Chapter 10 Notes – Fluids There are three common states of matter – solid, liquid, and gas. A solid maintains a fixed shape and fixed size. When a strong force is applied to that solid, in most cases, it keeps its shape and volume. A liquid does not have a fixed shape but takes the shape of its container. Its shape changes, but it takes a large force per area to significantly change its volume. Finally, a gas does not have a fixed shape or volume. Pressure significantly affects the volume and shape of a gas. Liquids and gases do not have a fixed shape, so therefore, they have the ability to flow – aka fluids. 10.1 Statics – Density and Specific Gravity We often mix the terms weight and density, but there is a significant difference – volume. It’s often said that iron weighs more than wood, but something is missing from that statement. They must be the same volume to compare. What one must really mean is that iron has a higher density than wood. Density – mass per unit volume (variable: ρ: greek symbol rho and units: kg/m3) Density equation: ρ= 𝑚 𝑉 (m = mass and V = volume) The densities of pure substances such as copper are constant throughout the object. There is a table 10-1 on page 276 of your book which gives a list of density values for pure substances at 0ºC and 1 atm. The pressures and temperatures are listed as constant because they do affect the density of a substance. Specific Gravity (SG) – (of a substance) is the ratio of the density of that substance to the density of water at 4.0ºC. There are no units for SG because it is a ratio. The density of water is 1.00 g/cm3 or 1.00 x 103 kg/m3. In order to calculate the SG for any substance, simply take the value of the density and divide by the density of water (matching the units). For example, the density of lead is 11.3 x 103 kg/m3, so the specific gravity for lead is 11.3. 10.2 Pressure in Fluids Pressure – force per unit area (variable: P and units: N/m2 or Pascal (Pa)) 𝐹 Pressure equation: P= 𝐴 (F = force and A = area) The units of N/m2 can be interchanged with Pascal, which is named after Blaise Pascal. A 60-kg person standing on a 500 cm2 area exerts a (60kg)(9.8m/s2)/(0.050m2) = 12 x 103 Pa or N/m2. A fluid exerts pressure in all directions. If the pressure in the fluid is not the same in all directions, then the fluid must be flowing in a particular direction. Think about scuba diving. When you dive down a certain depth, you can feel the pressure of the water all around you equally. Another important fact about a fluid at rest is that the force due to fluid pressure is always perpendicular to any surface it is in contact with. This concept is much like a normal force that we studied in physics 1. Due to Newton’s 3rd Law, the container of the liquid can only push back with and equal and opposite force that is perpendicular to the force of the fluid. Have you ever wondered how pressure changes as you go deeper into the ocean? I recently spent time snorkeling in Grand Cayman and the farther I dove down, the more my ear felt the effects of the pressure. When we dive down, we have the weight of the water above our head. W = mg – if we substitute the density equation in for mass, we are left with W = ρVg. The volume is simply the area (A) of water times the depth (h). We can substitute this in for volume – W = ρAhg. Now plug this into the pressure equation and we’re left with: P=ρgh (ρ = density of fluid, g = acceleration of gravity, h = depth of fluid or pressure head) 10.3 Atmospheric Pressure and Gauge Pressure If we look at the Earth’s atmosphere like a giant swimming pool, then the pressure is least at the top of the atmosphere and greatest at the surface of the atmosphere (or at the bottom of the swimming pool). The issue is that it’s very hard to measure from the top of the atmosphere, so we instead measure the difference in depth to compare pressures in the atmosphere. Atmosphere – unit of pressure on the Earth’s surface at sea level (1 atm = 1.013E5 N/m2 = 101.3 kPa) Another pressure commonly used in meteorology is the bar. 1 bar = 100 kPa How come humans aren’t crushed by the enormous pressure on the surface? Living cells have an internal pressure that matches the atmospheric pressure. An balloon exceeds atmospheric pressure so it can press against the atmosphere and keep its shape. Pressure Gauge – A device used to measure the pressure above the atmospheric pressure. i.e. – tire gauge In order to measure the absolute pressure, one must add the reading on the pressure gauge plus the atmospheric pressure. i.e. – 220 kPa tire gauge reading + 101 kPa = 321 kPa of absolute pressure. 10.4 Pascal’s Principle The Earth’s atmosphere exerts a pressure on all objects that it is contact with. Because of this, it’s important not to forget the Earth’s atmospheric pressure when calculating the pressure on a submerged object. If we calculate the water pressure on an object at 100m depth, P = ρgh = (1000 kg/m3)(9.8 m/s2)(100m) = 9.8E5 N/m2 or 9.7 atm This is the pressure due to the water, but the atmosphere sits on top of the water. Therefore we need to add the atmospheric pressure to calculate the total pressure on that object. Therefore the pressure on the object is really 10.7 atm (9.7 atm + 1 atm). Blaise Pascal, a French philosopher, noticed this and created a principle. Pascal’s Principle – pressure applied to a confined fluid increases the pressure throughout by the same amount. This principle is applied many places, but two examples are hydraulic brakes in a car and a hydraulic lift. The drawing above shows that the confined fluid (green) has the same pressure throughout. Therefore, when the driver pushes the brakes (Pressure In), the fluid presses against the brake cylinders (calipers) with equal pressure (Pressure Out). This causes the brake pads to clamp down on the disk (rotor) attached to the wheel. The trick used is that Pin = Pout so therefore Fout/Aout = Fin/Ain. If there is different area between the master cylinder and the calipers, then a little force on the brake pedal can put a great deal of force on the calipers. 10.5 Measurement of Pressure: Gauges and the Barometer There are many devices used to measure pressure. Manometer – U-shaped tube partially filled with liquid, usually mercury or water. The equation to describe the manometer is P = Po + ρgh where Po is the atmospheric pressure ρ is the density of the liquid, h is difference in level of liquid, and P is the pressure being measured. An example would be a Radon detector system. Other measurements of pressure include: 1 mm-Hg = 133 N/m2 = 1 torr (named after the scientist Evangelista Torricelli who invented the barometer). So many units of pressure can be a nuisance, so N/m2 = Pa can always be used as the standard units. Barometer – a manometer closed at one end, filled with mercury, and inverted into a bowl of mercury. If the pressure of the atmosphere drops, then the mercury inside the bowl will not have as much pressure on it and mercury inside the tube will drop. If the pressure increases, mercury will be pushed down in the bowl and up into the tube. The atmosphere can support a tube of Mercury 76 cm tall. If the mercury in the tube drops, it creates a vacuum in the top of the tube. 10.6 Buoyancy and Archimedes’ Principle Submerged objects in a liquid often appear to weigh less than they do when outside the fluid. You may be able to lift something heavy like a rock under the ocean that you wouldn’t be able to lift on land. If you lift the rock up above the surface of the water, it appears to become much heavier. NASA uses underwater diving to simulate spacewalks or gravity conditions on the moon. These are examples of buoyancy. The weight of the submerged object does not change. Instead there is an upward buoyancy force exerted by the liquid on the object. Divers or fish nearly have a buoyancy force equal to the gravitational force so they end up balancing. Wood has a greater buoyancy force than its weight, so it floats on the fluid. Since pressure increases as an object is submerged deeper, the pressure on the bottom of the object is greater than the pressure on the top of the object. This causes an upward buoyancy force (FB). If F1 is a fluid force at the top of a submerged object and F2 is a fluid force at the bottom of the same object then F2 – F1 = FB. If we substitute pressure P = F/A = ρgh into the equation, we get FB = ρgA(h2-h1) = ρgV since volume is equal to Ah. If we substitute in mass for density * volume, then we get FB = mfg where mf is the mass of the fluid. This means that the buoyant force is equal to the displaced mass of the fluid times gravity. Archimedes’ Principle – the buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by that object. Archimedes apparently thought of this in his bathtub while thinking he could determine if the king’s crown was made of gold or not. See examples 10-5 and 10-6. Any object that floats is less dense than water. This means that the weight of the object can’t displace a fluid that has more weight (assuming same volume). When an object floats, we can say FB = Fg (weight of object) – from this we can write a ratio of volume of object displaced/volume of full object = density of object/density of fluid. This will give a percentage or fraction of how much of the object is submerged under the fluid. 𝑉𝑑𝑖𝑠𝑝 𝜌𝑜 = 𝑉𝑜 𝜌𝑓 where Vdisp is the volume of fluid displaced by object and Vo is the entire volume of the object. 10.7 Fluids in Motion; Flow Rate and the Equation of Continuity Fluid Dynamics (hydrodynamics) – Fluids in motion Streamline or laminar flow – flow of fluid is smooth, so that neighboring layers of the fluid slide by each other smoothly. Turbulent Flow – Erratic, small, whirlpool-like circles that are called eddy currents Viscosity – Internal friction of a fluid while flowing – laminar flow has a low viscosity and turbulent has a high viscosity. A drop of dye can quickly show whether the flow is laminar or turbulent. Flow Rate – Mass of fluid that passes a given point per unit time We can substitute in density and volume for mass, and then break down volume into area and length. If water has a laminar flow through a pipe, we get 𝛥𝑚 𝜌𝛥𝑉 𝜌𝛥𝐴𝛥𝑙 = = = 𝜌𝐴𝑣 𝛥𝑡 𝛥𝑡 𝛥𝑡 The flow rate into the pipe is equal to the flow rate out. This means that if the pipe changes diameter, then ρ1A1v1 = ρ2A2v2 (density * Area of pipe * velocity of fluid) This is called the equation of continuity. If the fluid is incompressible (most liquids), then the density remains constant through the line. If this is the case, then, A1v1 = A2v2 Av (area * velocity) is a flow rate of volume per time or m3/s. This system is used for blood flow in the human body. Also, wheezing is a symptom of this principle of flow rate and Bernoulli’s Principle. The airways get so tiny that the velocity of air through them increases at a great rate. The velocity gets so high that it creates a very low pressure in the bronchioles. This causes the bronchioles to temporarily collapse, or wheeze. 10.8 Bernoulli’s Equation Bernoulli’s Principle (worked out by Daniel Bernoulli) – where the velocity of a fluid is high, the pressure is low, and where the velocity is low, the pressure is high. High pressure slows the fluid down and low pressure allows the fluid to accelerate and speed up. Bernoulli dervived his equation from the above scenario. He asked, how much work would be required to move fluid from area 1 to area 2? It turns out there are three magnitudes of work acting on the fluid. First: Fluid to left exerts a pressure on area 1 Second: Fluid to the right exerts a pressure on area 2 Third: Work is done by gravity since the fluid is moving uphill. W1 = F1Δl1 = P1A1Δl1 where P = Pressure, A = Area, and l = length of pipe W2 = F2Δl2 = -P2A2Δl2 (Note that there is a negative sign because the force points in the opposite direction of the motion of the fluid). W3 = -mg (h2-h1) This is the amount of work done due to gravity. The negative sign is there because if the pipe flows uphill, gravity does negative work on the system. The net work done on the fluid is W = W1 + W2 + W3 Now if we remember back to the work – energy principle from last year, the change in kinetic energy is equal to the net work done on the system. Now let’s put it all together. ΔKE = W1 + W2 + W3 ½ mv22 – ½ mv12 = P1A1Δl1 - P2A2Δl2 - mg (h2-h1) Divide through by volume and rearrange to get the equation below known as Bernoulli’s Equation: