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I'm having some difficulty with the following problem:

Derive a formula for the propulsion force containing $\rho, Q,S, u$ (pic below)

The propeller is moving left with the speed u. My only idea was to use Bernoulli as well as put an observer on the propeller and look into the flow from his perspective. I'd be grateful for verifciation of my solution as well as any hints

enter image description here

My attempt:

enter image description here

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    $\begingroup$ Have you looked at dimensional analysis either Rayleigh or Buckingham? This seems to be what your professor wants you to do. $\endgroup$
    – Solar Mike
    Jan 23, 2021 at 6:41

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I'll give you a hint, use the conversation of linear momentum equation.

For a fixed control volume $CV$: $$ \sum \vec{F} = \frac{d}{dt} \int_{CV} \rho \vec{V} dV + \int_{CS} \rho \vec{V}(\vec{V}_r . \vec{n}) dA$$

wher $\vec{V}_r = \vec{V} - \vec{V}_{CS}$ is the relative flow velocity exiting the control volume relative to the control surface.

Assuming a steady state case: $$ \sum \vec{F} =\int_{CS} \rho \vec{V}(\vec{V}_r . \vec{n}) dA = \dot{m} V_r$$

NOTE: your idea of using Bernoulli is justified if there was a difference in pressure between inlets and outlets of your control volume, and you needed to plug forces resulting from difference in pressure in the above Newton second law formulation, but since pressure at inlet = pressure at outlet = $p_{\infty}$, there is no need to use Bernoulli.

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  • $\begingroup$ Thank you for your reply! I should have been more precise. There is a difference in pressure between the area far away from the narrowing and near it - that's why I thought Bernoulli might come in handy. I've attached my solution above. I'd be really grateful if you could point out my mistakes :) $\endgroup$ Jan 23, 2021 at 8:42
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    $\begingroup$ @EngineerInProgress You're overcomplicating this :) . Of course there is a difference in pressure but you can avoid including the effects of velocity and pressure at the inlet of the propeller by simply extending your control volume to the region of air where $p = p_{\infty}$ and $v = 0$, then apply the equation to the control volume. Is this clear to you? $\endgroup$
    – Algo
    Jan 23, 2021 at 8:55
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    $\begingroup$ @EngineerInProgress Do you have access to Cengel's "Fluid Mechanics Fundamentals and Applications"? If so, check Chapter 6, section 2 "Choosing a Control Volume". $\endgroup$
    – Algo
    Jan 23, 2021 at 8:58
  • $\begingroup$ Good point... that would greatly simplify the calculations. I just want to make sure - even though overcomplicated - is my answer correct? Thank you for your time! @Algo $\endgroup$ Jan 23, 2021 at 18:59

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