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MEL 417 Lubrication. Reynold’s equation. Reynold’s equation for fluid flow. Assumptions: External forces are neglected (gravitational, magnetic etc.) Pressure is considered constant throughout the thickness of the film

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mel 417 lubrication

MEL 417 Lubrication

Reynold’s equation

reynold s equation for fluid flow
Reynold’s equation for fluid flow

Assumptions:

  • External forces are neglected (gravitational, magnetic etc.)
  • Pressure is considered constant throughout the thickness of the film
  • Curvature of the bearing surfaces are large compared to the oil film thickness
  • No slip at boundaries
  • Lubricant is Newtonian
  • Flow is laminar
  • Fluid inertia can be neglected
  • Viscosity is constant through the thickness of the film
newtonian fluid shear stress shear strain relationship
Newtonian fluid: shear stress-shear strain relationship

Shear rate

  • Linear dependence
  • Slope is 1/h

a

Shear stress t

reynold s eqn equilibrium of a fluid element forces in one dimension
Reynold’s eqn: Equilibrium of a fluid elementforces in one dimension

y

Shear force on top face

z

dy

Pressure force on right face

Fluid element

dz

Pressure force on left face

x

Shear force on bottom face

dx

x, y, z: Mutually perpendicular axes

p: pressure on left face, t: shear stress on bottom face in x direction

dx, dy, dz: elemental distances

fluid element equilibrium equations
Fluid element- equilibrium equations

Forces on left should match forces on right.

Therefore

Simplifying we get:

OR similarly

substituting using newton s law of viscosity
Substituting using Newton’s law of viscosity

In the z direction the pressure gradient is 0, therefore

According to Newton’s law for viscous flow

and

Whereu and v are the particle velocities in the x and y directions respectively

h is the coefficient of dynamic viscosity

pressure gradients
Pressure gradients

Therefore the pressure gradients in terms of only the viscosity and velocity gradients is

and

Assuming that the viscosity is constant

and

conditions
Conditions
  • p and hare independent of z (assumptions)
  • Therefore integrating
  • we get
  • Applying boundary conditions, u = U1 at z = h and u = U2 at z = 0 we get C2 = U2 and

so

z=h

h

z=0

volume flow rate
Volume flow rate

Substituting we get

And

Let the rate of flow (per unit width) in the x and y directions be qx and qy respectively

Therefore

and

reynold s equation for fluid flow between inclined surfaces
Reynold’s equation for fluid flow between inclined surfaces

Pressure profile

Top surface

pmax

Oil wedge

Upper surface is stationary

p = Pressure

Film thickness = h

When h = ho

p = pmax

therefore

h

ho

Bottom surface moves with velocity U

Bottom surface

and

reynold s equation in one dimension
Reynold’s equation in one dimension

When p = pmax, dp/dx=0, and h = ho

Therefore

Substituting we get

If r is the density of fluid, the mass flow rate in the x direction is

flow rate after substitution
Flow rate after substitution
  • Equation of continuity for 2 dimensions
  • In most bearing systems there is no flow in the y direction, therefore V1=V2=0. If surface1 is stationary then U1 is also 0. Then equations 3 and 4 reduce to

and

reynold s equation in 2 dimensions
Reynold’s equation in 2 dimensions

Substituting into the continuity equation we get

Which gives

velocity of flow at a fluid element
Velocity of flow at a fluid element

Velocity at back face

Velocity at top face

y

z

dy

Velocity at right face

Fluid element

dz

Velocity at left face

Velocity at front face

x

dx

Refer to book Principles of Lubrication by Cameron A

Velocity at bottom face

balancing in and out flow rates
Balancing in and out flow rates
  • The velocities entering the element are u, v, and w along x, y, and z directions respectively
  • The velocities leaving are correspondingly

, , and

Therefore the flow rates are:

In-udydz, vdxdz, and wdxdy

Out- , and

continuity equation in 3 dimensions
Continuity equation in 3 dimensions
  • As there are no source or sinks for fluid flow within the element and the volume remains constant, the total volume flowing in = total volume flowing out, per unit time

Therefore:

On simplifying we get:

Which is the continuity equation in three dimensions

If we retain the volume terms, we get:

Where qx, qy, and qz are the flow rates per unit width in the x, y, and z directions respectively

reynold s equation infinitely long bearing l d
Reynold’s equation- Infinitely long bearing (L>>D)
  • In this assumption, the pressure does not vary in the y direction
  • Therefore = 0 and the flow rate qy = 0
  • Assuming that only one surface moves, with a velocity U, we get (derived earlier)

and

where ho is the film thickness at max/min pressure

Diameter D

L

L>>D

infinitely long bearing l d
Infinitely long bearing (L >> D)

Pressure p can be obtained from the equation

Provided h can be expressed in terms of x

Therefore

Where C is a constant of integration. Two boundary conditions are required to obtain the values for ho and C. This can be obtained from knowledge of the start and end points of the pressure curve where p = 0

The pressure curve in the figure below ranges from x = 0 to x = B

Diameter D

L

Pressure curve

x = 0

x = B

reynold s equation infinitely short bearing d l
Reynold’s equation- Infinitely short bearing (D>>L)
  • In this case the length of the bearing is considered much shorter than the diameter
  • Therefore the pressure differential in the x – direction is considered 0 as it is much lower compared to the pressure differential in the y direction
  • We therefore get
  • The film thickness is assumed not to vary with x, therefore
  • Reynold’s equation in two dimensions then becomes

L = length of bearing

Diameter D

infinitely short bearing
Infinitely short bearing
  • On integration we get
  • Further integration gives
  • Where K1and K2 are constants of integration
  • These can be obtained by putting pressure = 0 at the edges of the bearing and pressure gradient = 0 at the middle of the bearing (assuming symmetry)

pmax

-L/2

+L/2

y

infinitely short bearing1
Infinitely short bearing
  • We therefore get and
  • The equation therefore becomes
  • If p = 0 other than when y = -L/2 or +L/2, either dh/dx=0 or h3is infinite
  • This fact is applied to journal bearings and dh/dx=0 at points of maximum and minimum film thickness
  • It is also applicable to narrow rotating discs
  • It is not applicable to thrust bearings
  • This theory is applicable when L/D<1/4 and infinitely long theory is applicable when L/D>=4