0:04

In this segment, I'm going to continue our discussion of fluid properties,

Â looking particularly at the property of surface tension.

Â So surface tension also, sometimes called capillarity is a force that

Â arises at the interface between either a liquid and a gas or

Â between two liquids, which are immiscible.

Â In other words, they don't mix.

Â And it arises, because of unbalanced cohesive forces

Â between the molecules, which occur near the interface.

Â And we can illustrate that with this simple idea here.

Â Let's suppose that we have an interface here between a liquid and a gas.

Â So this could be, for example, a water bubble in air.

Â If we look at some molecules say, these molecules here,

Â which are deeply embedded far away from the interface.

Â 0:58

Those molecules are subject to forces due to all of the molecules,

Â which surround them.

Â So these forces are pulling on this molecule in all directions.

Â However, all of these forces cancel out.

Â So there is no net force exerted on that molecule.

Â However, if we look at a molecule, which is closer to the interface here,

Â then that molecule is also being subject to forces, which are pulling in here.

Â But now we no longer have molecules, which are in the gas here to counteract them.

Â Therefore, they, those molecules are subject to a net inward force,

Â which is causing the interface to contract.

Â And this unbalanced force is what gives rise to the surface tension force.

Â We can explain that in this way.

Â That the surface tension force is an equal and opposite force,

Â normal to a cut, but parallel to the surface.

Â So, if I imagine a small portion of the surface here,

Â I have a force which is pushing or pulling that direction.

Â If I imagine this as a hypothetical cut here,

Â I have a force F, which is pulling on both sides of those cut and

Â it's perpendicular to the line and tangential or parallel to the surface.

Â And if the length of this cut is dL, then we can explain the surface force

Â by this simple equation that F is equal to sigma times the length of the cut,

Â dL where sigma is called the surface tension.

Â And surface tension must have dimensions of force per unit length.

Â For example, either Newtons per meter or pounds per,

Â per square foot as given in the extract form the FE manual here.

Â Surface tension, we can see commonly, for example,

Â it's the force which is holding up this water strider on the surface.

Â It's the only force, which prevents it from sinking through the water surface.

Â 3:08

Surface tension gives rise to droplets and for

Â a droplet to be in balance to balance the surface tension force,

Â there must be a slight excess pressure inside it.

Â For example, let's suppose I have a bubble of water in air and

Â I'll make a hypothetical cut through this water bubble and

Â divide it into two hemispheres.

Â 3:36

And the force acting on the hemisphere is shown in this diagram right here.

Â So, I'll suppose that the, the pressure outside here is p and

Â the pressure inside is p plus delta p.

Â So there was a slight pressure jump.

Â The pressure inside is slightly higher, where that pressure jump is ne,

Â is necessary to counterbalance the surface tension force.

Â 4:04

The surface tension force across here is sigma and

Â it acts all the way around the perimeter of the circle here and

Â the magnitude of that surface tension force is sigma multiplied

Â by the circumference of the bubble, which is 2 pi R.

Â So if I do a force balance here,

Â then the force on this hemisphere in this direction must be exactly equal

Â to the force pushing in this direction for this to be in equilibrium.

Â In other words, the internal force is p plus delta p multiplied by the area,

Â pi R squared is the force in this direction.

Â The forces in this direction are the pressure on the outside,

Â p times pi R squared plus the surface tension force sigma times 2pi R.

Â 5:11

To see the magnitude of this force, we can plug in some typical values.

Â For example, sigma for water in air is about 0.005 pounds per foot.

Â And if we have a small droplet of radius 200 of an inch and

Â we plug into that equation, we find that delta p is equal to 2 sigma over R,

Â which plugging in the number is equal to 6 pounds per square foot or

Â 0.042 pounds per square inch.

Â In other words, the pressure jump is very small.

Â So very often surface tension forces will be very small compared to

Â other forces unless the dimensions are very small.

Â But nevertheless, surface tension can be important in a number of phenomena.

Â For example, droplet formation in fuel injections in automobiles or

Â the way that sponges show, soak up water by capillary action.

Â Or burning candles by wicking or

Â drawing up melted wax through the through the wick

Â are all cases where surface tension becomes important.

Â 6:21

Another important concept is the idea of a contact angle where we have a droplet

Â of oil or mercury, for example sitting on top of a, of a solid surface.

Â And the liquid may bead up or spread out depending on the magnitude

Â of this angle beta where beta is called the contact angle.

Â 6:45

And the little bit of terminology, if beta, the contact angle is

Â less than 90 degrees, we say that the liquid wets the surface.

Â If beta is greater than 90 degrees as is shown in this sketch here,

Â we say that the liquid is nonwetting.

Â For example a bead of mercury in air sitting on a glass surface,

Â beta is about 140 degrees.

Â In other words, it's a nonwetting.

Â It beads up as shown in that sketch there.

Â Now some important phenomena of surface tension is capillary rise.

Â For example, if we have a very small diameter tube here like here,

Â then the surface tension force can cause the liquid to

Â rise up in the tube if the liquid wet the surface.

Â And the height that this rises to h, we can find out by very simple force balance.

Â If we apply a force balance to this column of fluid here,

Â the forces which are acting on that fluid are the weight of the fluid or

Â gravity, which is acting straight downwards and

Â the gravity force is equal to the specific weight of the fluid,

Â gamma multiplied by the volume, which is pi R squared times h.

Â This is balanced by the output surface tension force, which is 2 pi R sigma.

Â 8:18

And if we equate those two together,

Â we arrive at the very simple equation that the height

Â is equal to 4 sigma cosine beta over gamma times d,

Â which is the equation given in the FE manual here.

Â If the liquid is nonwetting, if beta is greater than 90 degrees,

Â then we have a depression,

Â then the liquid is actually depressed falls below the liquid water surface.

Â 8:51

Let's do an example on that.

Â A glass tube contains mercury, the surface tension is 0.52

Â newtons per meter and the contact angle is 140 degrees and

Â the mercury density is 13.600 kilograms per cubic meter.

Â What is the minimum tube diameter required to maintain the depression

Â h less than 10 millimeter?

Â Is it most nearly, which of these alternatives?

Â To solve this, we just apply the formula from the previous slide,

Â h equals 4 sigma cosine beta over gamma d.

Â But in this case, we want to solve this for the diameter.

Â So rearranging that equation, we get d is equal to 4 cosine beta over gamma h and

Â remembering that gamma is equal to rho g, that is equal to this expression here.

Â So we can substitute in here that this is equal to

Â 4 given the values and beta is 140 degrees.

Â H in this case is negative,because it's a depression, so

Â that is minus 0.01 on the bottom, but cosine for 140 is also negative.

Â So therefore, we arrive at the positive answer that h is equal to

Â 1.19 times 10 to the minus 3 meters or 1.19 millimeters.

Â So the best answer is B,

Â 1.2 millimeters.

Â