Elementary Lubrication Theory
★ Broad economic importance of lubrication
★ Viscous, constant density flows, boundaries close together $$ L \gg h \quad \rightarrow \quad \text{assume } h = \varepsilon L, \;\; \varepsilon \ll 1 $$
★ Viscous, constant density flows, boundaries close together $$ L \gg h \quad \rightarrow \quad \text{assume } h = \varepsilon L, \;\; \varepsilon \ll 1 $$
Continuity (incompressible) $$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$ x-momentum: $$ \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = - \frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{\mu}{\rho} \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) $$ y-momentum: $$ \frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = - \frac{1}{\rho} \frac{\partial p}{\partial y} + \frac{\mu}{\rho} \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) $$
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