# Calculation of Laminar Flow through Fracture

The calculation of laminar flow through fracture is very fundamental (), and it is very important for my research, also. Late last year, I derived the equation for both Newtonian fluid and non-Newtonian fluid by hand. Now I am going to make a note for that derivation for Newtonian fluid, which could be regarded as the answer to an exercise on Transport Phenomena, 2nd Ed. by Bird, R.B., et al., a classical textbook on this topic.

At first we need to describe the physical properties of the fracture model (as shown in Figure.1).

Figure 1. Physical Properties of Fracture Model

A Newtonian fluid is in laminar flow in a narrow slit formed by two parallel walls with length, L, a distance B (fracture width) apart. It is understood that the fracture height, B《W, so that “edge effects” are unimportant. Make a differential momentum balance, and obtain the following expressions for the momentum-flux and velocity distributions:

$\tau_{yz}=\left ( \frac{p_{0}-p_{L}}{L}\right)x$

$v_{z}=\frac{\left (p_{0}-p_{L}\right)B^{2}}{2\mu L}\left [ 1-\left ( \frac{x}{B} \right )^{2} \right ]$

Obtain the slit analog of the Hagen-Poiseuille equation.

$w=\frac{1}{12} \frac{\left (p_{0}-p_{L}\right)B^{3}W\rho }{\mu L}$