Chapter 9- Fluids What is a fluid? Matter is traditionally categorized into one of three states a solid, a liquid, or a gas. Up to this point, we have assumed that everything we have worked with has been in the solid state, where atoms or molecules are not free to move, and the substance maintains a fixed, unchanging shape and volume. In the liquid state, atoms or molecules are free to move, but the volume of the liquid remains constant. Lastly, in the gas state, atoms or molecules are free to move, and the volume of the gas may change (it can be compressed or expanded). We will now consider the mechanics of liquids and gases. Both liquids and gases can be classified as fluids, because they can easily flow and will alter their shape to fit their container. A solid does not fit into this category, as it has a definite shape and cannot flow. Other States of Matter There are actually two other states of matter that we do not traditionally consider in chemistry, but have significant applications in the realm of physics. The first of these is plasma, which is similar to the gas state, but will occur only under conditions with extreme heat. In the plasma state, electrons are stripped from the atom, and move freely through the material. The result is a mixture of positively and negatively charged ions, that move freely like a gas. As the charged particles move, they produce electromagnetic fields, which change as the particles move. These electromagnetic fields govern the motion of the plasma, and are unique to the plasma state. Physicists suggest that plasma is the original state of matter that was present after the Big Bang, when temperatures were too high to allow solid, liquid, or gas states to exist. Plasma is actually still the most prevalent state of matter in the universe, as our Sun, other stars, and intergalactic matter exist mostly in this state. On Earth, we see plasma when air is ionized during a lightening strike, or in the colorful displays of the aurora borealis. Commercially, the plasma state is now used in the display of plasma TVs, fluorescent lamps, and neon signs, to name a few examples. The fifth state of matter, a Bose-Einstein Condensate, is also known as a super fluid, and is the inverse of plasma in that it occurs at extremely low temperatures. Einstein first proposed the existence of this state of matter in 194, after reading a paper by Satyendra Nath Bose. At this time, Einstein was known in the science community, and he used his influence to get Bose paper published. Due to the level of difficulty in producing the low temperatures (about 1 μk) needed to observe a Bose-Einstein Condensate, Bose and Einstein were restricted to theoretical analysis of this state. Bose-Einstein Condensation was first experimentally observed in 1995 through an experiment by Eric Cornell and Carl
Wienman at the University of Colorado. In this experiment, Cornell and Wienman brought matter into the Bose-Einstein Condensate form for 10 seconds, and later received the Nobel Prize for this experiment in 001. In Bose-Einstein Condensates, atoms are indistinguishable from one another as they condense into a blob-like form. There are currently no useful applications of Bose-Einstein Condensation, although its existence may be further evidence to support the Big Bang Theory. Density The density of a substance is the amount of mass per unit of volume of a given substance We use the equation: ρ = m/v, where ρ (rho) is the variable used to represent density, m is mass, and V is volume. The standard units of density are kg/m 3. In a solid, the atoms or molecules are tightly packed together, so the mass of a certain volume of that substance would be relatively high. In the liquid state, the atoms/molecules have more energy and are able to move freely, so they spread apart from each other. As the particles have spread apart, there are fewer particles in the same volume of the substance, resulting in a lower mass per unit of volume. Measuring the density of the substance in liquid form, you would find that the density would have decreased from that of the solid. If that liquid then transitioned into the gas state, the particles have even more energy, move faster, and are more spread out. As a result, the density of the gas is now even lower than that of the liquid. If the pressure of a gas is increased, the volume of the gas will increase, yielding a lower density. The inverse is also true; if you decrease the pressure, volume decreases, and density increases. This is not the case for solids and liquids, as neither of these states can be compressed. Since the volume of the solid or liquid cannot change, the density will remain constant even with varying pressures. When fluids of various densities are mixed, the fluid with the lower density will float to the top, while the fluid with the higher density will sink. A classic example of this is mixing oil and water. Vegetable oil has a density between 915 kg / m 3 and 918 kg / m 3, while fresh water has a density of 1,000 kg / m 3. Since the oil is less dense, it will separate from the mixture and rise to the surface of the water, as is commonly observed in oil based salad dressings.
Pressure Pressure in Fluids Let us re-examine the atmospheric pressure that our bodies experience as we live on the surface of planet Earth. If we were to go to a higher altitude, do you think the pressure on our bodies would change? In fact, it does, illustrating the idea that as the depth of an object in a fluid increases, so does the pressure acting on that object. Remember that at the surface of the planet, we are actually at the bottom of the atmosphere, and we decrease our depth as we go higher. If you have ever experienced a steep take-off in an airplane, you may remember the feeling of having your ears pop. This popping is actually the result of a change in pressure- when the pressure outside of your ears has decreased, the pressure inside is greater, forcing the membrane to move outward slightly, which in turn reestablishes the equilibrium of pressure inside and outside of your ear. Another example that may be easier to visualize is that of a submarine under water. Let s say that the submarine has gone to a depth of h below the surface. The hull of the submarine is in effect supporting the weight of all the water above it. If we imagine a small area (A) on the top of the submarine, then the volume of the column of water above that part of the sub could be calculated by multiplying the area by the height: The mass of that column of water could then be calculated, since we know the density (ρ) and volume of that water: The pressure on the sub can be calculated using the pressure equation from earlier in the year, where the force is the weight of the water above the sub: We can substitute our equation for mass, and cancel out area in our equation: This simplifies to: V A h m V Ah P F A mg A P mg A Ahg gh A P gh
It is important to note that the pressure resulting from the weight of the water is not the only pressure acting on the submarine. We still need to take atmospheric pressure into account. The pressure calculated using P gh is known as gauge pressure. To calculate the total or absolute pressure (P0), add the gauge pressure and atmospheric pressure. Buoyant Force P = P0 + ρgh Have you ever noticed that a rock feels heavier to lift in air than it does in water? Based on what we have learned so far about forces, the rock in the water must have an additional force acting up on it. The upward force resulting from an object being wholly or partially immersed in a fluid is called the buoyant force. If an object is floating (Figure 1), the buoyant force pushing up is equal to the force of gravity pulling down; the object is in equilibrium. If the force of gravity is greater than the buoyant force, then the object will sink (Figure ). What do you think would be the result if the buoyant force was greater than the force of gravity? The object would shoot out of the fluid! Figure 1 - The force of gravity is equal to the buoyant force for a floating object1. When an object is totally submerged in a fluid, it will displace a volume of the fluid equal to the volume of the object. If only part of the object is submerged, the volume that is below the surface of the fluid is equal to the volume of the fluid that is displaced. The magnitude of the buoyant force acting on an object is equal to the weight of the volume of the fluid that is displaced by the object. This concept is known as the Archimedes Principle, named after the Greek mathematician, Archimedes. According to legend, Archimedes was charged with the task of determining whether the king s crown was made of pure gold or some other less expensive metal, without damaging the crown. The solution to this problem came to him while he was sitting in a bathtub, and experienced the buoyant force acting on his arms. Archimedes realized that he could use the buoyant force to measure the density of the crown, and compare his findings to the density of pure gold. Mathematically, Archimedes Principle can be represented with the following equation: FB = mfg Where FB is the buoyant force, mf is the mass of the displaced fluid, and g is the acceleration of gravity. Solving the density equation for mass, ρ = m/v m = ρv Figure 1 If the force of gravity is greater than the buoyant force, the object sinks
we can substitute the density equation into Archimedes Principle: FB = ρfvfg Remember that ρf is the density of the fluid, and Vf is the volume of the displaced fluid. Example: How much weight could a king sized mattress support before it sinks in fresh water? Let us first start assuming the king mattress is meters long, meters wide, and.45 m tall. The total volume of the mattress is: m m 0.45m 1.8m 3 Realize that buoyant force behaves a lot like the force of friction in that it responds to as strong as needed until it can t be any stronger. As the buoyant force increases, the mattress will sink deeper in the water. The maximum possible buoyant force will come when all of the mattress is submerged. In other words, FB is maxed when 1.8 m 3 of water is displaced. We now have FB = ρfvfg FB = (1000)(1.8 m 3 )(9.8) = 17,640 N So any weight above 17,640 N should sink the mattress. Floating Objects For cases when an object is floating, we can simplify our equilibrium equation into a very useful ratio. Remember that in equilibrium, the net force is zero, so the buoyant force must be equal to the force of gravity. FB = Fg Substituting in Archimedes Principle again, we see: ρfvf = ρovo This can be rearranged to make a density to volume ratio: ρf/ρo = Vo /Vf
Since Vf is the volume of the fluid displaced, it can never be greater than the volume of the object, as an object cannot take displace a volume larger than itself. Vf will be less than the volume of the object if the object is only partially submerged; the two volumes are equal if the entire object is under the surface of the fluid. For an object to float in a fluid, the object s density must be less than or equal to the density of the fluid in which it floats. Conceptual Question: If steel is denser than water, how does a steel boat float? Submerged Objects If an object is entirely under water, there is still a buoyant force acting on it, but it is less than the force of gravity. For instance, if a rubber raft that is floating on a pond was punctured, air would slowly leak out. As the raft deflates, the volume of the raft decreases, which in turn will decrease the volume of the water that is displaced. Eventually, when it is completely submerged, the volume of the raft will be equal to the volume of the displaced water. The net force acting on the raft will be the difference between the force of gravity (the weight) pulling down, and the buoyant force pushing up. Fnet = FB - Fg Fnet = ρfvfg ρovog Since both the water and the raft have the same volume, we can simplify this equation to: Fnet = (ρf ρo)vg According to this equation, the net force will depend on the densities of the raft and the water (or in general, the object and the fluid). If the density of the object is greater, the net force is negative the object will sink If the density of the object is less, the net force is positive the object will rise to the surface and float We can rearrange our buoyant force equation one more time to derive this final ratio: Fg (object)/fb = ρo / ρf
In words, the ratio of the force of gravity acting on the object to the buoyant force is equal to the ratio of the density of the object to the density of the fluid. Fluids in Motion Up to this point, we have only considered static, or non-moving, fluids. The study of fluid in motion is called fluid dynamics and can become very mathematically complicated. While we will consider the case where fluids are moving at a steady velocity for the purposes of this chapter, realize that real-life applications are more involved if you incorporate characteristics of the fluid, such as compression, viscosity, and rotational currents. A good starting point in examining fluids in motion would be to consider water that is allowed to flow through a pipe in a drainage system. Imagine the pipe starts with a diameter of 10 cm, and then narrows to a diameter of 6 cm. What would you expect to happen to the velocity of the water as it travels through the more narrow section of pipe? If you guessed that the velocity would increase, you are correct! We can calculate the velocity of water in a pipe of varying diameters using the equation of continuity, which states that the flow rate in a single pipe must be constant, regardless of the size of the pipe. We can calculate the flow rate of the pipe by multiplying the area of the circle inside the pipe by the velocity of the pipe at that position. If the diameter of the pipe changes, the area of the circle will change, which means the velocity will need to change in order to ensure continuity. We can write this as an equation: A 1 v 1 A v Let s look at the example above of a pipe that starts with a diameter of 10 cm that narrows to a diameter of 6 cm. If the water is travelling with a velocity of 5 m/s in the 10 cm section, how fast is it travelling in the 6 cm section of pipe? Looking at the continuity equation, we see that the first step is to find the cross-sectional area of the pipe in each location. The cross-sectional area can be calculated using A = πr. Substituting this into the continuity equation, we get: Solving for v: r 1 v1 r v m 10cm 5 r1 v1 r1 v1 s m v 13. 89 r r 6cm s
The velocity of the water does in fact increase! If you have ever used a garden hose, you may have observed this characteristic of continuity before. If the hose does not have a nossle on it, water will pour out at a certain velocity. When you put your finger over the hole, the area of the hole that water is still able to pour out of decreases, so the velocity of the water increases. If you block almost the entire hole, water will spray out with a very fast velocity! If you have interest in further investigating the physics of fluids, here is another useful resource worth exploring: http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html To find information related to this chapter, click on the Fluids link (Mechanics>Solids>Fluids) Bernoulli s Principle In 1738, Daniel Bernoulli used kinetic energy, potential energy and pressure considerations to write a complicated equation relating them. We won t include that here but we will consider the conceptual idea, known as Bernoulli s Principle which states: faster moving fluids exert a lower pressure. Additionally, pressure moves from high to low. There are several good examples of this. Have you ever felt pulled towards a truck as it passes you on the highway? This is because the truck makes a column of fast moving air and a column of lower pressure. Your car is pushed towards this lane. Another example is a shower curtain. Inside the shower, there exists faster moving fluid. Therefore, the pressure is lower inside the shower than out and the curtain is pushed inward. Sometimes in bad storms (hurricanes and tornadoes), roofs are popped off a houses. This is also because of Bernoulli s Principle. The fast moving winds create lower pressure than inside the house, which sometimes pops the roof off. One strategy is to actually leave your windows open to equalize the pressure. Airplane wings are another good example. Because of their shape, the air that goes over the top of the wing travels a greater distance in the same amount of time as the air underneath. Therefore, the faster moving fluid above the wing has a lower pressure than below the wing. In a sense, the whole airplane is pulled up; it really has little choice but to be forced upward despite its weight! One final example can be found in a scene from Finding Nemo. Recall when the turtles are riding the East Australian Current; the current is a fast moving fluid and should be a lower pressure than outside the current. This would mean it should be easy to get into the current but hard to get out. Indeed, the clip shows the baby turtle, Squirt, has to get thrown out but gets sucked back in on his own. Who knew Nemo took physics?