In this segment, I'm going to continue our discussion of fluid properties, looking particularly at the property of surface tension. So surface tension also, sometimes called capillarity is a force that arises at the interface between either a liquid and a gas or between two liquids, which are immiscible. In other words, they don't mix. And it arises, because of unbalanced cohesive forces between the molecules, which occur near the interface. And we can illustrate that with this simple idea here. Let's suppose that we have an interface here between a liquid and a gas. So this could be, for example, a water bubble in air. If we look at some molecules say, these molecules here, which are deeply embedded far away from the interface. Those molecules are subject to forces due to all of the molecules, which surround them. So these forces are pulling on this molecule in all directions. However, all of these forces cancel out. So there is no net force exerted on that molecule. However, if we look at a molecule, which is closer to the interface here, then that molecule is also being subject to forces, which are pulling in here. But now we no longer have molecules, which are in the gas here to counteract them. Therefore, they, those molecules are subject to a net inward force, which is causing the interface to contract. And this unbalanced force is what gives rise to the surface tension force. We can explain that in this way. That the surface tension force is an equal and opposite force, normal to a cut, but parallel to the surface. So, if I imagine a small portion of the surface here, I have a force which is pushing or pulling that direction. If I imagine this as a hypothetical cut here, I have a force F, which is pulling on both sides of those cut and it's perpendicular to the line and tangential or parallel to the surface. And if the length of this cut is dL, then we can explain the surface force by this simple equation that F is equal to sigma times the length of the cut, dL where sigma is called the surface tension. And surface tension must have dimensions of force per unit length. For example, either Newtons per meter or pounds per, per square foot as given in the extract form the FE manual here. Surface tension, we can see commonly, for example, it's the force which is holding up this water strider on the surface. It's the only force, which prevents it from sinking through the water surface. Surface tension gives rise to droplets and for a droplet to be in balance to balance the surface tension force, there must be a slight excess pressure inside it. For example, let's suppose I have a bubble of water in air and I'll make a hypothetical cut through this water bubble and divide it into two hemispheres. And the force acting on the hemisphere is shown in this diagram right here. So, I'll suppose that the, the pressure outside here is p and the pressure inside is p plus delta p. So there was a slight pressure jump. The pressure inside is slightly higher, where that pressure jump is ne, is necessary to counterbalance the surface tension force. The surface tension force across here is sigma and it acts all the way around the perimeter of the circle here and the magnitude of that surface tension force is sigma multiplied by the circumference of the bubble, which is 2 pi R. So if I do a force balance here, then the force on this hemisphere in this direction must be exactly equal to the force pushing in this direction for this to be in equilibrium. In other words, the internal force is p plus delta p multiplied by the area, pi R squared is the force in this direction. The forces in this direction are the pressure on the outside, p times pi R squared plus the surface tension force sigma times 2pi R. So very simply this results in this simple equation that delta p, the pressure jump is equal to 2 sigma divided by R, the radius. So the pressure jump gets bigger as the dimension of the bubble gets smaller. To see the magnitude of this force, we can plug in some typical values. For example, sigma for water in air is about 0.005 pounds per foot. And if we have a small droplet of radius 200 of an inch and we plug into that equation, we find that delta p is equal to 2 sigma over R, which plugging in the number is equal to 6 pounds per square foot or 0.042 pounds per square inch. In other words, the pressure jump is very small. So very often surface tension forces will be very small compared to other forces unless the dimensions are very small. But nevertheless, surface tension can be important in a number of phenomena. For example, droplet formation in fuel injections in automobiles or the way that sponges show, soak up water by capillary action. Or burning candles by wicking or drawing up melted wax through the through the wick are all cases where surface tension becomes important. Another important concept is the idea of a contact angle where we have a droplet of oil or mercury, for example sitting on top of a, of a solid surface. And the liquid may bead up or spread out depending on the magnitude of this angle beta where beta is called the contact angle. And the little bit of terminology, if beta, the contact angle is less than 90 degrees, we say that the liquid wets the surface. If beta is greater than 90 degrees as is shown in this sketch here, we say that the liquid is nonwetting. For example a bead of mercury in air sitting on a glass surface, beta is about 140 degrees. In other words, it's a nonwetting. It beads up as shown in that sketch there. Now some important phenomena of surface tension is capillary rise. For example, if we have a very small diameter tube here like here, then the surface tension force can cause the liquid to rise up in the tube if the liquid wet the surface. And the height that this rises to h, we can find out by very simple force balance. If we apply a force balance to this column of fluid here, the forces which are acting on that fluid are the weight of the fluid or gravity, which is acting straight downwards and the gravity force is equal to the specific weight of the fluid, gamma multiplied by the volume, which is pi R squared times h. This is balanced by the output surface tension force, which is 2 pi R sigma. And if we equate those two together, we arrive at the very simple equation that the height is equal to 4 sigma cosine beta over gamma times d, which is the equation given in the FE manual here. If the liquid is nonwetting, if beta is greater than 90 degrees, then we have a depression, then the liquid is actually depressed falls below the liquid water surface. Let's do an example on that. A glass tube contains mercury, the surface tension is 0.52 newtons per meter and the contact angle is 140 degrees and the mercury density is 13.600 kilograms per cubic meter. What is the minimum tube diameter required to maintain the depression h less than 10 millimeter? Is it most nearly, which of these alternatives? To solve this, we just apply the formula from the previous slide, h equals 4 sigma cosine beta over gamma d. But in this case, we want to solve this for the diameter. So rearranging that equation, we get d is equal to 4 cosine beta over gamma h and remembering that gamma is equal to rho g, that is equal to this expression here. So we can substitute in here that this is equal to 4 given the values and beta is 140 degrees. H in this case is negative,because it's a depression, so that is minus 0.01 on the bottom, but cosine for 140 is also negative. So therefore, we arrive at the positive answer that h is equal to 1.19 times 10 to the minus 3 meters or 1.19 millimeters. So the best answer is B, 1.2 millimeters.
