Reynolds Equation: Derivation and Solution

Reynolds Equation
Tribology Wikipedia > Reynolds Equation: Derivation and Solution

Reynolds equation is a partial differential equation which describes the flow of a thin lubricant film between two surfaces. It is derived from the Navier-Stokes equations and is one of the fundamental equations of the classical lubrication theory. It was first derived by Osborne Reynolds in 1886.

Derivation of Reynolds Equation

The principles of the theory are derived from the observation that the lubricant can be treated as isoviscous and laminar and the fluid film is of negligible curvature. The classical Reynold’s equation can be derived from the Navier-Stokes equations and the equation of continuity under assumptions of:

  • constant viscosity, Newtonian lubricant
  • thin film geometry
  • negligible body force
  • no-slip boundary conditions

Reynolds Equation

When these assumptions are applied, we obtain following equations (following Szeri):

(1)    \begin{eqnarray*} \frac{\partial p}{\partial x} = \mu\frac{\partial^{2} u}{\partial y^{2}} \\ \label{complete_sys2} \frac{\partial p}{\partial z} = \mu\frac{\partial^{2} w}{\partial y^{2}} \\ \label{complete_sys3} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \end{eqnarray*}

It is also assumed the lubricant is incompressible here. The third of these equations is the equation of continuity and will be used later in the derivation. The first two equations now can be integrated twice with respect to  y , since partial derivatives of pressure p dont vary across  y (assumption of thin film geometry):

(2)    \begin{eqnarray*} u = \frac{1}{2\mu}\frac{\partial p}{\partial x}y^2 + Ay + B \\ \label{complete_sys2} w = \frac{1}{2\mu}\frac{\partial p}{\partial z}y^2 + Cy + D \end{eqnarray*}

Boundary conditions should be applied:

(3)    \begin{eqnarray*} u = U_1, w = 0, y = 0, \\ \label{complete_sys2} u = U_2, w = 0, y = h, \end{eqnarray*}

where U_1 and U_2 represent the velocity of the bearing surfaces.
Evaluation of the integration constants leads to the following velocity distribution:

(4)    \begin{eqnarray*} u = \frac{1}{2\mu}\frac{\partial p}{\partial x}(y^2 - yh) + (1 - \frac{y}{h})U_1 + \frac{y}{h}U_2 \\ \label{complete_sys2} w = \frac{1}{2\mu}\frac{\partial p}{\partial z}(y^2 - yh) \end{eqnarray*}

There are only tree equations for four unknowns. This difficulty will be alleviated by integrating, in effect averaging, the equation of continuity across the film:

 \int_{0}^{h}- \frac{\partial v}{\partial y} dy = \int_{0}^{h} {\frac{\partial u}{\partial x}dy + \int_{0}^{h} \frac{\partial w}{\partial z}} dy

Integrating yields following Reynold’s equation for lubricant pressure:

(5)    \begin{eqnarray*} \frac{\partial}{\partial x}(\frac{h^3}{\mu}\frac{\partial p}{\partial x}) + \frac{\partial}{\partial z}(\frac{h^3}{\mu}\frac{\partial p}{\partial z})= 6(U_1+U_2)\frac{\partial h}{\partial x} + 12\frac{\partial h}{\partial t}, \end{eqnarray*}

Solution of Reynolds Equation

In general, the Reynolds equation has to be solved using numerical methods such as finite difference, or finite element. Depending on the boundary conditions and the considered geometry, however, analytical solutions can be obtained under certain assumptions.

For the case of a sphere on flat geometry (for rigid bodies) and steady-state case, the 2-D Reynolds equation can be solved analytically assuming Sommerfeld (also called half-Sommerfeld) cavitation boundary condition. This solution was proposed by a Nobel Prize winner Professor Kapitza. The Sommerfeld boundary condition, however is not accurate and this solution has to be used as an approximate.

In case of 1-D Reynolds equation, there are several analytical, semi-analytical and approximate solutions available. In 1916 Martin obtained a closed form solution for a minimum film thickness and pressure for a cylinder and plane geometry under assumptions of rigid surfaces. Derivation of the solution and the corresponding MATLAB software can be found here. Martin employed Swift-Stieber cavitation boundary conditions. This solution is not accurate in case of high loads (high pressure in the lubricant), when the elastic deformation of the surfaces contributes to the film thickness. Divergence of experimental and theoretical results by Martin for high loads leaded researchers to conclusion that elastic distortion plays a significant role in lubrication. In 1949, Grubin obtained a solution for so called elasto-hydrodynamic lubrication (EHL) line contact problem with certain simplifications, where he combined both elastic deformation and lubricant hydrodynamic flow. Although his solution did not satisfy both elastic and hydrodynamic equations of EHL, his analysis was recognized as particularly useful. Corresponding MATLAB code can be found here. A lecture on solution of Reynolds equation using Finite Difference Method is given in the lecture below.

Generalization of Reynolds Equation

Various of generalized Reynolds equations were derived to weaken the assumptions used to derive the classical form. For example, compressible, non-Newtonian lubricant behavior can be considered. In tribology, Reynolds equation is used to predict the thickness of the lubricant film, but also to predict the friction developed by the lubricant on the surfaces. Since many tribological contacts operate in highly loaded regime and thin films, the shear rates can be very high (in the order of  10^7-10^9 ). Many of the typical lubricants start to behave non-Newtonian in the contact conditions and therefore, Reynolds equation was generalized to the case of non-Newtonian lubricant.

Another generalization includes the slip boundary conditions. This form of the Reynolds equation is used to calculate film thicknesses and friction in textured surfaces or surfaces with high slip.

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