\(x\): coordinate along the bearing (m), \(x\in[0,a+b]\)
\(y\): coordinate across the film (m), \(y\in[0,h(x)]\)
\(h(x)\): film thickness, piecewise constant
\(b\): length of the narrow region; \(a\): length of the wide region (m)
\(h_b\): narrow gap (m); \(h_a\): wide gap (m), typically \(h_b \(U_1\): 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 \(x=a+b\) (Pa)

\[ h(x) = \begin{cases} h_b & 0 \le x \le b \\[4pt] h_a & b < x \le a+b \end{cases} \] \[ \frac{\partial}{\partial x}\!\left(\frac{h^{3}}{12\mu}\frac{\partial p}{\partial x}\right) = \tfrac{1}{2}\,U_1\,\frac{\partial h}{\partial x} \] Because \(h\) is constant in each segment, \(\partial h/\partial x = 0\), so \[ \frac{\partial}{\partial x}\!\left(\frac{h^{3}}{12\mu}\frac{\partial p}{\partial x}\right)=0 \;\Longrightarrow\; \frac{\partial p}{\partial x}=G_i\,\text{ constant} \] i.e. \(p(x)\) is piecewise linear: \(G_1\) on \([0,b]\), \(G_2\) on \([b,a+b]\). Pressure continuous (\(p\) is a state variable) and flow rate per unit width continuous: \[ Q = -\frac{h^{3}}{12\mu}\,\frac{\partial p}{\partial x}+\frac{U_1 h}{2}=\mathrm{const} \] With \(p(0)=p(a+b)=p_0\) the two unknowns \(G_1,G_2\) follow from \[ G_1\,b + G_2\,a = 0, \qquad -\frac{h_b^{3}}{12\mu}G_1+\frac{U_1 h_b}{2}=-\frac{h_a^{3}}{12\mu}G_2+\frac{U_1 h_a}{2} \] \[ G_1 = -\frac{6\mu\,U_1\,a\,(h_a-h_b)}{a\,h_b^{3}+b\,h_a^{3}},\qquad G_2 = -G_1\,\frac{b}{a} \] \[ p_{\max}-p_0 = G_1\,b = -\frac{6\mu\,U_1\,a\,b\,(h_a-h_b)}{a\,h_b^{3}+b\,h_a^{3}} \] \[ Q = \frac{U_1\,h_a\,h_b\,(a\,h_b^{2}+b\,h_a^{2})}{2\,(a\,h_b^{3}+b\,h_a^{3})} \] \[ u(x,y) = -\frac{1}{2\mu}\,\frac{\partial p}{\partial x}\,y\,(h-y) + U_1\,\frac{h-y}{h} \] \[ \tau_i = \mu\,\left.\frac{\partial u}{\partial y}\right|_{y=0} = -\frac{h_i}{2}\,G_i - \frac{\mu U_1}{h_i}, \quad i\in\{1,2\} \] The pressure profile is triangular, so \[ N = \int_0^{a+b}\!(p-p_0)\,\mathrm{d}x = \frac{(a+b)}{2}\,(p_{\max}-p_0) \] \[ T = \int_0^{a+b}\!\tau\,\mathrm{d}x = \tau_1\,b + \tau_2\,a \] \[ f = \frac{T}{N} \]