Non-dimensionalize $$ x^{★} = \frac{x}{L}, \quad y^{★} = \frac{y}{h} = \frac{y}{\varepsilon L}, \quad t^{★} = \frac{Ut}{L}, \quad u^{★} = \frac{u}{U}, \quad v^{★} = \frac{v}{\varepsilon U}, \quad p^{★} = \frac{p}{P_a} $$

• $P_a$ is atmospheric pressure
• $\varepsilon = \frac{h}{L}$ : passage's fineness ratio

Geometry $$ \frac{\partial}{\partial x} \sim \frac{1}{L} \;\;\ll\;\; \frac{\partial}{\partial y} \sim \frac{1}{h} $$

In dimensionless form
Continuity $$ \frac{\partial u^{★}}{\partial x^{★}} + \frac{\partial v^{★}}{\partial y^{★}} = 0 $$ x-momentum $$ \varepsilon^2 Re_L \left( \frac{\partial u^{★}}{\partial t^{★}} + u^{★} \frac{\partial u^{★}}{\partial x^{★}} + v^{★} \frac{\partial u^{★}}{\partial y^{★}} \right) = - \frac{1}{L} \frac{\partial p^{★}}{\partial x^{★}} + \varepsilon^2 \frac{\partial^2 u^{★}}{\partial x^{★2}} + \frac{\partial^2 u^{★}}{\partial y^{★2}} $$ y-momentum $$ \varepsilon^4 Re_L \left( \frac{\partial v^{★}}{\partial t^{★}} + u^{★} \frac{\partial v^{★}}{\partial x^{★}} + v^{★} \frac{\partial v^{★}}{\partial y^{★}} \right) = - \frac{1}{L} \frac{\partial p^{★}}{\partial y^{★}} + \varepsilon^4 \frac{\partial^2 v^{★}}{\partial x^{★2}} + \varepsilon^2 \frac{\partial^2 v^{★}}{\partial y^{★2}} $$