Definitions
\(x\): coordinate along the bearing (m), \(x\in[0,a]\)
\(y\): coordinate across the film (m), \(y\in[0,h(x)]\)
\(h(x)\): film thickness (m); \(h_0=h(0)\), \(h_1=h(a)\)
\(U_1\): velocity of the bottom (flat) plate (m/s); top plate fixed
\(\mu\): dynamic viscosity (Pa·s)
\(p_0\): ambient pressure imposed at \(x=0\) and \(x=a\) (Pa)
Film thickness (exponential gap)
\[ h(x) = h_0\,\mathrm{e}^{c x}, \qquad c = \frac{1}{a}\ln\!\frac{h_1}{h_0} \]
Reynolds equation (steady, incompressible, thin film)
\[ \frac{\partial}{\partial x}\!\left(\frac{h^3}{12\mu}\frac{\partial p}{\partial x}\right) = \tfrac{1}{2} U_1\,\frac{\partial h}{\partial x} \]
Flow rate per unit width (constant in steady state)
\[ Q = \int_0^{h}\! u\,\mathrm{d}y = -\frac{h^3}{12\mu}\frac{\partial p}{\partial x} + \frac{U_1 h}{2} = \mathrm{const} \]
\[ Q = \frac{3\,U_1\,h_0\,h_1\,(h_0+h_1)}{4\,(h_0^{2}+h_0 h_1 + h_1^{2})} \]
Pressure
\[ \frac{\partial p}{\partial x} = -\frac{12\mu Q}{h^3} + \frac{6\mu U_1}{h^2} \]
\[ p(x) - p_0 = -\frac{4\mu Q}{c}\!\left(\frac{1}{h_0^{3}}-\frac{1}{h(x)^{3}}\right) + \frac{3\mu U_1}{c}\!\left(\frac{1}{h_0^{2}}-\frac{1}{h(x)^{2}}\right) \]
Velocity profile
\[ u(x,y) = -\frac{1}{2\mu}\,\frac{\partial p}{\partial x}\,y\,(h-y) + U_1\,\frac{h-y}{h} \]
Shear stress on the flat plate \((y=0)\)
\[ \tau(x) = \mu\,\left.\frac{\partial u}{\partial y}\right|_{y=0} = -\frac{h}{2}\,\frac{\partial p}{\partial x} - \frac{\mu U_1}{h} \]
Integrated quantities (per unit width)
\[ N = \int_0^{a} (p - p_0)\,\mathrm{d}x \qquad\text{(load capacity)} \]
\[ T = \int_0^{a} \tau(x)\,\mathrm{d}x \qquad\text{(viscous drag on the flat plate)} \]
\[ f = \frac{T}{N} \qquad\text{(friction coefficient)} \]