Question 4.10: Simple Slider Bearing with a Given Volumetric Flow Rate...
Simple Slider Bearing with a Given Volumetric Flow Rate
| Concepts | Assumptions | Sketch |
| • Reynolds lubrication equation | • Steady unidirectional laminar flow in a very small gap |  |
| • Conservation of mass | • Negligible inertia terms | |
| • {\textrm{}}|\mathrm{h}_{2}-\mathrm{h}_{1}\mid\ll \mathrm{L} | • No end effects | |
| • Constant fluid properties |
Again, the basic problem solution is a combination of Couette flow and Poiseuille flow (see Eq. (4.44)):
where here
\rm\mathrm{h}\equiv\mathrm{h}(\mathrm{x})=\mathrm{h}_{1}+{\frac{\mathrm{h}_{2}-\mathrm{h}_{1}}{\mathrm{L}}}\,\mathrm{x}\qquad\qquad(E.4.10.2)The flow rate Q is:
And from Eq. (4.45)
The maximum pressure occurs at an x-location where dp/dx = 0, i.e.,
Thus,
\rm{x_{\mathrm{opt}}}={\frac{{h_{1}}\mathrm{L}}{{h_{1}}+{ h_{2}}}}\qquad\qquad({ E.4.l0.5b)}The sustaining force per unit width is:
\rm\mathrm{F}=\int_{0}^{\mathrm{{L}}}(\mathrm{p}-\mathrm{p}_{0})\,\mathrm{{d}}\mathrm{x}\qquad\qquad(E.4.10.6)where with Eq. (E.4.10.4) we obtain:
Tasks:
• Solve (E.4.10.7) and (E.4.10.6) with \rm\mathrm{h}({x})=\mathrm{h}_{1}+\frac{\mathrm{h}_{2}-\mathrm{h}_{1}}{\mathrm{L}}{ x}
• Plot p(x) and indicate \rm x_{opt} and \rm p_{max} for a reasonable geometry (see Graph a)
• Plot Eq. (E.4.10.1) profiles (see Graph b)
Graphs:
Comments:
(a) When \mathrm{x}=\mathrm{x}_{\mathrm{opt}}={\frac{\mathrm{h}_{1}\mathrm{L}}{\mathrm{h}_{1}+\mathrm{h}_{2}}}, p(x) reaches its maximum \rm p_{max} .
(b) The u(x,y)-distributions vary greatly relative to the x position; as x increases, the backflow region decreases.
