- #1
Ron Burgundypants
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I have a physics project at university, we designed an experiment to measure the effectiveness of Poiseuilles law in a 'quasi non-steady state'. Poiseuilles law, simply being the measurement of the flow rate of a fluid in a pipe, holding only under steady state though. So by quasi steady state I mean that there is some acceleration of the fluid, but only a small amount. I've attached a picture of the apparatus below. The general idea is that we use compressed air to pump the fluid up a measuring tube and use a high speed camera to measure the flow rate. By varying the radius of the measuring tube and the viscosity of the fluid we hope to find a range were poiseuilles law begins to breakdown.
So we did some experiments and after a while, realized that Poiseuilles law only matched actual results in the limits of low pressure and small radii (sub 10mm and 1bar). Which is kind of counter intuitive... If Force = Pressure x Area then by holding pressure constant and increasing the area we would have expected the Force to increase, by means of a faster flow rate. This wasn't the case, instead, at larger radii, we saw slower flow rates...
Our theory then, is that with increased radius we have increased volume and hence mass, so the inertial force begins to dominate and slows down the flow rate significantly and it deviates from our expected outcome. However, by deriving an equation to describe this, we ended up with a monstrosity and can't think of a way of solving it. This is were we would like some help. Below are some pictures showing the free body diagram of our revised model and the derivations of the equations that describe the motion of fluid in a pipe. If anyone has any suggestions on how to go about solving this equation I'd be most grateful. The idea is that a solution (if found), plotted on a graph might match our measured data. In the last equation we have pressure as a solution but we wish to find the solution to the function h(t)...
Thanks
So we did some experiments and after a while, realized that Poiseuilles law only matched actual results in the limits of low pressure and small radii (sub 10mm and 1bar). Which is kind of counter intuitive... If Force = Pressure x Area then by holding pressure constant and increasing the area we would have expected the Force to increase, by means of a faster flow rate. This wasn't the case, instead, at larger radii, we saw slower flow rates...
Our theory then, is that with increased radius we have increased volume and hence mass, so the inertial force begins to dominate and slows down the flow rate significantly and it deviates from our expected outcome. However, by deriving an equation to describe this, we ended up with a monstrosity and can't think of a way of solving it. This is were we would like some help. Below are some pictures showing the free body diagram of our revised model and the derivations of the equations that describe the motion of fluid in a pipe. If anyone has any suggestions on how to go about solving this equation I'd be most grateful. The idea is that a solution (if found), plotted on a graph might match our measured data. In the last equation we have pressure as a solution but we wish to find the solution to the function h(t)...
Thanks