124103 Lecture 14: Viscous Fluids

追加: 2026-06-03

[00:00:04]good morning everyone good to see you all um in this friday lecture friday class um yeah so welcome to lecture 14 of 124103 um this will be the third and last lecture on fluids so next week we'll go to a different topic um so i'll just give sort of a recap we started talking oh yeah before i begin just just to make sure that uh you can hear me and you can see the slides that i'm sharing oh yeah good cool all right so we started talking about fluids by by defining some some properties that all fluids have um like density pressure and buoyancy or buoyant force so all fluids will have those properties um and we can calculate the density the pressure and the buoyancy of the fluid so that was our very first lecture um and so regardless of whether the fluid is uh stationary or the move the fluid is moving it will have density pressure and buoyancy or buoyant force right and then in the previous lecture in in lecture 13 we then looked at how we can describe a fluid when it flows right so we looked at fluid flow and so we said that when the fluid flows we can describe we can describe its flow by looking at the flow rate q and the flow rate q is related to the volume of fluid that's flowing across a surface per unit time and so that fluid flow is also related to the cross-sectional area of where that fluid is flowing uh times the speed of flow and then we we learned related to fluid flow is that the amount of fluid that goes in into a certain cross-sectional area should be the same amount of fluid that goes out of um another cross-sectional area and that's the idea behind the continuity equation which relates how fast a fluid flows across different cross-sectional areas and then finally we looked at how pressure p and and the speed and the elevation of the fluid uh are are interrelated with each other um so yeah so the first lecture this refers so these are general general concepts they refer to all kinds of fluids regardless of whether the fluid is stationary or moving lecture 13 the topics we learned there and these equations these are also general concepts and general equations that can be used for any kind of fluid that is flowing right so in in the previous lecture we did not really distinguish between different kinds of fluids when we were talking about flow rate and continuity equation in bernoulli's equation so these concepts and these equations can be used to describe the flow of any kind of fluid whether the fluid is viscous or not so we did not distinguish between viscous and non-viscous fluids and so this lecture today uh deals exactly with uh the difference in fluid flow depending on the type of fluid so we will specifically look at how fluid flow is affected by the viscosity of the fluid so of course we will first define what is viscosity and from there we will be able to distinguish the difference between a viscous and non-viscous fluid so so after defining viscosity we can um we can look more deeply at a mathematical um relationship that describes viscosity that will enable us to um to see how viscosity will affect fluid flow and to do that we'll have to review what we learned in lecture 11 about stress and strain but particularly shear stress and strain and from there once we have a good grasp of what viscosity is we can then now look at how viscosity affects fluid flow and this is given by what is called possessed law it's yeah this word is really quite tricky um yeah but i i checked i checked google they say it jose's law is like french french scientist who who investigated viscous flow and then um to wrap up our discussion on fluids we will look at two kinds of flow turbulent flow and laminar flow and and see when we what are the conditions for getting um these types of flow all right okay so let's first this define viscosity viscosity um so we'll we'll first look at a conceptual definition of viscosity to sort of get a conceptual understanding and idea what viscosity is all about and then after that we'll look at the mathematical representation as i mentioned earlier so viscosity is um [Music] represented by this fancy letter n which is a greek symbol that we call eta so we call that eta and that refers to viscosity it's it's a property of the fluid um and it measures the resistance of um the fluid to flow uh yeah it measures the resistance of fluid flow essentially uh in in common speak in common language it's often referred to as the thickness of a fluid okay um so here have a look at this video uh which illustrates which shows different fluids um with different viscosities and and [Music] uh yeah so i'll just so can you hear the sound no okay um let's just see let's see what's going on um just just hang on i just need to um sorry i i i'll just stop the sharing for a moment welcome ladies and gentlemen to our first ever viscosity games where we find out which one of our liquids will flow fastest down our ramp of danger water dish soap vegetable oil honey suntan lotion and in first place we have water second place vegetable oil third place our dish soap fourth place honey and it turns out our suntan lotion was very old okay um so looking at that video which do you think yeah okay so those are the different fluids that were in the race so based on that video and maybe an intuition of what viscosity is which uh fluid do you think is the most viscous meaning the fluid with the highest viscosity right so i can see there a response of honey any other um responses of what you think is the most viscous right okay so viscosity um measures the resistance to flow or the thickness thickness of the fluid yeah so the thicker the fluid is the more difficult it is to flow because there is sort of this internal resistance in the fluid that that prevents it from flowing um and so among the among the different liquids uh water had had the least viscosity it was less viscous because it flowed faster your honey is very viscous so notice here um it flows very slowly but santan lotion the lotion especially uh did not even did not even leave its container so it's it's very very viscous so in in this case with these different fluids uh the lotion would have the highest viscosity and we say that it's the most viscous um among the liquids uh considered all right so the higher the viscosity the more difficult it is for the fluid to flow uh which means you need to exert more force to overcome the internal friction and we say that the fluid is viscous or what is often referred to as being thick okay um so that's the qualitative description of the viscosity now if we look at how we represent viscosity mathematically we'll have to go back to uh shear stress and strain this viscosity is then would then be related to how the fluid is deformed right uh because when the fluid will start to flow uh the the fluid essentially deforms at as it starts to flow and so to understand mathematically how viscosity affects the deformation of the fluid as it flows then let's go back to what we learned in lecture 11 about stress and strain okay stress stress being related to the force that you apply to an object and strain being related to the corresponding deformation that the object experiences because of the force that for the stress that you've applied but in particular it will be shear stress that will be relevant to fluids okay where shear stress uh is the kind of stress where the force is exerted parallel to the surface so a good example is when you put your hand on top of the of a book and if you slide your hand towards the right away from the binding of the book you are exerting force that's parallel to the pages of the book and what you will notice is that the book will start to deform with the top pages moving parallel moving forward in the same direction of your of your for of the force exerted by your hand whereas notice that um at the bottom [Music] the pages hardly moved at all so there's a little there there's little to no movement at the bottom of the book the bottom is of course fixed to the table or the desk and the desk is sort of providing the counter acting frictional force in the opposite direction of so you have this counter actually acting frictional force in the opposite direction of uh the pushing force of your hand and that and that prevents the the pages at the bottom of the book from moving forward you will also see that there's like this gradient in the deformation of the pages of the book so that the pages closest to the exerted force by the hand move farther um compared to the pages uh lower uh closer to the bottom of the book and the stress stress uh exerted on the book is related to the force that you applied divided by the surface area um over which you apply uh the the force and correspondingly the the deformation of the book is measured in terms of the strain okay um and the stress and the strain is related to the young you lose and young's mojulus we learned before is a property of the material property of the material that tells you how easily the material is deformed right so high high um young's modulus means that it's more difficult for the material to deform means smaller deformation or smaller strain um which essentially means if your young's modulus is big then um the strain is small the deformation is small okay so that's what happens to solids okay you have this shear stress sheer stress exerted parallel to the surface and you have this corresponding deformation that you can see in the solid objects now if we apply shear stress and strain to fluids it's a little bit more difficult to see um i mean the deformation of the fluid is a little bit more difficult to see because the fluid is sort of like continuous we don't see uh the individual particles or molecules making up the fluid unlike the pages of the book that we can easily see uh in in the previous example but this uh cartoon this this cartoon video that i'll show you shortly sort of shows us how the particles or molecules in the fluid behaves if we apply a shear stress on the fluid so as you watch the video um i'd like you to observe what will happen to the particles or the fluid molecules on top and at the bottom of the fluids um right so so i i'd like you to observe what will happen to the particles around uh the top part uh and going down uh to the bottom right so i'll show this video now and hopefully the sound will work this time imagine a boat is moving on water the boat water molecules on top layer are also moving all right um i hope you heard the sound all right what's the sound yep okay good yeah so um and i also hope that you were able to observe um let's do that again yeah and i also hope that you were able to observe how uh these particles you know these circles represent the molecules uh making molecules of the fluids okay notice the boat there is applying sort of the shear stress so you have a force that's being applied parallel to the surface of the fluid so that's giving you the shear stress okay which causes the the the movement of uh the particles uh essentially giving uh the deformation of of the fluid so let me just play that once again for those who were not able to uh observe that phenomenon and then we will discuss it more closely imagine a boat is moving on water with boat water molecules on top layer are also moving okay so notice um so if we if we look at a um particular reference point so say if we look at this um let me let me choose a different reference point so if we look at this reference point and let's see how the particles moved uh with respect to that reference point so say try to make that straighter yeah so say let's say this is the line where the particles uh are supposed to be aligned before you applied the shear stress okay you have of course the force to the right giving you the shear stress okay and this is our reference reference point so notice that um on top close to where the the shear stress was applied this particle here moved quite far so if you look at the displacement of that particle it was displaced quite far relative to the reference point and as you go deeper and farther away from where the sheer stress was being applied notice how the displacement becomes smaller and smaller and at the bottom the displacement is almost zero or approximately zero uh and we say that that particle at the bottom did not move at all okay so the displacement of these um water molecules or these these water particles as was a result of the shear stress the force that was exerted parallel to the water surface now if the if we look at the movement or the displacement of the particles over time then we will be able to say something about the velocity of the particles right recall that velocity [Music] from our very first lecture is equal to displacement over time right so so in this case which particle had uh the biggest velocity meaning which particle moved fastest is it the particle at the top or the particle at the bottom at the top right and and hopefully you also saw that in in the video um animation video yeah the particle at the top moved fastest because it it was closest to the sheer force it also had the biggest displacement and therefore it will have the biggest velocity and at the bottom it's sort of like the particle was at rest and it was it did not move um so you can sort of see this deformation of the fluid if you trace out if you trace out yeah the the displacement of individual water molecules you can sort of like this similar behavior in the deformation of the fluid similar to the deformation of the book textbook that we've seen earlier for for fluid for four solids i mean okay so looking at a more idealized experimental setup where you know you you sandwich a fluid between two plates um and you move the top plates here so you have this fluid that sandwich sandwiched between two plates and you move the top plate to the right you will exactly see um a change in the velocity of the fluid depending on where you're looking at so at the top of the fluid you have a faster velocity and as you go deeper uh the fluids you the velocity decreases um at the point where the plate is stationary you can see that the velocity uh is zero uh at that point okay so here um hopefully the key id i wanted to show and to illustrate here in this slide is that when you apply a shear force on the fluid the fluid will also sort of deform in this manner um and and the deformation is represented by the displacement of the individual water uh or fluid molecules uh and that deformation is related to the velocity of the movement of the fluids because displacement um is related to velocity uh by dividing it by time so if we look if we uh now make a mathematical equation that describes viscosity then we have the strain the shear strain which describes the deformation the deformation will be related to the displacement over time and displacement over time is just the velocity of the particles and it should also be related to the length or the position of the molecule so notice here the length this is the length that we are talking about and notice that the deformation depends on um the position relative to where the the shear stress was exerted so at the top part you'll have bigger uh deformation at the bottom part where l is bigger you have almost no deformation so the shear strain which describes the deformation should also be related to um the depth of where where you're looking at the strain and of course how fast the molecules are moving because then that describes the displacement of the particles okay um so that's the deformation and viscosity is defined as the shear stress over um shear strain so looking at analogies uh young's modulus describes how easily the solid object deforms for fluids it's viscosity that defines how easily uh the fluid deforms and therefore viscosity defines how easily the fluid flows and that is described by the shear stress divided by the shear strain stress is of course force divided by area and strain is as we've seen here is equal to velocity over length um we we we will really just look at we we won't look more closely into the direction so we can just use speed instead of velocity um to describe the strain um of the fluid okay and and just to wrap just to complete the discussion for viscosity the si unit for this viscosity is past calls uh divided by seconds it's sort of a strange unit but force the unit for force is newtons divided by the unit of area is meter squared divided by the unit of speed is meters per second squared divided by the length is meters um this will work out to be uh pascal so so on top here is calls um yeah and then it will work out to be seconds there um uh so this equation can be rearranged you mean this equation for viscosity uh yes this equation can be rearranged uh and we will do that in the next slide you can of course write this yeah because it's quite weird to have all those um divide signs so you can write this also maybe in a meter form force times length divided by velocity or just speed times area so this will be a simplified form for viscosity if you want to rearra if you want to remove like this divide signs um on top and below the divide signs it was written this way just to show because shear stress is force over area and shear strain is equal to um velocity over length but of course i think yeah i think you're right this is a much simpler um equation to deal with yeah so so yeah use this equation for viscosity um [Music] uh so if it was pascal divided by second um [Music] oh sorry i have so the unit for four velocities should just be meters per second meters per second square so let's try to work out yeah okay so so newtons per meter squared this one's this one is already pressure that's equal to pascals and then you have here yeah so this is where it's tricky now when when you're when you're doing this divide divide signs um in both the numerator denominator of a division as well this one the meters will actually cancel out and the seconds is in the bottom of the divide side so that will actually go up i think it will be easier to work out the units looking at this rearranged simplified form for viscosity so if we look at the units from here so force is newtons length is meters so times meters divided by um speed is meters um second to the minus one or meters per second and the area is meters squared okay so looking at this um so this meter will cancel out okay and then you have newton divided by meter that's pascals newton divided by meters squared is pascals so that newton divided by meters squared is pascals and then you have the second to the minus one here so if that will go up yeah that will be pascal's time seven yeah so it's like pascal's time seconds so that's why it's a little it looks like a strange unit it's past calls times [Music] time seconds for the of of viscosity right okay um so we have that equation for viscosity now if um based on uh viscosity we can classify uh fluids into two um we can classify them as non-newtonian fluids and newtonian fluids um in newtonian fluids the viscosity changes with stress and and strain so um this this means that the this means that because the viscosity changes it means that the fluid will thicken or can thicken or can become more viscous or the fluid can thin out or can become less viscose when stress or when force is applied to it and a good example of that is when you you know if you when you're baking and you mix flour or corn starch with water okay uh initially when you mix flour and cornstarch together you start from a thin fluid and but as you mix it continuously mix it together um then that mixture starts to thicken um and that's because you're as you mix you're applying more stress to the mixture and that changes the viscosity of the mixture making it more thick so it's also when you when you make slime um yeah i do that often with my with my daughter so you have to make to homemade slime you have glue um glue and um dishwashing soap so if you mix those two two together initially it's quite runny and liquidy but you actually continue to steer and mix them together then this line becomes to form and it becomes thicker becomes more viscous and that's a result of the stress uh that that was introduced by mixing mixing continuously so that's the idea behind non-newtonian fluids their viscosity changes okay for the case of corn starch and in slime the viscosity increases as they become more viscous when stress is applied for the example of thinning out or the fluid becoming less viscous as you you apply stress that's the idea behind quicksand right so they say that if you are if you encounter quicksand for example you step on the quicksand the best thing you can do is to stay stationary right so you probably heard that if you struggle and try to struggle and try to move about to get yourself out then you'll be in more trouble because you you it will be more difficult for you to get out the more you struggle and that's because quicksand is not newtonian and the more uh it's stressed the more you struggle and apply force and stress to it the more it thins out and the more yours you sink into quicksand okay for non-newtonian fluids the viscosity does not change regardless of whether you you keep mixing it or regardless of whether you you continue introducing stress to to the newtonian fluids um actually newtonian fluids are ideal there's in real life there really isn't any newtonian fluid but a good approximation air and water can can sort of be approximated to be newtonian fluids so yeah so i but i think yeah i think in this course we'll just mostly deal with we'll we'll approximate fluids to be newtonian so that the viscosity uh does not change um and the viscosity can be considered to be constant right so let's look uh more closely at how viscosity will affect fluid flow um viscosity this was the definition we can rearrange it as well and also simplify simplify the equation and the goal of rearranging this is to see how um the viscosity affects uh the fluid flow in terms of the force you need to apply to make the fluid flow as well as the speed of fluid flow so by rearranging the equation for viscosity this way so that you have a relationship between force and uh the viscosity eta we can see a direct proportionality direct proportionality between force and viscosity okay which means that again direct proportionality we've learned this before means that if one something increases the other thing should also increase which means that if viscosity increases then force should also increase okay the more viscous the fluid is the more force is needed the thicker the fluid is the the more force uh it's needed to cause it to flow um and and maybe a good example of this is um maybe um lotion or let's say tomato sauce okay uh tomato sauce in in in a bottle sometimes you squeeze the end of the bottle right tomato sauce is quite thick um and sometimes so you have to squeeze uh the bottle in order for the tomato sauce to flow out of the bottle you have to like exert this force and and the thicker the sauce is the more you need to squeeze squeeze um so that you overcome the viscosity or the thickness of the tomato sauce and and it will cause the the tomato sauce to flow right so um the force you need to apply is directly proportional to the viscosity of the fluid so likewise the speed um of fluid flow is directly proportional to the force which probably makes sense um because right the the faster the the more you pull on this top plate for example then the faster the water molecules on top will will move so so i think hopefully this this relationship between force and speed is rather logical the more you push the faster the flow will be right so then we've learned about viscosity um we've seen what viscosity is and how it's related to to the force and the flow of the fluids so for non-viscous fluids the fluids can flow uniformly so it's very thin um so there's there's not a lot of internal resistance in in the fluid so the fluid sort of flows uniformly with with a uniform speed throughout the fluid for but for a viscous fluid the velocity is non-uniform if we look at what we looked at here is is one plate is moving okay just the top plate is moving and the bottom plate is stationary and we've seen with the cartoon or the illustration of the boat and the water molecules that the fluid on top will flow faster and the fluid at the bottom sort of is close to stationary it's not moving because because the bottom is stationary now in fluid flow through a pipe okay it's like you have two stationary plates uh on top and at the bottom okay so if we put if we do the same analysis that we did previously with a single moving plane okay so the water or fluid molecules near the stationary plates will will sort of be stationary so they have zero velocity and they will be stationary they're sort of they they will probably move but only just a little bit but the fluid at the middle will have uh a faster or a a bigger velocity um that's where the force is also being applied close to the middle of of the pipe where the fluid is flowing so if you look at the profile of the velocity so um at the middle you have this fast velocity and and it's tapering down to almost zero velocity near the stationary plates so for viscous fluids you have this non-uniform distribution of velocity okay we also seen with the tomato sauce example that i i described earlier that for viscous fluids to flow um you will sort of have to apply some sort of pressure uh to cause to overcome the viscosity and and cause the fluid to flow so this is governed by posse's law so possess law says that you need to have a pressure difference between two points um to cause uh the fluid to flow and jose's equation is given here it describes the flow rate q still refers to the flow rate of the fluid but it now depends on the pressure difference so p1 minus p2 [Music] that's the pressure difference between uh two points in in the fluid um [Music] so r it's the radius of the pipe where the pipe is flowing um it's easy to imagine that bigger pipe will have more fluid more fluid can flow through it so you'll have a higher flow rate we've also seen that last time um viscosity is here so that means fluid flow is inversely proportional to the viscosity and we've been discussing that the thicker the fluid is um the less likely it is to flow so it will have a smaller flow rate the bigger the viscosities also the longer the pipe is the smaller the fluid flow okay so length is inversely proportional to the fluid flow or the flow rate and of course you have pressure there yeah so if it's easier for you you can also rearrange this equation to be pi r to the 4 multiplied with p 1 minus p 2 divided by 8 eta times l maybe this form of the equation will make it easier for you to rearrange possessed equation um for example if you wanted to find p1 minus p2 these equations are equivalent okay so let's look at an example of how we apply possessed equation and and how viscous fluids flow so drug is delivered to a patient by a syringe that is 5 cm long and with an internal diameter of 1 millimeter and the drug is being being delivered at the rate of 10 milliliters per minute we want to find the flow rate of the drug if the viscosity is 8.9 to the minus 4 pascal's time seconds and we want to find the pressure in the syringe the pressure here essentially p2 okay um let's assume that it's a newtonian fluid so the the viscosity is constant it does not change uh so determining the flow rate okay we've seen this equation for flow rate for viscous fluids you sort of need a lot of information to find the flow rate using this equation for one you need to know the pressure difference p1 minus p2 but in fact you don't know the pressure difference but because that's being asked in the second question so what should we use to determine flow rate can we determine flow rate well i put an answer there so it means it should be possible to determine the flow rate here but how do we determine the flow rate what is the flow rate okay um so the the concepts as i mentioned in the beginning of the lecture the concepts we learned in lecture 12 and lecture 13 you can use those concepts even if the fluid is viscous because those are general concepts that apply to all fluids okay so the definition of flow rate is the volume of fluid flowing per unit time right which is uh yeah volume divided by time which in fact is already given in the problem 10 milliliters per minute so me liters is a unit for volume melee is just the prefix 10 to the minus 3.

[00:46:22]so this milli means 10 to the minus 3 the prefix meaning and then the time is also so every minute 10 milliliters of drug is being delivered so that's essentially the flow rate it was already given except that it's not in sie so all you have to do is to convert the units to s i units so convert milliliters to cubic meters for it for volume um 1 liter is 10 to the minus 3 meter cube as we've seen in the previous lecture that's the conversion factor we need um merely of course this is the prefix milli which is equal to 10 to the minus 3 so you have 10 milli is 10 times 10 to the minus 3 so that's 10 milli and then um one liter using your conversion factor you can replace one liter with 10 to the minus three cubic meters and so you have the conversion from 10 milliliters to cubic meters here and of course for the time one minute is equal to 60 seconds so um substituting the numbers there you'll have a flow rate in cubic meters per second as shown here all right are we happy with that any questions okay are we happy with the solution for the flow rates of the drop yep good okay second question uh find the required pressure in the syringe okay so the reason why you push on the syringe is to apply a pressure across those two points you have the pressure in the syringe and the pressure on the arm okay the pressure will be different because of course you have um the blood pressure uh that you're considering in in the arm there so using prosaic's equation that's where pressure comes in right you need to apply um a pressure difference for flow to occur okay so we want pressure difference so we rearrange this equation so p1 minus p2 it's being multiplied with this whole pi r4 so we just divide divide both sides pi r4 okay ir4 then um it's p1 minus p2 is being divided by eight eta times l so we multiply by eight eta times l both sides and we have this rearranged equation um for p1 minus p2 and we know everything okay the length is given 5 centimeters everything should be in base units so it should be converted to meters the flow rate q we've solved that from the previous question the viscosity eta is given here it's already in base units and then the radius is the diameter is given as one millimeter and radius is half the diameter therefore radius is point uh um is 0.5 millimeters but convert that to meters uh by dividing by a thousand and so you have point zero zero zero five meters for the radius so you have everything just be careful that's that the radius is raised to the power four okay um it's easy to lose that when you're putting numbers in your calculator so just be aware of that so you'll have this pressure difference to be about 300 pascals right okay any question about the solution or are we happy to move on yep good okay so um for sales equation that's that's only applicable for viscous fluids now when fluids flow when when fluids flow um they can either behave uh certain in a certain way depending on how fast they flow and so we can classify flow um as laminar or turbulent depend depending on how fast the fluid is flowing and and how uh the fluid behaves the fluid molecules behave uh while they're flowing so in laminar flow as shown in the top figure here notice how um the stream lines are ordered and and they're sort of directed towards uh a single direction okay so the streamlines are are ordered and and therefore the pressure the density and the flow velocity of the fluid is the same at every point and we can only have laminar flow if the fluid flows at a low speed for example here [Music] when the stream is flowing calmly then you say that that's the laminar flow turbulent flow for example is what you observe when you have a waterfalls for example so that you have this more complicated um orientation of the the fluid molecules as they flow so the stream lines become more irregular and they form vortex like features vertices or virtual like features for example when you have rapids or water flows um so it all depends on how fast the fluid is flowing whether you have laminar or turbulent flow the question is what's the reference speed right okay so when we say turbulent uh flow occurs when fluid flows fast okay how fast is fast how fast is fast is defined by the reynolds number re so reynolds number r subscript e gives you the flow rate for when turbulent flow occurs and it depends on uh these variables uh it's if it's easier for you also you can write down ray nodes number as density times diameter times flow rate divided by the area times the um flow rate so i'm just checking if i actually have uh should have a slightly definition so rho that's the density density of the fluid that's flowing d is the diameter diameter of the pipe where the fluid is flowing a is the cross-sectional area um a ties of course the viscosity and q is the flow rate so flow rate flow rate that's a times b so that's where the speed comes in um when determining whether uh you have turbulent or laminar flow and if the raynaud's number is greater than 3000 that will give you turbulent flow um if the raynolds number is less than 2000 that will give you laminar flow uh between so there's a little bit of a gap between 2000 and 3000 that's sort of like the transition region where you have internship interchanging turbulent and laminar flow so to determine whether the flow is turbulent or laminar you'll have to calculate for the reynolds number uh given these parameters here right let's look at an example too too um okay so let me just see there's a question okay so can a syringe question use bernoulli's equation if it's not a viscous fluid the answer is yes so you can use bernoulli's equation bernoulli's equation can be used for viscous and non-viscous fluids but for pose equation it's you you can only just use possessed equation for viscous fluids you cannot apply meaning meaning you cannot apply possessed equation for non-viscous rules right um we we will not specify which equation you you need to use i think it's part of what what's being tested in the question but you can sort those you can sort of um deduce what question what equation to use based on what information we will give you so if we will um if if the if the question requires you to use possessed equation we will definitely give you the viscosity eta because possessed equation depends on viscosity eta um so that's a clue that you can use if we give you the viscosity eta then that means we want you to use fersae's equation or if we give you the length um the length of the pipe then that's also a key clue that you should use for says equation note that the viscosity and the length l they are not in bernoulli's equation um so can can we use bernoulli's for for for viscous fluid uh yes bernoulli's equation it's also valid for viscous fluids because bernoulli's equation relates um pressure speed and elevation so if you think about a viscous fluid if you change the elevation then the pressure would also change right and and the difference in pressure will will cause the fluid to flow so like the video of the competition between um the different kinds of fluids flowing down the ramp okay you're actually having a different elevation elevation at different points of the ramp for the fluid to flow so you have here a change in elevation you also have there a change in speed between two points in the ramp um and and you have their the density of the fluid so so thinking about that you can then use bernoulli's equation for honey for for for to analyze the flow of honey down the ramp for example um yeah so bernoulli's equation is a general equation that can be used for any kind of fluid but possessed equation is only specific to viscous fluids yeah i hope that makes sense more yeah okay good all right so so turbulent and and laminar flow let's look at an example of the types of questions relating to reynolds number and turbulent and laminar flow so in the previous example for the drug delivery we want to determine whether the flow of the drug is turbulent or laminar um if the density of the drug is is 1050 kilograms per cubic meter so from the previous previous um example we already saw for the flow rates of the drug uh out of the syringe the diameter of the syringe was also given in the previous uh scenario previous example and then the viscosity of the of the drug was was also given and then there's a follow-up question here okay um what will happen if the flow rate is increased but first let's look at the first question with the previous uh parameter so so i've already written it down here let's determine whether the flow is laminar or turbulent a reynolds number is this equation density times diameter times the flow rate divided by the cross-sectional area times the viscosity so area we can assume that the syringe is a cylinder so if you cut through the syringe and look at one end you'll see a cross-sectional area a cross-sectional shape of a circle and the area of that circle is pi r squared okay so the radius is half the diameter okay remember to convert that to so convert that to meter so point 0.001 meters for the diameter have that that will give you the radius so it's point zero zero five um for the radius squared giving you this area right here so we know everything is just a matter of substituting so the density of the drug times the diameter of the syringe it has to be meters so everything should be in base units the flow rates divided by the cross-sectional area multiplied with the viscosity it gives you a renal number of 251. so do you think the flow is turbulent or laminar yeah laminar so it's really just comparing the reynolds number that you've obtained to the criteria for turbulent and laminar flow so for turbulent flow that's when you have reynolds number more than three thousand okay the reynolds number is um less than two thousand so that will be so if the flow rate is increased to 250 milliliters by per minute so that means that's a factor of 25 because originally um [Music] originally the flow rate was uh 10 milliliters per minute right we could we just converted milliliters to meter cube and then minutes to seconds but originally the flow rate was 10 milliliters per minute so if you increase it increase it to 250 milliliters per minute then that's a factor of 25 you've increased the flow rate by a factor of 25 so how will the raynaud's number change so do you think it will still be laminar or turbulent so the reynolds number when the flow rate was 10 was 251.

[01:03:49]do you think it will still it will be laminar or turbulent flow if we increase q by a factor of 25.

[01:03:59]right turbulent good so we need to figure out what the new reynolds number is so from the previous question reynolds number is 25 q increases 25 times notice how q is directly proportional to the reynolds number right so if q is doubled the reynolds number should also double in this case q is 25 times bigger than before so the reynolds number should be 25 times bigger than um than before yeah yeah so turbulent good you need you need to do the equation so you either do this equation um again substituting density diameter uh then you q um divided by area times m or you can just you can use the previous reynolds number you got 251 and um increase that by a factor of 25 or multiply that by 25 then you get you then see that the reynolds number is more than 3 000 therefore it will be turbulent flow okay yeah so it's really just determining reynolds number and comparing it to the criteria for turbulent flow so since q is on top so q is on top um means that it's directly proportional to the reynolds number uh 6 to 90 it's quite close to the new reynolds number um so maybe it's just a rounding [Music] is it just a rounding thing so so if you let me just let me just double check that 251 so 51 times 25 does give you 6275 um so i wonder if it's just a rounding thing 1.67 [Music] yeah um so multiply 25 with two five one this one or which q so q q from the the original q so 10 milliliters um oh okay so if you multiply if you multiply the original q by 25 then you also still have to yeah i think it i think it's just the rounding of q so it's the rounding of q there that when you multiply that with 25 the rounding error increases um and then you still need to multiply that with uh density multiply that with the diameter and then divide that with yeah yeah okay okay so you got it so it's just it's just a rounding error but but what you got is actually quite close to the answer 6275 is is close to 6 to 90 so i yeah that's why it's just surrounding um in in the test or exam uh it shouldn't be a problem rounding should be a problem should it be a problem because if it's multiple choice uh we will give you numbers that are very far from each other and rounding errors shouldn't affect um the the correct answer um yep okay so there's here a question for more if it's on top is that directly proportional and multiply is it inversely proportional so if it's on top so reynolds number uh like this okay so it's all a matter of uh so so the it's implied that if you have a variable that's written just by itself that is divided by one but we don't write the divided by one anymore because anything you divide by one is the same as itself right so that means if you write reynolds the reynold re on the left in the same form as the equation on the right then then the equation will look like this strictly speaking you have um re divided by one that means you compare if you are doing direct proportionality or inverse proportionality you're comparing um like um the quantities on top with the the quantity on top on the left with the quantity on top on the right see so this is direct proportionality okay because they are both on top of the denominator sign so that means density is directly proportional to ray nodes uh diameter is directly proportional to uh our e and q is directly proportional to re so yeah um and then a and n they're inversely proportional so these things here inverse uh really proportional to the reynolds number and these ones here on top are directly proportional so um yeah so if you swapped a times eight for reynolds that oh okay so um yeah you you're right so if you swap that you're right if you swap aata row dq over reynold if you stop that it becomes like that but when you're comparing two quantities um and saying directly proportional inverse is proportional those two quantities should be at opposite sides of the equal sign okay so we can only conclude direct or inverse proportionality uh between quantities that are at opposite sides of the equal sign so in this case it should be the the equation should be in this form so that ray nodes is in one side and q is on the other side of the equal sign if we if we put the two quantities in the same side of the equal sign then it's hard to see um what will happen because the effect will cancel each other out um yeah and it's hard to see what will happen if you increase q then what you can solve is actually a times n if that makes sense with the second equation here what what we are solving is a times n and not the reynolds number so it's hard to see what will happen to the reynolds number if you change any of uh this density d or q okay so when you're comparing um direct or inverse proportionality the two uh quantities should be at opposite sides of the equal sign so that if you uh change q for example then you will be able to solve for reynolds number um yeah i hope that that's uh that's clear i'm not sure if that makes sense more so uh direct proportional means ray node goes up then rho d goes up yes yeah so directly proportional means reynolds go up goes up if rho times d times q will go up and yeah yeah yeah that's that's right and the row dq goes up when eta uh goes down and vice versa yeah yeah you're right so i think i think you got the idea there all right okay um so let's um yeah i think just so in this example the reynolds number um it's more than 3 000 so that will be a turbulent flow um oh yeah and then and that's that's the end of the lecture yeah okay all right um so that's the end of the lecture on on viscous fluids um we first defined what viscosity is and then we tried to look at the mathematical representation of viscosity and from there we were able to see how viscosity affects fluid flow and uh posse's equation tells us that for a viscous fluid to flow there has to be a pressure difference uh essentially providing that force and that will push uh the viscous fluid and cause it to flow and um the kind of flow depends on the reynolds number um calculating what the reynolds number is and comparing it to the criteria for turbulent flow which is greater than 3 000 reynolds number and laminar flow which is less than 2 000 um reynolds number okay so that wraps up our discussion on fluids the workshop um so so feel free to type in if you have any questions regarding fluids i'll keep the annotations and post that on the stream later on but the workshop workshop 14 relates to viscous who is what we've just been talking about so i will um i'll give you some time to look at workshop 14 and start working through the problems of course if you have any questions while we're working through the problems feel free to put them in the chat box i'll still be here monitoring the chat um maybe let's meet again and say how about at 11 30 do you think it's just 15 minutes okay 11 30. let's meet again at 11 30 uh and let's look at solutions to the workshop problems i won't end the zoom session so you'll stay here but i'll just end the 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