1. Energy Reduction Through Tribology-2
Red Sea University Faculty of Engineering Department of Mechanical Engineering
Energy Reduction Through
Tribology-2
Moataz Abdelgadir Ali Abdelgadir

2. Hydrodynamic Pad
The bottom surface, sometimes called the ‘runner’, is covered with lubricant and moves with a certain velocity.
The top surface is inclined at a certain angle to the bottom surface.
As the bottom surface moves it drags the lubricant along it into the converging wedge.
A pressure field is generated as otherwise there would be more lubricant entering the wedge than leaving it.

3. Reynolds Equation
The entire process of hydro-dynamic pressure generation can be described mathematically to enable accurate prediction of bearing characteristics

4. Reynolds Equation Body forces are neglected
Simplifying Assumptions
Body forces are neglected
Pressure is constant through the film
No slip at the boundaries
Lubricant behaves as a Newtonian fluid
Flow is laminar
Fluid inertia is neglected
Fluid density is constant
Viscosity is constant throughout the generated fluid film

5. Reynolds Equation Equilibrium of an Element
Consider a small element of fluid from a hydrodynamic film shown (in x-direction)
Assuming that dxdydz ≠ 0 (i.e. non zero volume),

6. Reynolds Equation (1) (2) (3) (4) Forces balance
For simplicity, assume that the forces on the element are acting initially in the ‘x’ direction only.
after simplifying gives
Similarly, in the ‘y’ direction
In the ‘z’ direction
since the pressure is constant through the film (Assumption 2)
(1)
(2)
(3)
(4)

7. Reynolds Equation Shear Stress
the shear stress ‘’ can be expressed in terms of dynamic viscosity and shear rates in the ‘x’ and the ‘y’ directions as follow
Substituting these equations into the equilibrium conditions (2) & (3) for the forces acting in the ‘x’ and ‘y’ directions, we obtain:
(5)
(6)
(6’)

8. Reynolds Equation
Equations (6) can be integrated since the η = const & η ≠ f(z), the integration is simple by separating the variables, for ‘x’ direction, we obtain.
the boundary conditions are
(7)
(8)

9. Reynolds Equation
By substituting these boundary conditions into (7) the constants ‘C1’ and ‘C2’ are calculated, and after simplifying, we obtain
In a similar manner a formula for velocity in the ‘y’ direction is obtained
(9)
(10)

10. Reynolds Equation
The three separate terms in any of the velocity equations represent the velocity profiles across the fluid film and they can be schematically drawn

11. Reynolds Equation Continuity of Flow
The lubricant flows into the column horizontally at rates of ‘qx’ and ‘qy’ per unit length and width respectively.
In the vertical direction, the flow is at in rate of ‘w0dxdy’ and out rate of ‘whdxdy
column of lubricant

12. Reynolds Equation (11) (12) (13) Continuity of Flow
The principle of continuity of flow requires that the influx of a liquid must equal its efflux from a control volume under steady conditions (i.e. =const.)
Simplifying
Where ‘qx’ and ‘qy’ are given by
(11)
(12)
(13)

13. Reynolds Equation Continuity of Flow
substituting for ‘u’ and ‘v’ from equations 9 & 10 and after simplifying yields the flow rate in the ‘x’ & ‘y’ directions:
Now Substituting these flow rates into the continuity equation (12) h defining U = U1 + U2 and V = V1 + V2 and assuming that there is no local variation in surface velocity in the ‘x’ and ‘y’ directions
(14)
(15)

14. Reynolds Equation Continuity of Flow
After further rearranging and simplifying yields the full Reynolds equation in three dimensions
the Reynolds equation in its full form is far too complex for practical engineering applications and some simplifications are required before it can conveniently be used.
(16)

15. Simplifications to the Reynolds Equation
Unidirectional Velocity Approximation
After further rearranging and simplifying yields the full Reynolds equation in three dimensions
V = 0. There are very few engineering systems, in which, for example, a journal bearing slides along a rotating shaft.
(17)

16. Simplifications to the Reynolds Equation
Steady Film Thickness Approximation
There is no vertical flow across the film, i.e. wh - w0 = 0. This assumption requires that the distance between the two surfaces remains constant during the operation
(18)

17. Simplifications to the Reynolds Equation
Isoviscous Approximation
For many practical engineering applications it is assumed that the lubricant viscosity is constant over the film, i.e. η = constant.
(19)

18. Simplifications to the Reynolds Equation
Infinitely Long Bearing Approximation
It is assumed that the pressure gradient acting along the ‘y’ axis can be neglected, i.e. ∂p/∂y = 0 and h ≠ f(y).
This assumption reduces the Reynolds equation to a one-dimensional form which is very convenient for quick engineering analysis. Since ∂p/∂y = 0, equation (19) simplifies to
(20)

19. Simplifications to the Reynolds Equation
Infinitely Long Bearing Approximation
Equation (20) can easily be integrated with the following B.C. “dp/dx = 0 at a corresponding film thickness denoted as ‘h”
(21)