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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):

Where:

**η** – dynamic viscosity of oil,

**v** – linear velocity of the laminar layer,

**y** - the axis perpendicular to the flow direction.

- The inertia forces resulted from 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.

Consider the equilibrium of a unit volume in the lubricant film.

The pressure forces act on the right and the left faces of the unit volume.

The shear forces resulting from the relative motion of the laminar layers act along the upper and the lower faces of the unit volume.

Assuming that there is no flow in **z** direction the equation of the equilibrium of the forces in the direction of the flow is as follows:

Substitution of **τ** from Newton’s law of viscosity (1) results in:

The velocity function is obtained by integrating the equation (2) with respect to **y**:

The constants of integration **C _{1}** and

Substituting

The total flow of the lubricant is:

Substituting

According to the assumption about incompressibility of the lubricant the flow

Differentiating the equation (5) with respect

This is Reynolds equation for one dimensional flow.

It can be used with the assumption of no flow in

If the flow in

Where:

Close form solution of Reynolds equation can not be obtained therefore finite elements method is used to solve it.

**Sommerfeld Solution**

The equation is solved with the assumption that there is no lubricant flow in the axial direction (**infinitely long bearing assumption**).

**Ocvirk Solution**

Ocvirk solution for **infinitely short bearing assumption** neglects circumferential pressure gradients (first term of Reynolds equation).

Where:

**C _{r}** – radial clearance

Where:

Sommerfeld and Ocvirk solutions are applicable only in the region of positive pressure.

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http://www.itescam.edu.mx/principal/sylabus/fpdb/recursos/r49656.PDF

http://nptel.iitm.ac.in/courses/IIT-MADRAS/Machine_Design_II/pdf/5_2.pdf

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http://www.stle.org/resources/lubelearn/lubrication/default.aspx

http://www.uni-due.de/imperia/md/content/werkstofftechnik/sk_hwt_05_lubrication_ss08.pdf

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