4. Fluids
Propeller and jet engines generate thrust by pushing air backward. In both cases, because the wing top is curved, air streaming over it must travel farther and thus faster than air passing underneath the flat bottom.
According to Bernoulli's equation, the slower air below exerts more force on the wing than the faster air above, thereby lifting the plane. Another example of Bernoulli's equation in action is the use of pitot tubes. These are specialized measurement devices that determine the speed of fluid flow by determining the difference between the static and dynamic pressure of the fluid at given points along a tube.
KEY CONCEPT 2
An object will float if its average density is less than the average density of the fluid it is immersed in. It will sink if its average density is greater than that of the fluid.
Atmospheric pressure changes with altitude. Residents of Denver (5280 feet above sea level) experience atmospheric pressure equal to 632 mmHg (0.83 atm), whereas travelers making their way through Death Valley (282 feet below sea level) experience atmospheric pressure equal to 767 mm Hg (1.01 atm).
Atmospheric pressure impacts a number of processes, including hemoglobin's affinity for oxygen and the boiling of liquids.
The streamlines indicate some, but not all, of the pathways for the fluid along the walls of the tube. You'll notice that the tube gets wider toward Q, as indicated by the streamlines that are spreading out over the increased crosssectional area. This leads us to consider the relationship between flow rate and the cross-sectional area of the container through which the fluid is moving. Once again, we can assume that the fluid is incompressible (which means that we are not considering a flowing gas).
Because the fluid is incompressible, the rate at which a given volume (or mass) of fluid passes by one point must be the same for all other points in the closed system. This is essentially an expression of conservation of matter: if x liters of fluid pass a point in a given amount of time, then x liters of fluid must pass all other points in the system in the same amount of time. Thus, we can very clearly state, without any exceptions, the flow rate (that is, the volume per unit time) is constant for a closed system and is independent of changes in cross-sectional area.
Pressure is a scalar quantity, and therefore has a magnitude but no direction. It is easy to assume that pressure has a direction because it is related to a force, which is a vector. However, note that it is the magnitude of the normal force that is used. No matter where one positions a given surface, the pressure exerted on that surface within a closed container will be the same, neglecting gravity. For example, if we placed a surface inside a closed container filled with gas, the individual molecules, which are moving randomly within the space, will exert pressure that is the same at all points within the container.
Because the pressure is the same at all points along the walls of the container and within the space of the container itself, pressure applies in all directions at any point and, therefore, is a scalar rather than a vector. Of course, because pressure is a ratio of force to area, when unequal pressures are exerted against objects, the forces acting on the object will add in vectors, possibly resulting in acceleration. It's this difference in pressure that causes air to rush into and out of the lungs during respiration, windows to burst outward during a tornado, and the plastic covering a broken car window to bubble outward when the car is moving. Note that when gravity is present, this also results in a pressure differential.
All fluids (except superfluids, which are not tested on the MCAT) are viscous to one degree or another; those with lower viscosities are said to behave more like ideal fluids, which have no viscosity and are described as inviscid.
Because viscosity is a measure of a fluid's internal resistance to flow, more viscous fluids will "lose" more energy while flowing. Unless otherwise indicated, viscosity should be assumed to be negligible on Test Day, thus allowing Bernoulli's equation (explained later in this chapter) to be an expression of energy conservation for flowing fluids.
Water striders are insects in that have the ability to walk on water. Water striders are able to glide across the water's surface without sinking, even though they are denser than water, because of a special physical property of liquids at the interface between a liquid and a gas. Surface tension causes the liquid to form a thin but strong layer like a "skin" at the liquid's surface. Surface tension results from cohesion, which is the attractive force that a molecule of liquid feels toward other molecules of the same liquid.
Consider the intermolecular forces between the separate molecules of liquid water. For those molecules below the surface, there are attractive intermolecular forces coming from all sides; these forces balance out. However, on the surface, the molecules only have these strong attractive forces from the molecules below them, which pulls the surface of the liquid toward the center. This establishes tension in the plane of the surface of the water; when there is an indentation on the surface (say, caused by a water strider's foot) then the cohesion can lead to a net upward force.
All fluids and solids are characterized by the ratio of their mass to their volume. This is known as density, which is a scalar quantity and therefore has no direction. The equation for density is
Density of water= 1 g/mL= 1 g/cm³
Absolute (hydrostatic) pressure is the total pressure that is exerted on an object that is submerged in a fluid. Remember that fluids include both liquids and gases.
Do not make the mistake of assuming that P always stands for atmospheric pressure. In open air and most day-to-day situations P is equal to 1 atm, but in other fluid systems, the surface pressure may be higher or lower than atmospheric pressure. In a closed container, such as a pressure cooker, the pressure at the surface may be much higher than atmospheric pressure. This is, in fact, exactly the point of a pressure cooker, which allows food to cook at higher temperatures. This is because the increased pressure raises the boiling point of water in the food, thus reducing the cooking time and preventing loss of moisture.
The respiratory system is also mediated by changes in pressure, and follows the same resistance relationship as the circulatory system. During inspiration, there is a negative pressure gradient that moves air into the lungs.
During expiration, this gradient reverses. An additional point to note is that when air reaches the alveoli, it has essentially no speed.
The weight of any volume of a given substance with a known density can be calculated by multiplying the substance's density by its volume and the acceleration due to gravity. This is a calculation that appears frequently when working through buoyancy problems on Test Day:
Fg = ρ V g
Fluids are characterized by their ability to flow and conform to the shapes of their containers. Solids, on the other hand, do not flow and are rigid enough to retain a shape independent of their containers. Both liquids and gases are fluids. The natural gas (methane) that many of us use to cook flows through pipes to the burners of our stove and ovens, and the air that we breathe flows in and out of our lungs, filling the spaces of our respiratory tract and the alveoli.
Fluids and solids share certain characteristics. Both can exert forces perpendicular to their surface, although only solids can withstand shear (tangential) forces. Fluids can impose large perpendicular forces; falling into water from a significant height can be just as painful as falling onto a solid surface.
For fluids that are incompressible—that is, fluids with volumes that cannot be reduced by any significant degree through application of pressure—a change in pressure will be transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. This is Pascal's principle.
For example, an unopened carton of milk could be considered an incompressible fluid in a closed container. If one were to squeeze the container, exerting an increased pressure on the sides of the milk carton, the applied pressure would be transmitted through the entire volume of milk. If the cap were to suddenly pop off, the resulting geyser of milk would be evidence of this increased pressure.
One way to conceptualize the buoyant force is that it is the force of the liquid trying to return to the space from which it was displaced, thus trying to push the object up and out of the water. This is an important concept because the buoyant force is due to the liquid itself, not the object. If two objects placed in a fluid displace the same volume of fluid, they will experience the same magnitude of buoyant force even if the objects themselves have different masses.
How can one determine how much of a floating object lies below the surface? To do this, one can make comparisons of density or specific gravity. Remember that an object will float, no matter what it is made of and no matter how much mass it has, if its average density is less than or equal to the density of the fluid into which it is placed. If we express the object's specific gravity as a percent, this directly indicates the percent of the object's volume that is submerged (when the fluid is pure water). For instance, the density of ice is so its specific gravity is 0.92. An ice cube floating in a glass of water has 92 percent of its volume submerged in the water—only 8 percent is sitting above the surface. Therefore, any object with a specific gravity less than or equal to 1 will float in water and any object with a specific gravity greater than 1 will sink in water. A specific gravity of exactly 1 indicates that 100 percent of the object will be submerged but it will not sink.
When you check the pressure in your car or bike tires using a device known as a gauge, you are measuring the gauge pressure, which is the difference between the absolute pressure inside the tire and the atmospheric pressure outside the tire.
In other words, gauge pressure is the amount of pressure in a closed space above and beyond atmospheric pressure. This is a more common pressure measurement than absolute pressure.
Turbulent flow is rough and disorderly. Turbulence causes the formation of eddies, which are swirls of fluid of varying sizes occurring typically on the downstream side of an obstacle. When the speed exceeds the critical speed, laminar flow becomes turbulent, generating eddies on the downstream side of the object.
In unobstructed fluid flow, turbulence can arise when the speed of the fluid exceeds a certain critical speed. This critical speed depends on the physical properties of the fluid, such as its viscosity and the diameter of the tube. When the critical speed for a fluid is exceeded, the fluid demonstrates complex flow patterns, and laminar flow occurs only in the thin layer of fluid adjacent to the wall, called the boundary layer. The flow speed immediately at the wall is zero and increases uniformly throughout the layer. Beyond the boundary layer, however, the motion is highly irregular and turbulent. A significant amount of energy is dissipated from the system as a result of the increased frictional forces. Calculations of energy conservation, such as Bernoulli's equation, cannot be applied to turbulent flow systems. Luckily, the MCAT always assumes laminar (nonturbulent) flow for such questions. The Reynolds number depends on factors such as the size, shape, and surface roughness of any objects within the fluid.
The principle that derives from the story is one of Archimedes' lasting contributions to the field of physics. Archimedes' principle deals with the buoyancy of objects when placed in a fluid. It helps us understand how ships stay afloat and why we feel lighter when we're swimming. The principle states that a body wholly or partially immersed in a fluid will be buoyed upwards by a force equal to the weight of the fluid that it displaces.
Just as Archimedes' body and his crown caused the water level to rise in the tub, any object placed in a fluid will cause a volume of fluid to be displaced equal to the volume of the object that is submerged. Because all fluids have density, the volume of fluid displaced will correspond to a certain mass of that fluid. The mass of the fluid displaced exerts a force equal to its weight against the submerged object. This force, which is always directed upward, is called the buoyant force.
When a fluid is moving, its flow can be laminar or turbulent. Laminar flow is smooth and orderly, and is often modeled as layers of fluid that flow parallel to each other. When the gravitational force is larger than the buoyant force, an object will sink.
Laminar flow is characterized by smooth flow lines around the object. The layers will not necessarily have the same linear speed. For example, the layer closest to the wall of a pipe flows more slowly than the more interior layers of fluid.
KEY CONCEPT 3
Low-viscosity fluids have low internal resistance to flow and behave like ideal fluids. Assume conservation of energy in low-viscosity fluids with laminar flow.
One application of Pascal's principle can be seen in hydraulic systems. These systems take advantage of the near-incompressibility of liquids to generate mechanical advantage, which, , allows us to accomplish a certain amount of work more easily by applying reduced forces.
Many heavy machines use hydraulics, including car brakes, bulldozers, cranes, and lifts.
Pressure is a ratio of the force per unit area. The equation for pressure is P=F/A
Other commonly used units of pressure are millimeters of mercury (mmHg), torr, and the atmosphere (atm). Millimeters of mercury and torr are identical units. The unit of atmosphere is based on the average atmospheric pressure at sea level. The conversions between Pa, mmHg, torr, and atm are as follows: 1.013 × 10 Pa = 760 mmHg ≡ 760 torr = 1 atm
KEY CONCEPT 1
Remember when applying Pascal's principle that the larger the area, the larger the force, although this force will be exerted through a smaller distance.
As blood flows away from the heart, each vessel has a progressively higher resistance; however, the total resistance of the system decreases because the increased number of vessels are in parallel with each other. Like parallel resistors in circuits, the equivalent resistance is therefore lower for the capillaries in parallel than in the aorta.
Return flow to the heart is facilitated by mechanical squeezing of the skeletal muscles, which increases pressure in the limbs and pushes blood to the heart, and the expansion of the heart, which decreases pressure in the heart and pulls blood in. Finally, the pressure gradients created in the thorax by inhalation and exhalation also motivate blood flow. Venous circulation holds approximately three times as much blood as arterial circulation. Heart murmurs, which result from structural defects of the heart, are heard because of turbulent blood flow.
Because the movement of individual molecules of a fluid is impossible to track with the unaided eye, it is often helpful to use representations of the molecular movement called streamlines.
Streamlines indicate the pathways followed by tiny fluid elements (sometimes called fluid particles) as they move. The velocity vector of a fluid particle will always be tangential to the streamline at any point. Streamlines never cross each other.
Before we cover Bernoulli's equation itself, let's approach a flowing fluid from two perspectives that we've already discussed. First, the continuity equation arises from the conservation of mass of fluids. Liquids are essentially incompressible, so the flow rate within a closed space must be constant at all points.
The continuity equation shows us that for a constant flow rate, there is an inverse relationship between the linear speed of the fluid and the cross-sectional area of the tube: fluids have higher speeds through narrower tubes.
Let's determine how such a lift could allow an auto mechanic to raise a heavy car with far less force than the weight of the car. We have a closed container that is filled with an incompressible liquid. On the left side of the lift, there is a piston of cross-sectional area A1 . When this piston is pushed down the column, it exerts a force with a magnitude equal to F1 and generates a pressure equal to P1 .
The piston displaces a volume of liquid equal to A1d1 (the cross-sectional area times the distance gives a volume). Because the liquid inside is incompressible, the same volume of fluid must be displaced on the right side of the hydraulic lift, where we find a second piston with a much larger surface area, A2 . The pressure generated by piston 1 is transmitted undiminished to all points within the system, including to A2 . As A2 is larger than A1 by some factor, the magnitude of the force, F2 , exerted against A2 must be greater than F1 by the same factor so that P1 = P2 , according to Pascal's principle.
Some fluids flow very easily, while others barely flow at all. The resistance of a fluid to flow is called viscosity (η). Increased viscosity of a fluid increases its viscous drag, which is a nonconservative force that is analogous to air resistance.
Thin fluids, like gases, water, and dilute aqueous solutions, have low viscosity and so they flow easily. Objects can move through these fluids with low viscous drag. Whole blood, vegetable oil, honey, cream, and molasses are thick fluids and flow more slowly. Objects can move through these fluids, but with significantly more viscous drag.
What this series of equations shows us is that hydraulic machines generate output force by magnifying an input force by a factor equal to the ratio of the cross-sectional area of the larger piston to that of the smaller piston.
This does not violate the law of energy conservation; an analysis of the input and output work in a frictionless system reveals that there indeed is conservation of energy. As mentioned above, the volume of fluid displaced by piston 1 is equal to the volume of fluid displaced at piston 2.
The circulatory system is a closed loop that has a nonconstant flow rate. This nonconstant flow is a result of valves, gravity, the physical properties of our vessels (elasticity, in particular), and the mechanics of the heart. In particular, the nonconstant flow can be felt and measured as a pulse. In addition to these features, there is a loss of volume from the circulation as a result of a difference between osmotic (oncotic) and hydrostatic pressures.
This fluid is eventually returned to the circulation as a result of lymphatic flow, but it is problematic for applications of the continuity equation. An important point to note is that despite these differences, blood volume entering the heart is always equal to blood volume leaving the heart during a single cycle.
The density of a fluid is often compared to that of pure water at 1 atm and 4°C, a variable called specific gravity. It is at this combination of pressure and temperature that water has a density of exactly The specific gravity is given by
This is a unitless number that is usually expressed as a decimal. The specific gravity can be used as a tool for determining if an object will sink or float in water.
When an object is placed in a fluid, it will sink into the fluid only to the point at which the volume of displaced fluid exerts a force that is equal to the weight of the object. If the object becomes completely submerged and the volume of displaced fluid still does not exert a buoyant force equal to the weight of the object, the object will accelerate downwards and sink to the bottom.
This will be the case if an object is more dense than the fluid it's in—a gold crown will sink to the bottom of the bathtub because it is more dense than water. On the other hand, an object that is less dense than water, such as a block of wood or an ice cube, will stop sinking (and start floating) because it is less dense than water. These objects will submerge enough of their volume to displace a volume of water equal to the object's weight.
You've probably heard some version of this story before: Archimedes, a physicist in ancient Greece, was tasked by his king to determine the metallic composition of a certain crown given to the king as a gift. Archimedes knew that he could do this by finding the crown's volume and mass, which would allow him to find its density and compare that density to those of known metals.
Weighing the crown would be easy enough, but he was having trouble finding a way to measure its volume without melting it down and ruining its workmanship. Then one day, while getting into his bath, the water that overflowed from the tub gave him the idea to submerge the crown in water and measure the volume of the displaced liquid.
A common application of Bernoulli's equation on the MCAT is the Venturi flow meter. As the tube narrows, the linear speed increases at point 2. Thus, the pressure exerted on the walls decreases, causing the column above the tube to have a lower height at point 2.
When considering Bernoulli's equation in this example, start by noting that the average height of the tube itself remains constant. Therefore, the ρgh term remains constant at points 1 and 2. Note that the h shown in Figure 4.6 is difference in height between the two columns at points 1 and 2, not h from Bernoulli's equation, which corresponds to the average height of the tube above a datum. As the cross-sectional area decreases from point 1 to point 2, the linear speed must increase according to the continuity equation. Then, as the dynamic pressure increases, the absolute pressure must decrease at point 2. With a lower absolute pressure, the column of fluid sticking up from the Venturi tube will be lower at point 2. This phenomenon is often called the Venturi effect.
Another force that liquid molecules experience is adhesion, which is the attractive force that a molecule of the liquid feels toward the molecules of some other substance. For example, adhesive forces cause water molecules to form droplets on the windshield of a car even though gravity is pulling them downward.
When liquids are placed in containers, a meniscus, or curved surface in which the liquid "crawls" up the side of the container a small amount, will form when the adhesive forces are greater than the cohesive forces. A backwards (convex) meniscus (with the liquid level higher in the middle than at the edges) occurs when the cohesive forces are greater than the adhesive forces. Mercury, the only metal that is liquid at room temperature, forms a backward meniscus when placed in a container.
KEY CONCEPT 4
While flow rate is constant in a tube regardless of cross-sectional area, linear speed of a fluid will increase with decreasing cross-sectional area.
Hydrostatics
is the study of fluids at rest and the forces and pressures associated with standing fluids.
Second, fluids that have low viscosity and demonstrate laminar flow can also be approximated to be conservative systems. The total mechanical energy of the system is constant if we discount the small viscous drag forces that occur in all real liquids. Combining these principles of conservation, we arrive at Bernoulli's equation:
where P is the absolute pressure of the fluid, ρ is the density of the fluid, ν is the linear speed, g is acceleration due to gravity, and h is the height of the fluid above some datum. Some of the terms of Bernoulli's equation should look vaguely familiar. The term 1/2 ρv^2 is sometimes called the dynamic pressure, and is the pressure associated with the movement of a fluid. This term is essentially the kinetic energy of the fluid divided by volume (ρ=m/v). The term ρgh looks like the expression for gravitational potential energy, and is essentially the pressure associated with the mass of fluid sitting above some position. Finally, let's consider how the absolute pressure fits into this conservation equation. If one multiplies the unit of pressure (N/m^2)by meters over meters, we obtain N x m/m^3=J/m^3. Pressure can therefore be thought of as a ratio of energy per cubic meter, or energy density. Systems at higher pressure have a higher energy density than systems at lower pressure. Finally, the combination of P + ρ gh gives us the static pressure, and is the same equation as that for absolute pressure (although h is used here to imply height above a certain point, whereas z was used earlier to imply depth below a certain point). Bernoulli's equation states, then, that the sum of the static pressure and dynamic pressure will be constant within a closed container for an incompressible fluid not experiencing viscous drag. In the end, Bernoulli's equation is nothing other than a statement of energy conservation: more energy dedicated toward fluid movement means less energy dedicated toward static fluid pressure. The inverse of this is also true—less movement means more static pressure. One example of this principle that you may have previously encountered is how the shape of an airplane's wing helps generate lift,
With laminar flow through a pipe or confined space, it is possible to calculate the rate of flow using Poiseuille's law:
where Q is the flow rate (volume flowing per time), r is the radius of the tube, ΔP is the pressure gradient, η (eta) is the viscosity of the fluid, and L is the length of the pipe. This equation is rarely tested in full; most often, MCAT passages and questions focus on the relationship between the radius and pressure gradient. Note that the relationship between the radius and pressure gradient is inverse exponential to the fourth power—even a very slight change in the radius of the tube has a significant effect on the pressure gradient, assuming a constant flow rate.
While the flow rate is constant, the linear speed of the fluid does change relative to cross-sectional area. Linear speed is a measure of the linear displacement of fluid particles in a given amount of time. Notably, the product of linear speed and cross-sectional area is equal to the flow rate. We've already said that the volumetric rate of flow for a fluid must be constant throughout a closed system. Therefore, Q = ν1 A1 = ν1 A1
where Q is the flow rate, ν and ν are the linear speeds of the fluid at points 1 and 2, respectively, and A and A are the cross-sectional areas at these points. This equation is known as the continuity equation, and it tells us that fluids will flow more quickly through narrow passages and more slowly through wider ones.
