1. Hydrodynamic theory of lubrication
Dr. Om Prakash Singh
Associate Professor, IIT (BHU), Varanasi

2. Hydrodynamic lubrication
Also called fluid-film, thick-film, or flooded lubrication
A thick film of lubricant is interposed between the surfaces of bodies in relative motion
There has to be pressure buildup in the film due to relative motion of the surfaces
Fluid friction is substituted for sliding friction
Coefficient of friction is decreased
Prevalent in journal and thrust bearings

3. Parallel surfaces Direction of motion of top plate
Velocity of top plate = u
Shear force F
Top layer of fluid moves with same velocity as the plate y
Velocity profile
(same throughout)
Velocity of bottom plate = 0
A is area of the plate
Lubricant
There is no pressure buildup in the fluid due to relative motion
It remains constant throughout influenced only by the load
As load increases the surfaces are pushed towards each other until they are likely to touch

4. Navier-Stokes Equation for simple flow cases
Exact solution of
Navier-Stokes Equation for simple flow cases

5. The Navier-Stokes Equations
Conservation of momentum
Navier-Stokes equations can be written in vector form as
The three scalar Navier-Stokes equations and the continuity equation constitute the four equations that can be used to find the four variables u,v,w, and p provided there are appropriate initial and boundary conditions.
The equations are nonlinear due to the acceleration terms, such as u∂u/∂x on the left-hand side; consequently, the solution to these equation may not be unique. For example, the flow between two rotating cylinders can be solved using the Navier-Stokes equations to be a relatively simple flow with circular streamlines; it could also be a flow with streamlines that are like a spring wound around the cylinders as a torus; and, there are even more complex flows that are also solutions to the Navier-Stokes equations, all satisfying the identical boundary conditions.

6. The Navier-Stokes Equations
Conservation of momentum
The Navier-Stokes equations can be solved with relative ease for some simple geometries.
But, the equations cannot be solved for a turbulent flow even for the simplest of examples; a turbulent flow is highly unsteady and three-dimensional and thus requires that the three velocity components be specified at all points in a region of interest at some initial time, say t =0.
Such information would be nearly impossible to obtain, even for the simplest geometry.
Consequently, the solutions of turbulent flows are left to the experimentalist and are not attempted by solving the equations.

7. N-S Equation for simple cases
Couette Flow between a Fixed and a Moving Plate
Using Navier-Stokes equation, derive the equation of velocity of the moving plate

8. Solution
Consider two-dimensional incompressible plane (/z = 0) viscous flow between parallel plates a distance 2h apart, as shown in Fig.. We assume that the plates are very wide and very long, so that the flow is essentially axial,u  0 but v = w = 0. The present case is Fig. a, where the upper plate moves at velocity V but there is no pressure gradient. Neglect gravity effects. We learn from the continuity equation that

9. Thus there is a single nonzero axial-velocity component which varies only across the channel. The flow is said to be fully developed (far downstream of the entrance). Substitute u = u(y) into the x-component of the Navier-Stokes momentum equation for two-dimensional (x, y) flow:
Most of the terms drop out, and the momentum equation simply reduces to
The two constants are found by applying the no-slip condition at the upper and lower plates:

10. Therefore the solution for this case (a), flow between plates with a moving upper wall, is
This is Couette flow due to a moving wall: a linear velocity profile with no-slip at each wall, as anticipated and sketched in Fig. a.

11. N-S Equation for simple cases
Flow due to Pressure Gradient between Two Fixed Plates
Determine velocity profile
Case (b) is sketched in Fig.b. Both plates are fixed (V= 0), but the pressure varies in the x direction. If v= w= 0, the continuity equation leads to the same conclusion as case (a), namely, that u= u(y) only. The x-momentum equation changes only because the pressure is variable:
Also, since v= w= 0 and gravity is neglected, the y- and z- momentum equations
lead to

12. Thus the pressure gradient is the total and only gradient:
Why did we add the fact that dp/dx is constant? Recall a useful conclusion from the theory of separation of variables: If two quantities are equal and one varies only with y and the other varies only with x, then they must both equal the same constant.
Otherwise they would not be independent of each other.
Why did we state that the constant is negative? Physically, the pressure must decrease in the flow direction in order to drive the flow against resisting wall shear stress.
Thus the velocity profile u(y) must have negative curvature everywhere, as anticipated and sketched in Fig. b.

13. The solution to above Eq.is accomplished by double integration:
The constants are found from the no-slip condition at each wall:
Thus the solution to case (b), flow in a channel due to pressure gradient, is
The flow forms a Poiseuille parabola of constant negative curvature. The maximum velocity occurs at the centerline y= 0:

14. Hydrodynamic lubrication
Lift force
Force normal to surface
Top surface
Top surface
Drag force
Drag force
Oil wedge
Oil wedge
Direction of movement
of oil wedge
Bottom surface
Bottom surface
Surfaces are inclined to each other thereby compressing the fluid as it flows.
This leads to a pressure buildup that tends to force the surfaces apart
Larger loads can be carried
Watch YouTube Video:

15. In journal bearings Shaft/journal Top surface Oil wedge Oil wedge
Bottom surface
Oil wedge forms between shaft/journal and bearing due to them not being concentric

16. In Journal bearing- process at startup
Shaft/journal
e = eccentricity
Bearing
Stationary journal
Instant of starting (tends to climb up the bearing)
While running (slips due to loss of traction and settles eccentric to bearing)
Because of the eccentricity, the wedge is maintained

17. The regimes of lubrication...
As the load increases on the contacting surfaces three distinct situations can be observed with respect to the mode of lubrication, which are called regimes of lubrication:
Fluid film lubrication
Elastohydrodynamic lubrication
Boundary lubrication

18. Hydrodynamic lubrication- characteristics
Fluid film at the point of minimum thickness decreases in thickness as the load increases
Pressure within the fluid mass increases as the film thickness decreases due to load
Pressure within the fluid mass is greatest at some point approaching minimum clearance and lowest at the point of maximum clearance (due to divergence)
Viscosity increases as pressure increases (more resistance to shear)

19. Bearing characteristic number
Since viscosity, velocity, and load determine the characteristics of a hydrodynamic condition, a bearing characteristic number was developed based on the effects of these on film thickness.
Increase in velocity increases min. film thickness
Increase in viscosity increases min. film thickness
Increase in load decreases min. film thickness
Therefore
Viscosity x velocity/unit load = a dimensionless number = C
C is known as the Bearing Characteristic Number
The value of C, to some extent, gives an indication of whether there will be hydrodynamic lubrication or not

20. The regimes of lubrication…
As the load increases on the contacting surfaces three distinct situations can be observed with respect to the mode of lubrication, which are called regimes of lubrication:
Fluid film lubrication
Hydrostatic lubrication: outside pressure is applied to maintain pressure and lubricating film
Hydrodynamic lubrication: motion of the contacting surfaces, and the exact design of the bearing is used to pump lubricant around the bearing to maintain the lubricating film. If design wears, lubricant film breaks down.

21. The regimes of lubrication
As the load increases on the contacting surfaces three distinct situations can be observed with respect to the mode of lubrication, which are called regimes of lubrication:
Elastohydrodynamic lubrication
Mostly in non-confirming surfaces or in higher load conditions
The body suffers elastic strains at the contact
Motion of the contacting bodies generates a flow induced pressure, which acts as the bearing force over the contact area
Viscosity changes (rises) due to high pressure; Contact between raised solid features, or asperities, can occur, leading to a mixed-lubrication or boundary lubrication regime.

22. The regimes of lubrication…
As the load increases on the contacting surfaces three distinct situations can be observed with respect to the mode of lubrication, which are called regimes of lubrication:
Boundary lubrication:
Bodies come into closer contact at their asperities; the heat developed by the local pressures causes a condition which is called stick-slip and some asperities break off.
At the elevated temperature and pressure conditions chemically reactive constituents of the lubricant react with the contact surface forming a highly resistant tenacious layer, or film on the moving solid surfaces (boundary film) which is capable of supporting the load and major wear or breakdown is avoided.

23. Hydrodynamic lubrication- characteristics
Film thickness at the point of minimum clearance increases with the use of more viscous fluids
With same load, the pressure increases as the viscosity of fluid increases
With a given load and fluid, the thickness of the film will increase as speed is increased
Fluid friction increases as the viscosity of the lubricant becomes greater

24. Hydrodynamic lubrication theory
A theoretical analysis of hydrodynamic lubrication was carried out by Osborne Reynolds.
The equations resulted from the analysis has served a basis for designing hydrodynamically lubricated bearings.
The following assumptions were made by Reynolds in the analysis:
The lubricating fluid is Newtonian - the flow is laminar and the shear stress between the flow layers is proportional to the velocity gradient in the direction perpendicular to the flow (Newton’s law of viscosity): 𝜏=𝜇 𝑑𝑢 𝑑𝑦 Where: η – dynamic viscosity of oil, u – linear velocity of the laminar layer, y - the axis perpendicular to the flow direction.
The inertia forces resulted from the accelerated movement of the flowing lubricant are neglected.
The lubricating fluid is incompressible.
The pressure of the fluid p is constant in the direction perpendicular to the laminar flow: dp/dy=0 (assumption of thin lubrication film).
The viscosity of the fluid is constant throughout the lubrication film.
Mass of the lubrication remain constant; no side leakages

25. h(radial clearance)
Journal
u = 0(stationary bearing U
Oil y
(radial) x
(tangential) z
(axial)
Assumptions
The inertia forces resulted from the accelerated movement of the flowing lubricant are neglected.
The lubricating fluid is incompressible.
The pressure of the fluid p is constant in the direction perpendicular to the laminar flow: dp/dy=0 (assumption of thin lubrication film).
The viscosity of the fluid is constant throughout the lubrication film.
Mass of lubrication remain constant

26. Conservation equations
Mass conservation (Continuity equation)
X-momentum conservation equation
Y-momentum conservation equation
Energy conservation equation
Vertical velocity v = 0 (velocity normal to the rotation)
v/y = 0  u/x = 0 (from continuity)  u = f(y) only
p/y = 0 (from Y-momentum as all other terms becomes zero as v = 0)
 p = f(x) only  p/x =dp/dx

27. The final balance of forces
(heat generation not considered
Hence,
𝑑𝑝 𝑑𝑥 =𝜇 𝜕2𝑢 𝜕𝑦2

28. The velocity profile The boundary conditions are:
At y = 0, u = U (bearing speed) and at x = 0, p = p0 (bearing entrance)
At y = h, u = 0 (no slip) and at x = l, p = p0 (bearing end)
Both the ends are assumed to be at same pressure and rightly so as fluid does not leak
Integrate twice and applying BCs, we get,
𝑢=𝑈 1− 𝑦 ℎ − ℎ2 2𝜇 𝑑𝑝 𝑑𝑥 1− 𝑦 ℎ 𝑦 ℎ
At the point of maximum pressure, dp/dx = 0, hence
𝑢=𝑈 1− 𝑦 ℎ
Note: the velocity at maximum pressure depicts that the velocity profile along y is linear at the location of maximum pressure.
The gap at this location may be denoted as h*.

29. Mass of lubrication The total flow of the lubricant is: 𝑄= 0 ℎ 𝑢𝑑𝑦
𝑢=𝑈 1− 𝑦 ℎ − ℎ2 2𝜇 𝑑𝑝 𝑑𝑥 1− 𝑦 ℎ 𝑦 ℎ
Using
𝑄= 1 2𝜇 𝑑𝑝 𝑑𝑥 ℎ3 3 − ℎ 𝑈 ℎ ℎ2− ℎ2 2
Or,
(This equation can be used to find pressure gradient or pressure difference between the bearing ends using pressure BCs. )
𝑄= 𝑈ℎ 2 − 1 12𝜇 𝑑𝑝 𝑑𝑥 ℎ3

30. Hydrodynamic lubrication equation
Reynolds equation for one-dimensional (1D) flow.
According to the assumption about incompressibility of the lubricant the flow Q does not change in x direction:
𝑑𝑄 𝑑𝑥 =0
Differentiating the lubrication mass conservation equation with respect x results in:
𝑑𝑄 𝑑𝑥 = 𝑈 2 𝑑ℎ 𝑑𝑥 − 𝑑 𝑑𝑥 ℎ3 12𝜇 𝑑𝑝 𝑑𝑥 =0
Or,
𝜕 𝜕𝑥 ℎ3 𝑑𝑝 𝑑𝑥 =6𝜇𝑈 𝑑ℎ 𝑑𝑥
Note the use of partial derivative.

31.  Hydrodynamic lubrication equation
Reynolds equation for two-dimensional (2D) flow.
If the flow in z direction is taken into account (bearings with side leakage of the lubricating fluid) then the analysis results in Reynolds equation for two dimensional flow:
𝜕 𝜕𝑥 ℎ3 𝑑𝑝 𝑑𝑥 + 𝜕 𝜕𝑧 ℎ3 𝑑𝑝 𝑑𝑧 =6𝜇𝑈 𝑑ℎ 𝑑𝑥
Where:
h – local oil film thickness,
 – dynamic viscosity of oil,
p – local oil film pressure,
U – linear velocity of journal,
x - circumferential direction.
z - longitudinal direction. 
Close form solution of Reynolds equation can not be obtained therefore finite elements method (FEM) or CFD is used to solve it.
Analytical solutions of Reynolds equation exist only for certain assumptions

32. Pressure distribution in thin lubricating film
Thin film of oil, confined between the interspace of moving parts, may acquire high pressures up to 100 MPa which is capable of supporting load and reducing friction.
The salient features of this type of motion can be understood from a study of slipper bearing.
The slipper moves with a constant velocity U past the bearing plate. This slipper face and the bearing plate are not parallel but slightly inclined at an angle of .
A typical bearing has a gap width of mm or less, and the convergence between the walls may be of the order of 1/5000.
It is assumed that the sliding surfaces are very large in transverse direction so that the problem can be considered two-dimensional.

33. In journal bearings  Top surface h1 Oil wedge h(x) h2  U
Bottom surface
Top surface  h2 h1
h(x) U 

34. Pressure distribution in the thin lubricating film
Mass flow is given by
𝑄= 𝑈ℎ ℎ − 1 12𝜇 𝑑𝑝 𝑑𝑥 ℎ3
Or,
𝑃𝑥=12𝜇 𝑈 2ℎ2 − 𝑄 ℎ3
Here Px is dp/dx
Integrate above equation with x, we have
𝑑𝑝 𝑑𝑥 𝑑𝑥=6𝜇𝑈 𝑑𝑥 ℎ1−𝛼𝑥 2 −12𝜇𝑄 𝑑𝑥 ℎ1−𝛼𝑥 3 +𝐶3
Or,
𝑝= 6𝜇𝑈 𝛼(ℎ1−𝛼𝑥) − 6𝜇𝑄 𝛼 ℎ1−𝛼𝑥 2 +𝐶3
Where  = (h1-h2)/l and C3 is a constant

35. Pressure distribution in the thin lubricating film
Since the pressure must be the same (p = p0), at the ends of the bearing, namely, p = p0 at x = 0 and p = p0 at x=l, the unknowns in the above equations can be determined by applying the pressure boundary conditions. We obtain,
𝑄= 𝑈ℎ1ℎ2 ℎ1+ℎ2
𝐶3=𝑝0− 6𝜇𝑈 𝛼(ℎ1+ℎ2)
and
With these values inserted, the equation for pressure distribution becomes
𝑝−𝑝0= 6𝜇𝑈𝑥(ℎ−ℎ2) ℎ2(ℎ1+ℎ2)
It may be seen from above Eq. that, if the gap is uniform, i.e. h = h1=h2, the gauge pressure will be zero.
Furthermore, it can be said that very high pressure can be developed by keeping the film thickness very small.
Figure shows the distribution of pressure throughout the bearing.

36. Load bearing capacity, shear stress and drag force
The total load bearing capacity per unit width is
𝑇= 0 𝑙 𝑝−𝑝0 𝑑𝑥
The shear stress at the bearing plate is:
𝜏0=−𝜇 𝜕𝑢 𝜕𝑦 |𝑦=0
The drag force required to move the lower surface at speed U is expressed by
𝐷= 0 𝑙 𝜏0𝑑𝑥

37. Application of Hydrodynamic equation in IC Engine
Thermo-elasto-hydrodynamic piston secondary motion model
Many physics simultaneously interact with each other in piston–cylinder assembly.
Extreme temperature and pressure on piston top, friction and hydrodynamic pressure on piston skirt and connecting rod and piston inertial forces makes the piston secondary motion a complex process.
Hence the profile of the thin oil film thickness (h) between the piston and liner is a function of many parameters.
Accordingly, h is expressed in the following form,
where h0 is the time varying clearance, hs is piston profile, hth is the thermo-mechanical deformation of piston–cylinder, hP is surface deformation due to pressure, 1 and 2 are the roughness amplitude of piston and cylinder wall respectively.

38. Application of Hydrodynamic equation in IC Engine
Thermo-elasto-hydrodynamic piston secondary motion model
The average Reynolds in the elasto-hydrodynamic lubrication (EHL) analysis is given by
where x and y are pressure flow factor, s is shear flow factor,  is composite r.m.s roughness, U is sliding speed, g is oil viscosity and p is hydrodynamic pressure.
In thermal EHL, the elastic deformation of the bounding skirt and cylinder surfaces due to hydrodynamic pressure and thermal effects are investigated.

39. Heat generation due to viscous dissipation
When dealing with extremely viscous flows of the type encountered in lubrication problems or the piping of crude oil, the energy equation is improved by taking into account the internal heating due to viscous dissipation,
No viscous dissipation
Modified form of energy equation
With viscous dissipation
In three dimensions, the viscous dissipation function  is expressed as follows:

40.  Heat generation due to viscous dissipation =0 due to continuity
For 2D problems, w velocity and velocity with z gradient drops out

41. 1. Problem on viscous dissipation
Consider the laminar boundary layer frictional heating of an adiabatic wall parallel to a free stream (U,T) as in figure. Modeling the flow as one with temperature-independent properties and assuming that the Blasius velocity solution holds i.e thin boundary layer assumption holds, use scaling arguments to show that the relevant boundary layer energy equation for this problem is

42. Solution

45. 2. Problem on viscous dissipation
Using scale analysis on above boundary layer equation with viscous dissipation, determine the wall temperature rise T as a function of U and fluid properties for two cases
convection ~ heat generation balance,
for what values of Pr these scaling is valid in (a) ? and
conduction ~ heat generation balance and
for what values Pr the scaling in (c) valid?
in which case: (a) or (b) thermal boundary layer would be thick?
Draw thermal and hydrodynamic boundary layers for both the cases.

53. End