Governing Equation for Arbitrary-Shaped Channel

In a Poiseuille flow, the fluid is driven through a long, straight, and rigid channel by imposing a pressure difference between the two ends of the channel. Originally, Hagen and Poiseuille 

Based on the dimensional analysis of the previous section, we can assume the nonlinear term (V - V V) = 0 for a high aspect ratio channel. The vanishing force in y-z surface indicates that only the jc-component of velocity is nonzero. The steady-state N-S equation becomes 

Since the y- and z-component of the velocity field are zero, it follows that = 0 and = 0, and consequently the pressure field only depends on x, that is, P(r) = P(x). Using this result, the x-component of N-S equation (2.32b) becomes 

it is seen that the left-hand side is a function of y and z while the right-hand side is a function of x. The only possible solution is thus that the two sides of the N-S equation is equal to the same constant. However, a constant pressure gradient implies that the pressure must be a linear function of x, and using the boundary conditions for the pressure we obtain 

we obtain the second-order partial differential equation as 

---