3.2.4.14. Lubrication Equation

Reynolds’ lubrication equation is a reduced-order model for fluid flow in thin, confined regions. This equation is commonly used for manufacturing applications, such as bearings, coating processes, and nanomanufacturing processes. Lubricating flows also occur in high-speed machinery, such as pumps. It is only defined for reduced-order elements, such as shells.

The lubrication equation implemented here is given by

(3.70) $-\pt{h} + \div\vector{q} + \vector{B}_u\bcdot\grad h_u - \vector{B}_l\bcdot\grad h_l = 0$

where $\vector{q}$ is the lubrication flow rate, given by

(3.71) $\vector{q} = -\frac{h^3}{k \mu}\grad p_\mathrm{lub} + \frac{h}{2}(\vector{B}_u + \vector{B}_l).$

In these equations, the independent variable is the lubrication pressure, $p_\mathrm{lub}$ . Other variables are the lubrication region height (thickness) $h$ , velocity of the upper confining surface $\vector{B}_u$ , velocity of the lower confining surface $\vector{B}_l$ , the upper wall height $h_u$ , the lower wall height $h_l$ , turbulent parameter $k$ , and viscosity $\mu$ . The lubrication height is defined as $h=h_u-h_l+\vector{n}\bcdot\vector{d}$ , where $\vector{n}$ is the shell normal vector and $\vector{d}$ is the mesh displacement. This equation is detailed in [13].

Using equation (3.70), the G/FEM residual form is

(3.72) $\symRes_V^i & = - \underbrace{\int\limits_\Vol \pt{h}\phi^i \dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \vector{q}\bcdot\grad\phi^i \dV}_\mathrm{DIFF} + \underbrace{\int\limits_\Vol ( \vector{B}_u\bcdot\grad h_u - \vector{B}_l\bcdot\grad h_l ) \phi^i \dV}_\mathrm{BOUND} \\ & + \int\limits_\Surf \vector{n}\bcdot\vector{q} \phi^i \dS = 0$