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Consider a showerhead that releases water at a constant flow rate and temperature. What is the relationship between the pressure of water coming out of the shower and the volume of water collected in a bucket after $t$ seconds? Is Bernoulli's equation relevant here? Or should I just experimentally test it? I’m wondering if there is any theoretical work on this.

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If you mean the water pressure in the pipes, then the flow rate is likely proportional to the square root of that pressure, and the volume fill time is inversely proportional to that. I.e.

$$\mathrm {time} \propto \frac{1}{\dot V} \propto \frac{1}{\sqrt{P}} $$

I say "likely" because the assumption is flow through an orifice based on Bernoulli's principle:

$$\Delta P = \frac 1 2 \rho u^2 $$

Which experience says would model your scenario well.

The proportionality factor depends on the exact flow resistance of the shower head restriction, which you would need to find empirically.

(I noticed you mentioned "constant flow rate." If you are truly assuming that, then your fill time of any fixed volume is of course fixed. Furthermore when you say "pressure" of the stream I think you mean velocity. The pressure is always 1 atmosphere.)

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  • $\begingroup$ Thank you. Yes, time is fixed. By pressure, I mean something like $50$ psi, measured immediately after it comes out of the shower. I'm considering showers in which you can manually vary the water pressure. $\endgroup$
    – callum
    Sep 16, 2022 at 3:33
  • $\begingroup$ The pressure immediately after it comes out of the shower is always atmospheric, Unless the flow is supersonic like in a water jet cutter $\endgroup$
    – RC_23
    Sep 16, 2022 at 3:34
  • $\begingroup$ I'm considering showers in which you can manually vary the water pressure i.e. it shoots water at a greater force. Apologies if my terminology is incorrect, I'm only a high school student. $\endgroup$
    – callum
    Sep 16, 2022 at 3:35
  • $\begingroup$ If you are turning the knob and adjusting the Intensity of the flow, It means the thing you are varying is the pressure inside the pipe, which varies the velocity of the stream outside the pipe $\endgroup$
    – RC_23
    Sep 16, 2022 at 3:37
  • $\begingroup$ Is there a way to determine the pressure based on velocity? $\endgroup$
    – callum
    Sep 16, 2022 at 3:38

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