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Recent questions in Poiseuille's Law
Fluid MechanicsAnswered question
Aedan Gonzales Aedan Gonzales 2022-05-08

Steady isothermal flow of an ideal gas
So I have a steady isothermal flow of an ideal gas through a smooth duct (no frictional losses) and need to compute the mass flow rate (per unit area) as a function of pressures at any two different arbitrary points, say 1 and 2. I have the following momentum equation in differential form:
ρ v d v + d P = 0
where v is the gas the flow velocity and P is static pressure. The mass flow rate per unit cross section G, can be calculated by integrating this equation between points 1 and 2. This is where it gets confusing. I do the integration by two ways:
1) Use the ideal gas equation P = ρ R T right away and restructure the momentum equation:
v d v + R T d P P = 0
, integrate it between points 1 and 2 and arrive at:
v 1 2 v 2 2 + 2 R T l n P 1 P 2 = 0
Since the flow is steady, I can write G = ρ 1 v 1 = ρ 2 v 2 , again use the ideal gas law to write density in terms of gas pressure and finally arrive at the mass flow rate expression:
G 2 = 2 l n P 2 P 1 R T ( 1 P 1 2 1 P 2 2 )
2) In another way of integrating (which is mathematically correct), I start by multiplying the original momentum equation by ρ to get
ρ 2 v d v + 1 R T P d P = 0
write ρ 2 v = G 2 / v, integrate between points 1 and 2 to arrive at
G 2 l n v 2 v 1 = P 1 2 P 2 2 2 R T
Using the ideal gas law the velocity ratio can be written as the pressure ratio to finally arrive at the mass flow rate equation
G 2 = P 1 2 P 2 2 2 R T l n P 1 P 2
Both the expressions are dimensionaly sound and I know that the second expression is the correct one. My question is, whats wrong with first expression.

As you proceed with your studies of fluid mechanics or need to apply Physics for some mechanical experiment, Poiseuille's law example problems will be one of the questions that you will have to solve. In basic terms, it is a law related to hemodynamics based on the volumetric flow rate sensitivity. For example, this law is used to explain why constricted capillaries lead to higher blood pressure as one of the examples. This is where Poiseuille's law equation can help you explain the fluid mechanics principles. See how such problems are addressed both verbally and graphically in the presented problems.