Question

I have a simple, easy to solve, Poiseuille type of problem, to which I added a second question, in which viscosity is zero, so I can solve it using Bernoulli. I am arriving a greater flux using Pouseuille than Bernoulli, which is to me counterintuitive. Here is the problem and a summary of my solution:

From a jar, and through a circular tube, water flows that is poured into the atmosphere through a needle 3 cm in length and with an internal radius of 0.36 mm. When the jar is at a height of 1.5 m, a) what is the flow rate of fluid that comes out through the needle? (density of the fluid, 1.05 g cm–3; viscosity of the fluid, 1.3 cp). b) what would be the flow rate from the needle if we did not take into account the viscosity of the plasma?

a) I solve it using Poiseuille's equation, considering ΔP=rhoghA: Q=πR⁴/8μLΔP=π(0.3610⁻3)⁴/(81.310⁻3310⁻21.0510³9.811.5)=2.61*10⁻6 m³/s

b) If there's no viscosity, we can use Bernoulli's equation between the top of the jar and the exit of the needle, assuming the velocity of the fluid on top of the jar vA=0 m/s since it is much wider than the needle: Patm+rhoghA=Patm+1/2rhogvB², which results in vB=(2ghA)⁰.5=5.42 m/s, which then I use to find the flux, Q=vS=vπR²=2.21*10⁻6 m³/s

How can the flux when I use Poiseuille's equation be greater than when I use Bernoulli? Shouldn't it be the other way around? Thank you

Answer 1

The Poiseuille's equation and Bernoulli's equation are two different equations used to solve fluid flow problems. The Poiseuille's equation takes into account the viscosity of the fluid, which affects the flow rate, while Bernoulli's equation assumes zero viscosity. Therefore, it is not counterintuitive that the flux when using Poiseuille's equation is greater than when using Bernoulli's equation.

Step-by-step explanation:

The Poiseuille's equation and Bernoulli's equation are two different equations used to solve fluid flow problems. The Poiseuille's equation is based on the flow of a viscous fluid through a tube, while Bernoulli's equation is based on the conservation of energy in fluid flow.

When solving the problem using Poiseuille's equation, you considered the viscosity of the fluid, which affects the flow rate. However, when using Bernoulli's equation and assuming zero viscosity, the flow rate is higher because there is no resistance due to viscosity.

So, it is not counterintuitive that the flux when using Poiseuille's equation is greater than when using Bernoulli's equation because the two equations are based on different principles and assumptions.