mathrm{N} / \mathrm{m}^2]\)
\(\mathbf{u}\) vector velocity field \([\mathrm{m} / \mathrm{s}]\)
\(\mathbf{U}_1, \mathbf{U}_2\) vector surface velocities \([\mathrm{m} / \mathrm{s}]\)