1 00:00:16,580 --> 00:00:21,710 Welcome to this lecture number 8 on this NPTEL course on fluid mechanics. For fluid mechanics; 2 00:00:21,710 --> 00:00:28,180 for under graduates students in chemical engineering and until the last lecture we discussed fluids 3 00:00:28,180 --> 00:00:34,610 under static conditions, the variation of pressure in a fluid under static conditions, 4 00:00:34,610 --> 00:00:39,010 the forces that are experienced by objects submerged in a fluid under static conditions 5 00:00:39,010 --> 00:00:44,870 and so on. Before, I move on to the new topic, I am going to quickly recapitulate the key 6 00:00:44,870 --> 00:00:47,110 results that we discussed so far. 7 00:00:47,110 --> 00:00:54,110 So, under fluid statics, we first saw that if a fluid is under static conditions, you 8 00:00:57,780 --> 00:01:04,780 should take any volume element. This is a volume element in a fluid. Then, the forces 9 00:01:06,920 --> 00:01:12,799 that are exerted by the fluid that is present outside on the fluid that is present inside 10 00:01:12,799 --> 00:01:19,799 is purely normal and it is acting in a compressive sense. So, the force that is exerted for the 11 00:01:22,830 --> 00:01:29,380 fluid that is present outside on the fluid is present inside is denoted by R. And if 12 00:01:29,380 --> 00:01:35,390 you take any particular point on this volume element, this force is a function of the unit 13 00:01:35,390 --> 00:01:42,390 normal. So, since the; so, if you take any point on this surface, the unit normal point 14 00:01:43,980 --> 00:01:50,420 outwards that it is from inside to outside. The force on a static fluid element acts from 15 00:01:50,420 --> 00:01:57,250 outside to inside. So, this force is written is written as p 16 00:01:57,250 --> 00:02:03,340 times minus n because minus n is the direction at which the force is acting and p is the 17 00:02:03,340 --> 00:02:08,390 magnitude of the force. Well, it is more appropriate and this is the magnitude of compressive force 18 00:02:08,390 --> 00:02:14,060 per unit area acted upon by the fluid outside on the fluid that is inside, on the surface 19 00:02:14,060 --> 00:02:19,530 that separates outside and inside. So, this compressive force per unit area, the magnitude 20 00:02:19,530 --> 00:02:26,530 of it is called the pressure in a fluid. And we saw that this pressure has units, pascals. 21 00:02:29,010 --> 00:02:35,069 So, 1 Pascal is 1 Newton per meter square. 22 00:02:35,069 --> 00:02:42,069 And then we said that if you have a fluid that is under the influence of a gravitational 23 00:02:42,730 --> 00:02:49,730 field. So, we took a coordinate system x, y and z and gravity acts in the direction 24 00:02:52,640 --> 00:02:59,640 of minus z direction. So, if you look at; if you want to write the gravity vector. It 25 00:03:01,470 --> 00:03:08,069 is minus g times k. The g is the acceleration due to gravity, it is 9.8 meter per second 26 00:03:08,069 --> 00:03:14,470 squared and the surface of the earth and k is the direction of the positive z axis. Minus 27 00:03:14,470 --> 00:03:19,610 k is the direction of negative z axis. So, g is minus, the g vector is minus g times 28 00:03:19,610 --> 00:03:24,830 k. So, we have a fluid that is present under the influence of a gravitational field and 29 00:03:24,830 --> 00:03:31,830 we said that by taking tiny volume element of a fluid. We can show that the fundamental 30 00:03:35,470 --> 00:03:42,470 equation of hydrostatics is minus del p plus rho g is equal to 0. 31 00:03:43,459 --> 00:03:50,459 So, this is the fundamental equation for a fluid under static conditions. This the physical 32 00:03:51,360 --> 00:03:58,250 interpretation for this equation is as it is very simple rho g is the force per unit 33 00:03:58,250 --> 00:04:05,250 volume, volume due to gravity. This is the weight of the tiny volume element and this 34 00:04:07,770 --> 00:04:14,770 is del p is the net pressure for force minus del p. So, net pressure force per unit volume 35 00:04:17,660 --> 00:04:22,660 acting on the fluid element and since a fluid element is at rest the sum of all the forces 36 00:04:22,660 --> 00:04:26,340 must be 0 and that is what gives rise to this simple balance. 37 00:04:26,340 --> 00:04:33,340 Now, normally this equation, if you remember that this is very general equation in the 38 00:04:34,770 --> 00:04:41,770 sense that this is valid for any coordinate system. Not necessarily the rectangular coordinate 39 00:04:43,789 --> 00:04:50,789 system depicted in this cartoon valid for any coordinate system. So, but if you want 40 00:04:54,499 --> 00:04:58,559 to solve a problem you refer this equation with respect to this partition or a rectangular 41 00:04:58,559 --> 00:05:05,559 coordinates x y z and g pointing like this. So, we found that minus dp dz minus rho g 42 00:05:11,779 --> 00:05:18,779 is 0 or dp dz is minus rho g. 43 00:05:19,659 --> 00:05:24,999 So, if you want to integrate this equation we have to assume certain things about density. 44 00:05:24,999 --> 00:05:29,159 If you assume incompressible flow, sorry, incompressible fluids for which density is 45 00:05:29,159 --> 00:05:36,159 independent of pressure, rho is independent of pressure p. Then we can and since g is 46 00:05:41,680 --> 00:05:46,360 a constant both rho and g are constants in this equation. So, we can easily integrate 47 00:05:46,360 --> 00:05:53,360 this. So, you get integral dp between any 2 points p1 or p0 and p is minus rho g integral 48 00:05:57,169 --> 00:06:04,169 z naught to z dz. So, p minus p0 is rho g times z0 minus z. Now, in most applications 49 00:06:09,539 --> 00:06:16,539 you have a pool of liquid that is water let say and that is open to an atmosphere, air 50 00:06:19,210 --> 00:06:24,419 at atmospheric pressure. So, this is p atmosphere, this is the free 51 00:06:24,419 --> 00:06:31,419 surface that separates the liquid from air, water from air. For example, so at this point 52 00:06:33,909 --> 00:06:40,909 the pressure is p atmosphere, it is known. So, our coordinate system is align like this, 53 00:06:42,710 --> 00:06:49,710 x y z, where z is increasing in this direction. So, if you refer z naught in this equation 54 00:06:50,110 --> 00:06:57,110 as the free surface z equals to z naught and any point z, is this distance is z naught 55 00:07:02,099 --> 00:07:05,629 minus z and we can call that as h. 56 00:07:05,629 --> 00:07:12,629 h is the depth of the liquid from the free surface. In which case, we can rewrite this equation as p minus p0 atmosphere because 57 00:07:24,499 --> 00:07:31,499 that is the value the pressure at z equals to z0. p minus p atmosphere is rho gh or p 58 00:07:33,809 --> 00:07:40,809 is p atmosphere plus rho gh. This is a fundamental equation valid for incompressible fluids because 59 00:07:45,229 --> 00:07:52,229 we assume rho is constant and it is valid when gravity is in the direction of minus 60 00:07:55,770 --> 00:08:02,679 k. So, these are the two key assumptions and then this equation is very useful in solving 61 00:08:02,679 --> 00:08:08,639 many problems. One example is what we did was to show, suppose 62 00:08:08,639 --> 00:08:15,639 you have a free surface and then you have a solid surface, solid planar solid surface. 63 00:08:17,689 --> 00:08:22,839 And then we are interested in finding the force, net force, effective force on this 64 00:08:22,839 --> 00:08:29,839 solid surface and that force will act normally what is the magnitude of force which we call 65 00:08:33,959 --> 00:08:38,990 FR and what is the line of action of the force? 66 00:08:38,990 --> 00:08:45,990 So, both of this we calculated by simply. So, this is a cross section of a planar surface 67 00:08:47,779 --> 00:08:53,300 like this. So, this is the planar surface and you take a tiny area element and then 68 00:08:53,300 --> 00:09:00,300 we find what is the; that is called as dA. The pressure the force acting is p dA and 69 00:09:00,490 --> 00:09:05,710 then you integrate this you get the total force FR. So, it is as simple as said and 70 00:09:05,710 --> 00:09:11,390 then by equating the moments we found what is the line of action the last lecture. The 71 00:09:11,390 --> 00:09:18,390 next application of this simple formula that p is p atmosphere plus rho gh was to calculate 72 00:09:18,830 --> 00:09:25,830 the force on a submerged object. Suppose, you have a solid object that is submerged 73 00:09:26,470 --> 00:09:32,320 in a fluid and this is atmospheric pressure. The fact is that because gravity is acting 74 00:09:32,320 --> 00:09:39,210 in this direction, the force the pressure on this side will be smaller than the pressure 75 00:09:39,210 --> 00:09:45,710 on this side because of this column of liquid that increases the pressure by rho gh. 76 00:09:45,710 --> 00:09:52,710 Then, we saw that by taking a tiny cylindrical volume element and we saw that the net force 77 00:09:55,160 --> 00:09:59,770 acts upwards because of the fact that the pressure is more than that the pressure here. 78 00:09:59,770 --> 00:10:05,950 And by taking many such tiny cylindrical volumes over the entire volume, we saw that the net 79 00:10:05,950 --> 00:10:12,950 force acting upwards, the direction upwards to the gravity vector. That is the upward 80 00:10:13,030 --> 00:10:20,030 direction upwards on a submerged object is simply rho of the liquid, density of the liquid 81 00:10:26,640 --> 00:10:33,640 times g times the volume of the liquid that is been displaced by the solid. Here the entire 82 00:10:34,820 --> 00:10:39,360 volume is submerged. So, the entire volume of the solid itself volume of the liquid that 83 00:10:39,360 --> 00:10:46,360 is displaced by the solid. If the object is partially submerged like here only this part 84 00:10:55,090 --> 00:11:02,090 is submerged, then is a floating object. By the by the way this force is called the 85 00:11:03,180 --> 00:11:10,180 buoyancy force and this principle is Archimedes principle. So, for a floating object, the 86 00:11:14,420 --> 00:11:20,100 displaced volume is let us call it as V displaced. So, if on object is floating that means that 87 00:11:20,100 --> 00:11:25,260 it is not completely sinking into the liquid. Gravity is acting down, if M is the mass of 88 00:11:25,260 --> 00:11:30,300 the solid and g is the acceleration due to gravity, Mg is the weight of the object, solid 89 00:11:30,300 --> 00:11:37,300 object, weight of the solid. This must be balanced by V displaced times rho times g. 90 00:11:40,340 --> 00:11:47,340 Now, the mass of the solid is nothing but, rho solid times V solid, the volume of the 91 00:11:48,910 --> 00:11:55,910 solid times g is equal to V displaced time rho liquid times g. This is condition for 92 00:11:57,580 --> 00:12:04,580 floating, condition for a body to be at equilibrium for a body to be floating in a liquid surface. 93 00:12:12,820 --> 00:12:19,820 So, this is Archimedes principle. What I will do next is to illustrate this with a straightly 94 00:12:22,630 --> 00:12:28,840 more involved example. So, let us try to apply principle for a following problem. You have 95 00:12:28,840 --> 00:12:35,840 a huge you know swimming pool let us say huge body of water. This is the water level is 96 00:12:38,660 --> 00:12:45,660 water and you have an object let say a raft is floating and over the raft there is a barrel, 97 00:12:50,640 --> 00:12:57,640 a cylindrical barrel. This is a raft, this is a barrel. So, this is water so this is 98 00:12:59,670 --> 00:13:06,670 gravity is acting in this direction and let us say the displaced volume is V. In this 99 00:13:09,120 --> 00:13:11,820 case, let us call this situation case A. 100 00:13:11,820 --> 00:13:18,820 Let us consider two other situations, a second situation is where you have same barrel and 101 00:13:21,560 --> 00:13:28,560 raft, but the barrel and the raft are floating separately. They are partially submerged, 102 00:13:34,440 --> 00:13:41,440 both are floating. So, let us call the displaced volume as V raft and this is V barrel. So, 103 00:13:44,740 --> 00:13:51,740 their floating separately let us call this case B. And the third a situation is like 104 00:13:53,410 --> 00:14:00,410 so, you have free surface of the water and then you have the barrel, but sorry, you have 105 00:14:06,060 --> 00:14:09,480 the raft and the barrel is such that it is completely submerged. 106 00:14:09,480 --> 00:14:16,480 So, let us call this displaced volume. We are in the both cases of the raft. So, this 107 00:14:20,920 --> 00:14:27,920 is the displaced volume of the barrel, let us call this case C. Now, in comparison this 108 00:14:30,370 --> 00:14:36,910 three cases, what is the level of water of case B and case C with respect to the lever 109 00:14:36,910 --> 00:14:41,700 of water in the swimming pool in case A? That is question that we are going to answer by 110 00:14:41,700 --> 00:14:43,570 using the Archimedes principle. 111 00:14:43,570 --> 00:14:50,570 So, let us consider case A. Here, the both in case A, in both barrel and raft are floating, 112 00:15:03,150 --> 00:15:10,150 but the displaced volume is only due to the raft. By using Archimedes principle, the net 113 00:15:17,200 --> 00:15:24,200 force downwards is given by the sum weights of the raft and the barrel times g. 114 00:15:29,060 --> 00:15:33,810 This must be balanced if this combination is to be floating, then this must be balanced 115 00:15:33,810 --> 00:15:40,810 by buoyancy upwards. The buoyancy force upwards which is nothing but, rho water, which I denote 116 00:15:42,610 --> 00:15:49,610 as rho. This is the density of water times Vr, let us call it just V that is the notation we used times g. So, condition for 117 00:15:59,850 --> 00:16:06,850 the floating in the first case, case A is Mr plus Mg, Mbg is so rho vg. In case B, barrel 118 00:16:11,380 --> 00:16:18,380 floats separately, the raft and barrel float separately. That means that M raft times g 119 00:16:31,110 --> 00:16:38,110 is V raft times rho g and M barrel times g is V barrel times rho g. This two must separately 120 00:16:42,890 --> 00:16:49,890 hold for both this things to float separately. We can add this two equations Mr plus Mb times 121 00:16:53,360 --> 00:17:00,360 g is equal to Vr plus Vb times rho g. Let us call this equation 2, let us call this 122 00:17:04,020 --> 00:17:11,020 equation 1. When I compare these two equations, this implies 123 00:17:11,100 --> 00:17:18,100 that and Vr Mr plus Mb is same on the left side. So, rho Vg must be equal to the right 124 00:17:18,900 --> 00:17:25,900 side must also be the same must be equal to rho Vr plus Vb times g. So, if I cancel rho 125 00:17:27,039 --> 00:17:34,039 density of water and acceleration due to gravity. I get that V is Vr plus Vb. That is in case 126 00:17:34,070 --> 00:17:40,879 B, where the barrel and raft are floating separately. Here, the net displaced volume 127 00:17:40,879 --> 00:17:46,950 is the same as the total displaced volume in case A. So, no change in the water height, 128 00:17:46,950 --> 00:17:53,950 in the level of water in the swimming pool. That is the conclusion; we can come to just 129 00:17:57,149 --> 00:18:00,279 by applying Archimedes principle to these two cases individually. 130 00:18:00,279 --> 00:18:07,279 Let us go back to case C. Now, in case C should remember that the raft is floating. So, if 131 00:18:11,090 --> 00:18:15,909 the raft is floating then the net weight downwards must be balanced by buoyancy force acting 132 00:18:15,909 --> 00:18:22,909 upwards. Rho is a density of water, but this is raft floating condition, but the barrel 133 00:18:24,850 --> 00:18:31,850 as sunk. That means that the net downward weight must be greater than the buoyancy force, 134 00:18:35,149 --> 00:18:39,899 this is the entire volume of the barrel g. 135 00:18:39,899 --> 00:18:46,899 So, if I add these two equations then I get Mr plus Mbg must be greater than or equal 136 00:18:47,809 --> 00:18:54,809 to rho Vr plus Vbg, let us call this equation 3. Let me rewrite equation 1, which is Mr 137 00:18:58,429 --> 00:19:05,429 plus Mvg, this is equation 1. That I just derive few minutes back is equal to rho Vg. 138 00:19:09,059 --> 00:19:14,450 This means if this is the case, this implies that if the left side of this is greater than 139 00:19:14,450 --> 00:19:21,450 this. This means that, this is identically equal to this. So, rho Vg is greater than; 140 00:19:22,899 --> 00:19:29,899 so, this is a g here greater than rho Vr plus Vbg or V is greater than Vr plus Vb. So, this 141 00:19:36,080 --> 00:19:43,080 is equation 1 from case 1, where this is the displaced volume when the raft and the barrel 142 00:19:47,080 --> 00:19:54,080 are floating in this is case A. Now, in the case C only the raft is floating. This is 143 00:19:55,169 --> 00:19:58,830 the displaced volume of the raft entire barrel is sunk. 144 00:19:58,830 --> 00:20:05,830 So, this implies that the level will fall, this is case C water level fall in case C 145 00:20:08,960 --> 00:20:14,190 because displaced volume here is less than the total volume when the raft and barrel 146 00:20:14,190 --> 00:20:18,570 both were floating. So, compare to these two cases this water level will be smaller because 147 00:20:18,570 --> 00:20:23,749 a displaced volume here is small smaller compared to the displacement volume in case A. So, 148 00:20:23,749 --> 00:20:30,749 this is slightly counter intuitive a result because you have a situation where this barrel 149 00:20:30,809 --> 00:20:35,820 is completely sunk and you may intuitively or instinctively think that this water level 150 00:20:35,820 --> 00:20:40,720 will raise here compare to case here, but by careful application of Archimedes principle, 151 00:20:40,720 --> 00:20:47,350 we can show that it is not the case and fact it is opposite. Now, one last topic fluid 152 00:20:47,350 --> 00:20:53,720 statics, so far, I have been discussing fluids under static conditions under the influence 153 00:20:53,720 --> 00:20:55,340 of gravitational field. 154 00:20:55,340 --> 00:21:01,860 But in many chemical engineering applications, you encounter a fictitious body force, call 155 00:21:01,860 --> 00:21:08,860 the centrifugal force which happens; this force happens because of rotation. Suppose, 156 00:21:09,999 --> 00:21:16,999 you have a bowl of, a bowl of liquid like water and you rotate it. This whole bowl is 157 00:21:18,049 --> 00:21:24,119 rotated at a high speed. Let us say omega. Now, if you have liquid in it and it is not 158 00:21:24,119 --> 00:21:31,119 completely filled, then this liquid the centrifugal force if you remember from mechanics acts 159 00:21:32,159 --> 00:21:39,159 from the radius radial access to outwards. So, this is the direction of centrifugal force. 160 00:21:42,110 --> 00:21:49,110 So, the centrifugal force on any fluid element tends to through the fluids towards outside. 161 00:21:49,840 --> 00:21:56,159 So, you may imagine that if, the bowl is not fully filled, then this water, initially the 162 00:21:56,159 --> 00:22:03,159 water level will be like this. It is partially filled. Now, after rotating this water will 163 00:22:04,369 --> 00:22:10,570 just be thrown close to the surface and this situation will happen when gravity is very 164 00:22:10,570 --> 00:22:17,570 very small, a small compare to the centrifugal force. When the centrifugal force is so large, 165 00:22:19,419 --> 00:22:23,600 when the rotation speeds are so large, then this then this liquid will be completely thrown 166 00:22:23,600 --> 00:22:30,600 towards the end of the towards the rim of the container, the bowl. In this case, the 167 00:22:30,649 --> 00:22:37,649 entire mass of liquid will rotate like a fluid. Now, imagine this is the axis and this is 168 00:22:40,889 --> 00:22:47,889 the wall. This is the wall of the bowl and let us call this radius as r2 and this is 169 00:22:50,999 --> 00:22:57,999 the interface, this is liquid. So, I am just trying to blow this region up here and let 170 00:23:01,889 --> 00:23:08,889 us call the distance of radio distance of the interface from axis as r1. Now, the centrifugal 171 00:23:11,600 --> 00:23:17,960 force if you take any tiny fluid element. Well, it is actually annular, it is a annular 172 00:23:17,960 --> 00:23:24,960 element. So, because the geometry cylindrical. So, the fluid element will be annular, it 173 00:23:26,090 --> 00:23:32,289 is a ring like element. So, if you take any fluid element there is a unbalance centrifugal 174 00:23:32,289 --> 00:23:37,590 force acting readily outwards. So, if the fluid is rotating like a rigid 175 00:23:37,590 --> 00:23:42,019 object. There is no relative motion between two fluid elements, then you can think of 176 00:23:42,019 --> 00:23:46,519 the entire think as so it is a solid like motion. So, it is under static conditions 177 00:23:46,519 --> 00:23:52,080 even though it is moving like a rigid body, there is no relative deformation. So, there 178 00:23:52,080 --> 00:23:56,129 is no shear stresses. So, the only stresses are very normal to the fluid elements, so 179 00:23:56,129 --> 00:24:00,639 the pressure. So, the pressure must be vary accordingly to balance the body force due 180 00:24:00,639 --> 00:24:05,669 to centrifugal forces. So, how does one calculate this? The differential 181 00:24:05,669 --> 00:24:12,669 force on an element, on a volume element differential centrifugal force. Centrifugal force on a 182 00:24:21,509 --> 00:24:27,359 volume element is centrifugal force goes as mass times the radial distances squared times, 183 00:24:27,359 --> 00:24:32,940 sorry, the angular velocity square times the radial distance, Mr omega square. So, if you 184 00:24:32,940 --> 00:24:38,769 take a differential volume element. So, differential mass dm times omega square is the angular 185 00:24:38,769 --> 00:24:44,429 velocity square times r. This is the angle omega is the angular velocity in radians per 186 00:24:44,429 --> 00:24:51,429 second. So, this is the mass of the volume, tiny volume of the differential volume. 187 00:24:53,940 --> 00:25:00,940 If you take a cylindrical volume element it is mass is dm is the volume is 2 pi r. So, 188 00:25:04,970 --> 00:25:10,730 this is r any at any radial distance r. You consider a slice, cylindrical slice and at 189 00:25:10,730 --> 00:25:17,730 this height be b or the cylindrical element. So, 2 pi rb is the surface area of the cylinder 190 00:25:19,179 --> 00:25:25,549 at in a cylinder r times dr which is a thickness. This is tiny thickness we are considering. 191 00:25:25,549 --> 00:25:32,549 This is the volume times the density, this is the mass this is dm. So, dF is nothing 192 00:25:38,450 --> 00:25:45,450 but, 2 pi rb times dr rho times omega square r. This is the differential force on a volume 193 00:25:50,869 --> 00:25:57,869 element due to centrifugal forces due to rotation. Now, this must be balanced by differential 194 00:26:00,419 --> 00:26:07,080 pressure. So, if you take a tiny slice, a cylindrical slice there is a differential 195 00:26:07,080 --> 00:26:11,929 force that is acting like this. So, the pressure build up must act such that it balances; this 196 00:26:11,929 --> 00:26:17,119 is the pressure acting in this direction. So, pressure must tend to counteract this 197 00:26:17,119 --> 00:26:23,720 centrifugal force that acts in this direction. So, the pressure must balance change in pressure 198 00:26:23,720 --> 00:26:30,720 must balance this centrifugal force which tends to; now, let me simplify this slightly. 199 00:26:32,590 --> 00:26:39,590 So, it is 2 pi r square omega square b rho dr. That is the force. So, the pressure is 200 00:26:47,619 --> 00:26:54,619 force divided by area, area is 2 pi rb. This is the force pressure is force divided by 201 00:26:55,239 --> 00:26:59,109 area. So, let me cancel 2 pi rb. 202 00:26:59,109 --> 00:27:06,109 So, dp is nothing but, 2 pi b then one r cancels is rho r omega square dr. Integral dp if I 203 00:27:15,700 --> 00:27:19,409 want to find the pressure distribution or difference between any two points and integrate 204 00:27:19,409 --> 00:27:26,119 this over the two points between p1 and p2, between any two radial locations r1 and r2. 205 00:27:26,119 --> 00:27:33,119 No longer small let us r dr this implies p2 minus p1 is rho omega square by 2 r2 square 206 00:27:38,649 --> 00:27:45,649 for minus r1 square. This is a kinetic decision. So, this is a important result where pressure 207 00:27:48,070 --> 00:27:53,330 variation is now, a happening in a rotating fluid where the body forces due to centrifugal 208 00:27:53,330 --> 00:27:57,739 forces and not due to the gravity. And this is in the limit when the centrifugal forces 209 00:27:57,739 --> 00:28:04,409 are large compare to the gravitational forces. So, we have neglected gravity compare to centrifugal 210 00:28:04,409 --> 00:28:11,409 forces. So, just as gravity acts on a objects. So, 211 00:28:18,159 --> 00:28:23,749 in a gravitational field centrifugal forces can also act on object and the centrifugal 212 00:28:23,749 --> 00:28:30,749 force will tend to accelerate particles with higher density or element with higher density. 213 00:28:30,769 --> 00:28:34,289 And they will be thrown to the wall because their magnitude of the centrifugal forces 214 00:28:34,289 --> 00:28:41,289 are larger and these are used in a many separations. So, if you want to separate two liquids, two 215 00:28:41,649 --> 00:28:48,289 immiscible liquid with different densities. One way is to take this two liquids and put 216 00:28:48,289 --> 00:28:53,249 them in a centrifuge which is basically be a bowl that is rotated with very high speed. 217 00:28:53,249 --> 00:28:57,779 And then because of the centrifugal action the liquid of higher density will be thrown 218 00:28:57,779 --> 00:29:03,559 towards the wall and the liquids liquid of a lower density will be more towards the center. 219 00:29:03,559 --> 00:29:08,269 And then you can simply separate this two away just based on dense difference. 220 00:29:08,269 --> 00:29:11,669 The same thing can be done due to gravity with the help of gravity also. That is called 221 00:29:11,669 --> 00:29:18,669 gravity base separation, but the driving force which is because of acceleration to gravity 222 00:29:19,460 --> 00:29:26,460 g is very small. It is fixed rather, but in case of centrifugal forces we can vary the 223 00:29:28,139 --> 00:29:33,820 centrifugal acceleration at our will by changing the angular velocity of rotation and you can 224 00:29:33,820 --> 00:29:40,820 achieve faster rates of separation between these two fluids. So, this a really completes 225 00:29:40,909 --> 00:29:47,909 my emphasis on fluid statics. So, the next topic I am going to discuss is Fluid kinematics. 226 00:29:59,139 --> 00:30:06,139 So, firstly what is kinematics? In general in any subjects; any subject that deals with 227 00:30:11,840 --> 00:30:17,109 a mechanics whether it is solid mechanics or fluid mechanics or particle mechanics whatever 228 00:30:17,109 --> 00:30:22,779 subject you have it. Mechanics deals with; as I told in the beginning forces and the 229 00:30:22,779 --> 00:30:29,779 motion that cause by forces. So, mechanics is broadly divided into two parts one is kinematics. 230 00:30:30,429 --> 00:30:37,429 Kinematics is the subject that deals with description of motion without worrying about 231 00:30:45,200 --> 00:30:52,200 or without reference to the forces that cause them. So, that is the first thing is to be 232 00:31:04,340 --> 00:31:08,129 able before understanding how forces cause motion? 233 00:31:08,129 --> 00:31:13,269 The first step to first understand how to describe the motion per say the motion itself 234 00:31:13,269 --> 00:31:20,049 and once we have the necessary tools to describe the motion, then we can go ahead and study 235 00:31:20,049 --> 00:31:27,049 how forces cause motion. That subject is called dynamics. Dynamics relates motion to forces. 236 00:31:32,519 --> 00:31:39,519 So, there are these are two branches of mechanics and so, firstly we will have to understand 237 00:31:43,129 --> 00:31:50,129 how to describe fluid motion? So, we will do kinematics of fluid flows. 238 00:31:52,109 --> 00:31:58,720 Because in this course, we are interested in motion of fluids. So, we will first discuss 239 00:31:58,720 --> 00:32:05,720 kinematics of fluid flows, then we will proceed to dynamics. Now, a useful, so firstly, let 240 00:32:08,399 --> 00:32:13,190 us try to understand how to how; what are the various options we have to describe fluid 241 00:32:13,190 --> 00:32:18,200 flows? Now, remember that we have already taken the continuum route or the continuum 242 00:32:18,200 --> 00:32:24,409 approach where in we are saying that the fluid is a continuous medium. And you can identify 243 00:32:24,409 --> 00:32:31,139 each and every point in the fluid and ascribe unique properties to points such as velocity, 244 00:32:31,139 --> 00:32:36,090 pressure, temperature, density and what not. All kinds of properties can be attributed 245 00:32:36,090 --> 00:32:43,090 to each and every point in the fluid and these various variables such as pressure density 246 00:32:44,849 --> 00:32:48,940 and so on. They are smoothly varying functions of spatial coordinates and time. 247 00:32:48,940 --> 00:32:55,940 This is the essential crocks of continuum hypothesis. Now, how do I, what do I mean 248 00:32:56,679 --> 00:33:03,679 by a point in a fluid? So, in this context the notion of what is called a hypothetical 249 00:33:10,549 --> 00:33:17,549 fluid particle helps in the continuum picture. What is the fluid particle? Well, imagine 250 00:33:20,049 --> 00:33:27,049 let say you have a box containing a liquid like water and it is stationary initially. 251 00:33:29,919 --> 00:33:36,919 Let us say, we take a colored dye and then mark a fluid here and then take another color 252 00:33:37,879 --> 00:33:44,879 dye mark a fluid here. Take another color dye and mark a fluid here, take a dye of another 253 00:33:45,169 --> 00:33:52,169 color and mark a fluid here and so on. Imagine that you are marking fluids with colored 254 00:33:52,190 --> 00:33:57,960 dye and let us assume that dye molecules do not diffuse. So, that the dye stays good I 255 00:33:57,960 --> 00:34:02,609 mean it does not if you drop a color. Color liquid like ink of course, you know that it 256 00:34:02,609 --> 00:34:07,059 is going to dissolve a diffuse and water, but let us assume the diffusivities are so 257 00:34:07,059 --> 00:34:12,590 small for our timescales of interest here that you can imagine that the dye does not 258 00:34:12,590 --> 00:34:18,919 dissolve at all. We can dye; imagine the dye to be insoluble in the liquid that we are 259 00:34:18,919 --> 00:34:25,919 considering. Now, this is the time initially at time t equal to 0. So, you are imagining 260 00:34:26,720 --> 00:34:31,880 that at each and every point that you can resolve in your scale of measurements. You 261 00:34:31,880 --> 00:34:36,030 can ascribe you can point a dye and you can visualize it is motion. 262 00:34:36,030 --> 00:34:43,030 So, this is the time t equals to 0. At a later time all this points may move. For example, 263 00:34:43,370 --> 00:34:49,980 this is blue color dye, let say this blue color dye element will move. So, let us use 264 00:34:49,980 --> 00:34:56,980 pink color for; at a later time. So, I will draw the later time picture with a open circle. 265 00:34:58,730 --> 00:35:05,730 So, time t equal to 0 is closed or filled circles. A later time at time t it is open 266 00:35:16,660 --> 00:35:23,660 circles. That is I am going to indicate the motion of various points in the fluid and 267 00:35:26,650 --> 00:35:32,520 the fluid is moving due to presumably due to application of some forces. So, essentially 268 00:35:32,520 --> 00:35:39,520 we have a bunch of points that are marked by dye, color dye and we are assuming that 269 00:35:40,360 --> 00:35:46,410 the dye molecules do not diffuse and so, that they just stay put wherever they are and at 270 00:35:46,410 --> 00:35:52,440 time t equals to 0. When there is no force forcing on the fluid and at a later time due 271 00:35:52,440 --> 00:35:58,000 to application of some forcing the fluid is under motion and then the fluid is under motion 272 00:35:58,000 --> 00:36:01,760 all these points will move some other points, some other locations. 273 00:36:01,760 --> 00:36:08,760 All these colored points will move to some other locations for example, this may go to 274 00:36:08,900 --> 00:36:15,900 here and so on. And red dot, sorry, red dot will move here. This orange dot will move 275 00:36:19,350 --> 00:36:26,350 here and this yellow dot will move here. So, these are the locations of these various points. 276 00:36:30,210 --> 00:36:37,210 Notice that I am using close circles for initial time the location is initial time of various 277 00:36:40,190 --> 00:36:47,190 dye point, dye elements and I am using open circles for just for the sake of clarity exaggerating 278 00:36:49,640 --> 00:36:56,640 the size of these points, so that you can see them easily. So, this can happen that 279 00:36:57,990 --> 00:37:04,990 various points at were the initially located. At some locations denoted by close circles 280 00:37:05,360 --> 00:37:11,620 we will move eventually upon fluid motion to some other locations. 281 00:37:11,620 --> 00:37:18,620 Now, these can be thought of as the location or this let me just. These points can be thought 282 00:37:22,960 --> 00:37:29,960 of as fluid particles. Such dyes parts spots which can be used to identify location of 283 00:37:34,370 --> 00:37:41,370 fluids at various points in the fluid at initial time and the subsequent motion can be thought 284 00:37:41,820 --> 00:37:48,820 of as a fluid particle. And within the continuum hypothesis a fluid element can fluid can be 285 00:37:49,950 --> 00:37:56,420 compressed of infinitely a fluid is infinitely smooth. There is no discreteness in a fluid. 286 00:37:56,420 --> 00:38:01,700 So, you can resolve a fluid to any link scale you want. So, there are infinitely large numbers 287 00:38:01,700 --> 00:38:07,600 of a fluid particles correspond to each and every point in space within the continuum 288 00:38:07,600 --> 00:38:12,830 hypothesis. And just by way of an illustration. I use 289 00:38:12,830 --> 00:38:19,830 the notion of coloring the fluid with a dye element. A dye, a substance to visualize the 290 00:38:20,020 --> 00:38:25,480 motion, but idea is you can think of it as a mathematical framework where at time t equals 291 00:38:25,480 --> 00:38:32,480 to 0. You have various locations in the fluid which are marked by their initial locations 292 00:38:33,270 --> 00:38:39,350 and at a later time due to application of forces all these points will start moving 293 00:38:39,350 --> 00:38:43,340 and they will tend to occupy various different locations in general. 294 00:38:43,340 --> 00:38:50,340 So, this is the notion of a fluid particle. So, in general at time t you have a fluid 295 00:38:53,680 --> 00:39:00,680 particle which is moving at a later time. This is time t equals to 0. Initially, that 296 00:39:02,790 --> 00:39:09,790 is there is no motion initially and upon application of some forcing. This point moves in spatial 297 00:39:12,660 --> 00:39:18,040 coordinate you always have let say an x y z coordinate with respect to which we are 298 00:39:18,040 --> 00:39:25,040 describe in a motion that that implicit. So, just to be complete let me just put coordinate 299 00:39:25,100 --> 00:39:31,410 system there. So, there is a point that is located at time t equals to 0 and it is moving 300 00:39:31,410 --> 00:39:36,800 due to application of forces to some other location. This is the current location let 301 00:39:36,800 --> 00:39:43,800 us say at time t. This is what is called a particle trajectory. 302 00:39:44,190 --> 00:39:51,190 But, there are infinitely large number of such points. Let me just show it some other 303 00:39:54,250 --> 00:40:01,250 color. So, there are large number of such points. Because you can assign that to each 304 00:40:02,240 --> 00:40:08,490 and every point and space a point and all these points will start moving upon application 305 00:40:08,490 --> 00:40:15,490 of forces. So, one way to describe fluid motion is to consider all these points, label all 306 00:40:19,030 --> 00:40:26,030 these points at their initial location before application of forces. So, let us label all 307 00:40:28,800 --> 00:40:35,800 points in fluid based on their initial positions. 308 00:40:43,320 --> 00:40:50,320 So, let us call the initial positions of a point as so, if you put a coordinate system 309 00:40:55,350 --> 00:41:02,350 any point will be denoted by a vector x. So, this is x y and z coordinate system. So, x 310 00:41:04,090 --> 00:41:11,090 p0, x0 stands for initial time t equals to 0 and p denotes the fluid particles the subscript 311 00:41:17,960 --> 00:41:24,960 p denotes a fluid particle. So, x p0 this a vector. So, it is comprised of three co-ordinates 312 00:41:32,610 --> 00:41:39,610 x p0, y p0, zp 0 in a Cartesian coordinate system. So, this is nothing but, the location 313 00:41:48,700 --> 00:41:54,980 of the trajectory of the location of the particles position of the particle at time t equal to 314 00:41:54,980 --> 00:42:01,980 0 is x p0. This is the initial location of various particles, various fluid particles. 315 00:42:14,440 --> 00:42:21,440 Now, upon influence of forces this the location of these particles at time t will change in 316 00:42:30,180 --> 00:42:37,180 general of various particles will change due to fluid motion. Now, this x p of t the current 317 00:42:45,230 --> 00:42:52,230 location is a function of the initial locations. So, once I tell you what are the initial locations 318 00:42:54,680 --> 00:43:00,080 various particles, the current location of the various particles will be a function of 319 00:43:00,080 --> 00:43:05,980 the initial location. Because a point that was here will move here, a point that was 320 00:43:05,980 --> 00:43:11,710 here, will move to some other location. This point can never move here, unlike this point 321 00:43:11,710 --> 00:43:18,230 will never move here in a given set of fluid p in a specified flow. Each point which started 322 00:43:18,230 --> 00:43:25,230 out initially, at x p0 will eventually, lead reach a unique x p at time t. So, the current 323 00:43:26,830 --> 00:43:33,830 position of various fluid particles fluid particles will be function of the initial 324 00:43:40,410 --> 00:43:47,410 positions. Now, if I know this functional form sometimes 325 00:43:51,830 --> 00:43:58,830 I just people just write this as x p of t is x p of x p naught and time that is the 326 00:44:01,520 --> 00:44:08,520 function itself is denoted by the same symbol. Now, if I have this functional form how the 327 00:44:09,660 --> 00:44:15,000 current portion varies with initial portion and time then I can calculate the velocity 328 00:44:15,000 --> 00:44:22,000 of a fluid particle at time t. Velocity of a fluid particle x p naught at time t a fluid 329 00:44:27,300 --> 00:44:32,650 particle is identified by its location at time t equals to 0. So, we use some other 330 00:44:32,650 --> 00:44:38,690 color here. At time t equal to 0, a fluid particle is at x p naught and this particle 331 00:44:38,690 --> 00:44:45,690 moves here, some other particle which was here would move there at time t. So, each 332 00:44:47,050 --> 00:44:54,050 particle, this particle time t here is labeled by it is location at time equals to 0. So, 333 00:44:59,230 --> 00:45:02,740 each particle is labeled by their initial locations. 334 00:45:02,740 --> 00:45:09,630 So, this is the velocity of the particle which was at time x p0 at time t equal to 0 and 335 00:45:09,630 --> 00:45:13,360 the velocity of that particle at time t is nothing but, the rate of change of its position 336 00:45:13,360 --> 00:45:20,360 dx p by dt keeping x p0 constant. That is your fixing the same particle, that you are 337 00:45:22,760 --> 00:45:26,950 following the same label in some sense and you are asking what is the velocity at time 338 00:45:26,950 --> 00:45:32,950 t valid is simply the rate of change of it is position vector which is x p. Now, such 339 00:45:32,950 --> 00:45:39,950 a description is called; such description of fluid flow where in your identifying various 340 00:45:41,610 --> 00:45:47,370 particles by their locations at time t equals to 0. And merely following the positions of 341 00:45:47,370 --> 00:45:52,020 various particles as a function of time is called the Lagrangian description. 342 00:45:52,020 --> 00:45:59,020 It is called the Lagrangian description in fluid flow. What is the Lagrangian description? 343 00:46:02,280 --> 00:46:09,280 Well, here the independent variables are the position the initial position of various particles 344 00:46:15,710 --> 00:46:22,710 that is the label and time. So, all properties such as suppose velocity of a particle is 345 00:46:23,760 --> 00:46:28,990 a function of it is initial position and time. So, what is this? This is the velocity of 346 00:46:28,990 --> 00:46:35,920 a particle which was at x p0 at time t equal to 0. At and as it particle moves at a later 347 00:46:35,920 --> 00:46:42,920 time what is it is velocity? You are following various particles and then your enquiry what 348 00:46:43,170 --> 00:46:47,480 is the velocity? What is the acceleration? What is the density? What is a pressure? What 349 00:46:47,480 --> 00:46:51,140 is their temperature and so on? 350 00:46:51,140 --> 00:46:57,610 So, independent variables in a Lagrangian description are, so, if you have any function 351 00:46:57,610 --> 00:47:04,060 any property it is given as a function of the initial position of the particle which 352 00:47:04,060 --> 00:47:11,010 is a essentially serving as the label of the particle and time. Now, so not just for flow 353 00:47:11,010 --> 00:47:18,010 variables, even if can think of temperature in a fluid, in a moving fluid. This is a temperature 354 00:47:19,040 --> 00:47:26,040 of a particle which was at a time t equal to 0 at x p0 at a later time t. So, various 355 00:47:26,720 --> 00:47:32,510 particles as they move their property such as density, temperature, concentration and 356 00:47:32,510 --> 00:47:39,160 velocity and many other things will change, but you can describe the change based on their 357 00:47:39,160 --> 00:47:42,720 initial co-ordinates. This is the essential idea behind Lagrangian description. 358 00:47:42,720 --> 00:47:49,720 So, once I have this motion, how the trajectory of the particle changes various particles 359 00:47:55,820 --> 00:48:02,820 change as function of time? Then I get velocity of a particle which was at x p0 at time t 360 00:48:03,670 --> 00:48:10,670 equal to 0. At a later time t is nothing but, the rate of change of it is position by keeping 361 00:48:12,910 --> 00:48:19,910 x p0 same that is your following the same particle. Now, acceleration of a particle 362 00:48:24,030 --> 00:48:31,030 is nothing but, the rate of change of its velocity keeping the same particle such a 363 00:48:34,460 --> 00:48:41,460 description is also called sometime as the Material description or the Lagrangian description. 364 00:48:50,200 --> 00:48:57,200 Now, this is a one of way of describing the fluid motion, but this is not the only way 365 00:49:00,270 --> 00:49:04,710 or this is neither is this is the most useful way. What is normally done in fluid mechanics 366 00:49:04,710 --> 00:49:11,710 is what is called the Eulerian description? So, what is a Eulerian description? Here, 367 00:49:20,840 --> 00:49:27,840 we place a lab coordinate frame in our lab x, y, z and measure various properties such 368 00:49:29,350 --> 00:49:36,350 as velocity in the fluid as a function of three spatial laboratory coordinates and time. 369 00:49:37,800 --> 00:49:44,560 So, velocity is measured not as a function of various particles not by following the 370 00:49:44,560 --> 00:49:51,560 particles fluid particles. But by merely saying what is suppose I put 371 00:49:51,710 --> 00:49:58,520 flow measuring velocity meters at speed various points in space and then you measure velocity 372 00:49:58,520 --> 00:50:03,690 at each and every point as a function of time. So, that description is the Eulerian description. 373 00:50:03,690 --> 00:50:10,150 This is also called a Spatial description for obvious reasons because we are describing 374 00:50:10,150 --> 00:50:17,150 a part, a properties of the fluid such as velocity and acceleration, pressure, density, 375 00:50:19,170 --> 00:50:23,960 temperature as a function of by following the various particles. So, in the Lagrangian 376 00:50:23,960 --> 00:50:29,400 description by labeling a particle, when I say temperature of a particle, what I mean 377 00:50:29,400 --> 00:50:33,260 is that as I follow the particle how is the temperature changing with time. That is the 378 00:50:33,260 --> 00:50:37,030 Lagrangian description. In the Eulerian description, we are not following 379 00:50:37,030 --> 00:50:41,910 the particle any more, fluid particles any more. We are simply keeping a stationary frame 380 00:50:41,910 --> 00:50:46,420 of reference, a lab frame of reference and this could be stationary or it could move 381 00:50:46,420 --> 00:50:51,680 with a constant velocity that depends on the nature of the problem. For simplicity, let 382 00:50:51,680 --> 00:50:56,430 us keep a fixed coordinate system in our lab. And then we can measure various quantities 383 00:50:56,430 --> 00:51:02,700 such as velocity or temperature or pressure at various points with respect to this coordinate 384 00:51:02,700 --> 00:51:09,670 system as a function of time. And report quantities like how does a temperature change at various 385 00:51:09,670 --> 00:51:15,450 points in the fluid as a function of time. So, what is T of x, t? At a given location, 386 00:51:15,450 --> 00:51:22,450 if I fix x. So, if I have a 3 Cartesian co-ordinate system. I can fix x. At a fixed x, how does 387 00:51:23,810 --> 00:51:29,530 a temperature change as a function of time? Or at a given time how does a temperature 388 00:51:29,530 --> 00:51:36,120 vary as a function of x, y and z? So, this is called the spatial description in fluid 389 00:51:36,120 --> 00:51:43,120 mechanics. Now, such quantities are called fields. This is called the velocity field, 390 00:51:46,390 --> 00:51:52,660 where the velocity is expressed as a function of a 3 spatial coordinates and time. This 391 00:51:52,660 --> 00:51:56,910 is called the temperature field. 392 00:51:56,910 --> 00:52:03,910 Now, what this description does is that suppose, I put a thermometer; suppose, I have fluid 393 00:52:05,980 --> 00:52:12,980 that is flowing, I put the thermometer. So, this a thermometer and I measure T at a given 394 00:52:19,850 --> 00:52:26,850 location let us call this x and time. What this is measuring is at a time T is measuring; 395 00:52:31,160 --> 00:52:36,290 whatever so, the fluid is continuously flowing. A fluid particle with will occupy this spatial 396 00:52:36,290 --> 00:52:43,290 location x at a time T and T of x, t will be the temperature of that fluid particle 397 00:52:43,510 --> 00:52:50,510 denoted by this pinkish violet circle which happens to occupy this spatial location at 398 00:52:52,630 --> 00:52:58,730 time t. At a later time, t plus delta t a slightly 399 00:52:58,730 --> 00:53:05,730 later time at the same spatial location. This point would have moved let us say here, this 400 00:53:08,040 --> 00:53:15,040 point which was here at x, that are moved here and some other point would come and occupy 401 00:53:15,040 --> 00:53:22,040 the location x. This is the same location x, so, let us call this vector x. So, different 402 00:53:22,980 --> 00:53:29,980 fluid particles the key to understand here is the different fluid particles will occupy 403 00:53:30,680 --> 00:53:35,940 spatial positions at the same spatial position at different times by virtue of fluid motion. 404 00:53:35,940 --> 00:53:42,730 Because a fluid is continuously moving or flowing. So, what the thermometer will measure 405 00:53:42,730 --> 00:53:49,730 at a given spatial location it is merely a record of the temperature values at that location 406 00:53:49,900 --> 00:53:56,290 as a function of time. And this does not correspond to the temperature of fluid particles. So, 407 00:53:56,290 --> 00:54:02,080 the Eulerian description gives a very very completely different view point compared to 408 00:54:02,080 --> 00:54:08,850 Lagrangian description. In the Lagrangian description you would follow the same particle 409 00:54:08,850 --> 00:54:14,420 as a function of time. So, whereas in the Eulerian description, you 410 00:54:14,420 --> 00:54:19,000 are not following the same particle. You are merely sitting at a same point and space and 411 00:54:19,000 --> 00:54:23,780 you are recording various properties such as temperature in this particular instance. 412 00:54:23,780 --> 00:54:30,220 So, we are measuring different the properties of various fluid particles not the same fluid 413 00:54:30,220 --> 00:54:36,140 particle. So, we will stop here and will continue in the next lecture further. We will see you 414 00:54:36,140 --> 00:54:43,140 in the next lecture.