There is a fully developed Poiseuille flow $v_z(x, y)$ in a long channel with a rectangular cross-section of side lengths 2a and 2b driven by a constant pressure gradient $\frac{dP}{dz} = -\mu K$, where $\mu$ is the shear viscosity and K is a positive constant. The aspect ratio of the cross-section is $a = a/b$, and without loss of generality, assume that $a < b$ so that $0 \le a \le 1$.
(a) Formulate the problem using dimensionless coordinates X and Y, such that $0 \le X \le 1/a$ and $0 \le Y \le 1$.
Note that we only need to model the upper right quadrant because of mirror symmetry about the x = 0 and y = 0 planes. How should the dimensionless velocity $\Theta(X, Y)$ be defined if we don't know the velocity scale? (Hint: What combination of parameters give a quantity with units of velocity?)
(b) Use the FFT method to find $\Theta(X, Y)$. A well-behaved series is obtained for all a if Y is the basis function variable and if a superposition of the form $\Theta(X, Y) = F(Y) + \Psi(X, Y)$ is used, where F(Y) is the solution for the $a = 0$ parallel-plate limit without side walls.
(c) Because the mean velocity U is proportional to the pressure gradient $\frac{dP}{dz}$, it's convenient to define a dimensionless channel resistance $\Omega$
$\Omega(a) = \frac{D_H^2}{\mu U} |\frac{dP}{dz}|$ where $D_H = \frac{4a}{(1 + a)}$ is the hydraulic diameter of the channel. $\Omega$ is a function of only the shape of the channel cross-section, and therefore allows comparisons among arbitrarily shaped channels. Derive the expression for $\Omega(a)$.
