$$
Re_L = \frac{\rho U L}{\mu},
\quad \Lambda = \frac{\mu U L}{P_a h^2}
\equiv \text{bearing number} \equiv \frac{\text{viscous force}}{\text{pressure force}}
$$
$$
\Lambda \sim \text{near unity}, \quad Re_L \ \text{finite}, \quad \varepsilon \ll 1
$$
$$
\varepsilon^2 Re_L \to 0
\quad \Rightarrow \quad x-y \ \text{momentum:} \quad
0 \cong -\frac{\partial p}{\partial x} + \mu \frac{\partial^2 u}{\partial y^2},
\quad 0 \cong -\frac{\partial p}{\partial y}
$$
$$
\Rightarrow \quad
u(x,y,t) \cong \frac{1}{\mu} \frac{\partial p(x,t)}{\partial x} \frac{y^2}{2} + A y + B
$$
Boundary conditions
- @\(y = 0 : \quad u = \bar{U}_0(t)\)
- @\(y = h(x,t):\quad u = \bar{U}_h(t)\)
$$
\Rightarrow u(x,y,t) \;\cong\;
-\frac{h^2(x,t)}{2\mu} \frac{\partial p(x,t)}{\partial x}
\frac{y}{h(x,t)}\left(1 - \frac{y}{h(x,t)}\right)
+ \big(U_h(t) - U_0(t)\big)\frac{y}{h(x,t)}
+ U_0(t)
$$
★ Balancing viscous & pressure forces → parabolic velocity profile in cross-stream
★ $p(x,t)$ within the gap has not yet been determined
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