Submitted by

Mohan Kulkarni

Souvik Math

 

(To be verified)
 

 

HYDRODYNAMIC LUBRICATION  :

 

 

Pumping Action

 

When dry, friction will cause the journal to try to climb bearing inner wall(FIG 1).When lubricant is introduced, the —climbing action (FIG II)   and the viscosity of the fluid will cause lubricant to be drawn around the journal creating a film between the journal and bearing .The lubricant pressure will push the journal to the other  side of the bearing (FIG III). The pressure is developed because of the converging wedge formed between the journal and the bearing.

 

Assumptions made in deriving the Reynold’s equation:

 

 

 

  1. Body forces are neglected, which means that there is no outside field of force acting on the lubricant, such as -- gravitational or magnetic forces etc. This is not true if magnetic and electric fields play a part such as in magneto hydrodynamics.
  2. The pressure is considered to be constant through out the thickness of the film. As the oil film is very thin of the order of a few thousands of an inch at most, the pressure can not vary significantly across it. When the elastic properties of the lubricant are considered that may not hold, but no work has demonstrated that with normal fluids this assumption is invalid.
  3. The curvatures of the bearing surfaces are considered to large compared with the oil film thickness. The surface velocities need not therefore be taken as varying in direction.
  4. There is no slip at boundaries, which means that the velocity of the surface is the same as the velocity of the lubricant in contact with the surface.
  5. The lubricant obeys Newton’s law of viscosity
  6. Flow is laminar. This is true with bearings of normal dimensions and speeds, but in cases of large journals with high speeds the flow is turbulent.
  7. The fluid inertia can be neglected. It has been observed that even at higher Reynold’s no. of 1000 or so, the pressures are only modified by about 5%, if inertia is considered.
  8. Viscosity of the lubricant is taken as constant through out the thickness of the film. This assumption is made for mathematical simplicity
  9. There is no flow in Z  i. e. axial direction of the bearing.

 

 

 

 

 

 

 

Derivation of the Reynold’s equation:

The coloured rectangular area from the figure above is zoomed and its 3D view is taken  below  for force calculation :

Text Box: .τ dx dz

 

           

 

 

            From force equilibrium we get

                       

                        ∂p / ∂x =          ∂τ / ∂y        …………….   (1)

 

 

            From    Newton’s law of viscosity

 

 

            .                       τ =  µ   ∂u / ∂y

 

       from eqn . (1) we get

 

                        ∂p / ∂x    =  µ   ∂ 2u / ∂y2   ……………..(2)

 

                       ∂p / ∂x  =  ( dp / dx)  ;   as  pressure is not varying in y and z direction.

                       

                        ∂ 2u / ∂y2 = 1/ µ ( dp / dx)

 

Integrating

 

                        u =  1/ µ ( dp / dx) y2 /2 + C1 y + C2

 

  putting boundary conditions

 

            At y = 0, u = 0

           

C2  =  0

At y = h , u = - U

                   

C1 =  - U / h – h / 2 µ ( dp / dx)

 

 

 u =  1/ µ ( dp / dx) y2 /2 –{ U / h – h / 2 µ ( dp / dx)}y

 

Flow Rate Q =  u dy

                    =  -  Uh/2  - { (h 3 /12µ )( dp / dx)}

             

Considering the flow to be incompressible ,

     . dQ/ dx    =  0 

  

    hence,     -  U/ 2 ( dh /dx  )   - d / dx  { (h 3  /12 µ )( dp / dx) }  =0

 

 

so  

  d / dx { ( h 3 / µ )( dp / dx)}   =    -  6 U ( dh /dx )