Volume flow rate and equation of continuity

everything we've done so far has been stationary fluids or static fluids and we've been dealing with static pressure because all of the you know we we were trying to figure out what happens when everything's in a steady state so now let's work on what happens when the fluid is actually moving so let's imagine a pipe let me draw a pipe let's say one end of the pipe has a larger area than the other end or at least a different area let me draw so this is one end of the pipe and that's the other end of a pipe I didn't draw it that well and let's say it's filled with some fluid some liquid actually in our example so you know there's just a bunch of there's just a bunch of liquid in this fluid right and let's say let's see let's say this area at the entrance is the area let's call that the area in area in that's the area the area of the opening into the pipe and let's call this area area out area of the opening coming out of the pipe now let's think about what happened what happens if this this liquid is actually moving now let's say it's moving into the pipe with the velocity V in right let's think about how much volume moves into the pipe after T seconds so after T seconds if you think about it every you'd have this much area and then if you think about what was right here it will then be moved to the right by how much well what's we could just go back to our our basic kinematic form in the distance is equal to rate times time so the distance something travels is equals equals velocity times time so after T seconds so T seconds after T seconds whatever fluid was here and it would have an area of about that much whatever fluid was there would have traveled how much to the right it would have traveled let's assume that the the type doesn't change too much in in in diameter or radius from here to here it would have traveled velocity times time so V in times time it could be meters or whatever whatever my our length units are so after T seconds essentially this much water has traveled into the pipe right you could kind of imagine a cylinder of water here and once again I know I made it look like it's getting wider the whole time let's assume that it's which doesn't change that much over over the T seconds or whatever units of time we're working in right so what is the volume of this cylinder of water so the volume in the volume in over those seconds right over the T seconds is equals to equal to the area or kind of the the top or the left-hand side of the cylinder let me draw the cylinder no more vibrant colors you can figure out what the vaada we're trying to figure out the volume of so it equals this side the the left side of the cylinder the input area times the length of the cylinder and that's the velocity the fluid times the time that we're measuring times the input velocity times time all right that's the amount of volume that came in and now if that volume came into the pipe and once again we we learned several videos ago that the definition of a liquid is a fluid that's incompressible right so it's not like you know no fluid could come out of the pipe and and all of the fluid just get squeezed the same volume of fluid would have to come out of the pipe right so that must equal the volume out the volume out right so whatever whatever comes into the pipe has to equal the volume coming out of the pipe and one assumption we're assuming is in this in this fraction of time that we're dealing with is also that there's no friction in this in this liquid or in this fluid actually you know it doesn't it's not turbulent and it's not viscous a viscous fluid is really just something that has a lot of friction with itself and that it won't just naturally move without any resistance and so when something is not viscous and has no resistance with itself and just moves freely without any turbulence that's called laminar flow and that's just a good word to do to know about and that's the opposite of viscous flow and you know different things have different viscosities will probably do more on that like syrup has a very syrup or peanut butter has a very very high viscosity even glass actually is a fluid with very very high viscosity while I don't know I think there's there are some kinds of compounds and magnetic fields that you could create that our perfect laminar flow but this is this is kind of a perfect situation but anyway in these circumstances the volume in because the fluid can't be compressed it's incompressible has to equal the volume out right well what's the volume out over that period of time well similarly we could draw this bigger cylinder right that's the area out and after T seconds what what has how much water has come out well whatever water was here at the beginning of our time period will now be some distance to the will have come out right and we can kind of imagine a cylinder here and what is the the I guess in this case the width of the cylinder well it's going to be the velocity that the liquid is coming out on the right hand side or this is a capital V right capital V is for volume lowercase V is for velocity so it's going to be the output velocity that's a lowercase V times the same time right so that's so what is the volume that has come out in our time T well it's just going to be this area times this width so the output volume over that same period of time is equal to the output area of this pipe times the output velocity times time and once again I know I keep saying this but this is kind of the big aha moment is that in that amount of time this cylinder the volume and the cylinder has to equal the volume in this cylinder so maybe that's you know it's not as wide or something like that but their volumes are the same you can't get more or less water you can't get more water here all of a sudden then then what's going in and likewise you can't put more water into the left side than what's coming out of the right side because it's incompressible so these two volumes equal each other so we know that the input surface area of the or the area of the opening onto the left hand of the pipe times the input velocity times the duration of time we're talking about is equal to the output area times the output velocity times the duration of time we're talking about and if we want you know the time is both on its the same time on both sides of this equation so we could say that the input area times the input velocity is equal to the output area times the output velocity and this is actually called in in fluid motion this is called the equation of continuity and it leads to some interesting things we'll do some problems with it in a second but one thing I want to introduce at this point as well is what is the volume per second because this is also something we're going to deal with in a second and in probably the next video because I'm about to run out of time so we said in T seconds we have this amount of volume coming in and it's the same this coming in is coming out so what is the volume per second well it's this big capital VI per amount of time and we call that flux and we'll learn a lot about flux especially when we start doing vector calculus but flux is just how much of something crosses a surface in amount of time how much of volume crosses a surface in amount of time so in this case the surface is this the top of this the left-hand side of the cylinder and we're saying how much cross is an amount of time we figured out it's that input volume which crosses in every T seconds and this is called flux flux you've probably heard the flux capacitor and in back the future and maybe we can think about what they were trying to hint at but let's see if we can if we can use flux and the these ideas to come up with some other interesting equations let me see if I have enough time so we know that the volume per time is equal to flux this is a big V beat is equal to flux and actually the variable people generally use for flux is R and of course it's it's in meters cubed per second is its units we also know that the input area area input times the input velocity as the lowercase V is equal to the output area times the output velocity such as our output velocity and this is all the equation of continuity in it it holds true whenever we have laminar flow and now oh actually I'm about to run out of time so in the next video I'm actually going to use some of this information to figure out how much power is there in a system where we have fluid going through a pipe see you soon