MCAT Physics and Math Review

Chapter 4: Fluids

4.3 Fluid Dynamics

As the term suggests, fluid dynamics is the study of fluids in motion. This is perhaps one of the most fascinating areas of physics because its applications to real life are everywhere. Many aspects of our world, from water delivery to our homes to blood flow through our arteries and veins, can be analyzed and explained (at least in part) by the principles of fluid dynamics. The MCAT presents a relatively simplified version of the topic, making important assumptions such as rigid-walled containers and uniform density of fluids.

VISCOSITY

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.

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.

The SI unit of viscosity is the pascal–second 

KEY CONCEPT

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.

LAMINAR AND TURBULENT FLOW

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, as shown in Figure 4.3.

Figure 4.3. Laminar Flow around an Object Sinking in a Fluid 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.

Poiseuille’s Law

With laminar flow through a pipe or confined space, it is possible to calculate the rate of flow using Poiseuille’s law:

Equation 4.9

where Q is the flow rate (volume flowing per time), r is the radius of the tube, Δis 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.

Turbulence and Speed

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, as shown in Figure 4.4.

Figure 4.4. Turbulent Flow around an Object in a Wind Chamber 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.

For a fluid flowing through a tube of diameter D, the critical speed, νc, can be calculated as

Equation 4.10

where νc is the critical speed, NR is a dimensionless constant called the Reynolds number, η is the viscosity of the fluid, and ρ is the density of the fluid. The Reynolds number depends on factors such as the size, shape, and surface roughness of any objects within the fluid.

STREAMLINES

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.

Figure 4.5. Streamlines The stream’s cross-sectional area increases from P to Q.

Figure 4.5 shows a fluid within an invisible tube as it passes from P to Q. 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 cross-sectional 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 xliters 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.

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 = ν1A1 = ν2A2

Equation 4.11

where Q is the flow rate, ν1 and ν2 are the linear speeds of the fluid at points 1 and 2, respectively, and A1 and A2 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. Therefore, in Figure 4.5 earlier, while the flow rate at points P and Q are the same, the linear speed is faster at point P than point Q.

KEY CONCEPT

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.

BERNOULLI’S EQUATION

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.

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:

Equation 4.12

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  is sometimes called thedynamic 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  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  by meters over meters, we obtain  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 + ρgh 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, as shown in Figure 4.6.

Figure 4.6. Aerodynamics of an Airplane

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.

A common application of Bernoulli’s equation on the MCAT is the Venturi flow meter, as shown in Figure 4.7.

Figure 4.7. 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.

Example:

An office building with a bathroom 40 m above the ground has its water supply enter the building at ground level through a pipe with an inner diameter of 4 cm. If the linear speed at the ground floor is  and at the bathroom is  determine the cross-sectional area of the pipe in the bathroom. If the pressure in the bathroom is 3 × 105 Pa, what is the required pressure at ground level?

Solution:

The cross-sectional area of the pipe in the bathroom is calculated using the continuity equation, where point 1 is the ground level and point 2 is the bathroom:

The pressure can be found from Bernoulli’s equation:

MCAT Concept Check 4.3:

Before you move on, assess your understanding of the material with these questions.

1.    Define the following terms:

·        Dynamic pressure:

·        Static pressure:

·        Pitot tube:

·        Viscosity:

·        Laminar flow:

·        Turbulence:

2.    How do the following concepts relate to one another: Venturi effect, Bernoulli’s equation, and continuity equation? What relationship does each describe?

3.    What variables does flow rate depend on?