Question

(a) since the flow rate is proportional to the pressure difference, show that Poiseuille's law can be written in the form \(\Delta P=I R,\) where \(I\) is the volume flow rate and \(R\) is a constant of proportionality called the fluid flow resistance. (Written this way, Poiseuille's law is analogous to Ohm's law for electric current to be studied in Chapter \(18: \Delta V=I R,\) where \(\Delta V\) is the potential drop across a conductor, \(I\) is the electric current flowing through the conductor, and \(R\) is the electrical resistance of the conductor.) (b) Find \(R\) in terms of the viscosity of the fluid and the length and radius of the pipe.

Short answer

Question: Show that Poiseuille's law can be written in the form \(\Delta P = IR\) and find the value of \(R\) in terms of the viscosity of the fluid, the length of the pipe, and the radius of the pipe. Answer: Poiseuille's law can be written in the form \(\Delta P = IR\) where \(I\) is the volume flow rate, \(R\) is the fluid flow resistance, and \(\Delta P\) is the pressure difference. The value of \(R\) in terms of the viscosity of the fluid, the length of the pipe, and the radius of the pipe is: $$R = \frac{8 \eta L}{\pi r^4}$$

Step-by-step solution

Derive Poiseuille's law from the given information

First, let's recall that Poiseuille's law states that the volume flow rate \(Q\) is proportional to the pressure difference \(\Delta P\): $$Q \propto \Delta P$$ We can rewrite this equation in terms of a constant of proportionality, \(k\): $$Q = k\Delta P$$

Rewrite Poiseuille's law in the form \(\Delta P = IR\)

In order to write Poiseuille's law in the desired form, we need to identify the volume flow rate \(I\) and the fluid flow resistance \(R\). From the above equation, we can write: $$\Delta P = \frac{Q}{k}$$ Comparing this to the desired form \(\Delta P = IR\), we can identify \(I\) as the volume flow rate \(Q\). The fluid flow resistance \(R\) can be written as the inverse of the constant of proportionality \(k\): $$R = \frac{1}{k}$$ Then, the desired form of Poiseuille's law becomes: $$\Delta P = Q R$$

Find R in terms of viscosity, length, and radius of the pipe

Now we need to find the value of \(R\) in terms of viscosity, length, and radius of the pipe. We can use Poiseuille's law, which, for a cylindrical pipe, states: $$Q = \frac{\pi r^4}{8 \eta L} \Delta P$$ Here, \(r\) is the radius of the pipe, \(\eta\) is the viscosity of the fluid, and \(L\) is the length of the pipe. We can rewrite this equation to find the constant \(k\): $$k = \frac{\pi r^4}{8 \eta L}$$ Since we have already found that \(R = \frac{1}{k}\), we can now find the value of \(R\): $$R = \frac{8 \eta L}{\pi r^4}$$

Summary of the results

We have shown that Poiseuille's law can be written in a form analogous to Ohm's law: $$\Delta P = IR$$ where \(I\) is the volume flow rate, \(R\) is the fluid flow resistance, and \(\Delta P\) is the pressure difference. We have also found the value of \(R\) in terms of the viscosity of the fluid, the length of the pipe, and the radius of the pipe: $$R = \frac{8 \eta L}{\pi r^4}$$