Topic 09 – Fluid Mechanics
Lesson Learning Outcomes
- Define buoyancy and buoyant force
- Define pressure
- Define fluid
- Explain how a fluid exerts forces on an object moving through it
- Identify the components of fluid forces
- Define drag force
- Distinguish between surface drag and form drag
- Define lift force
- Explain Bernoulli’s principle
- Explain the Magnus effect
- Identify the various factors that determine the effect fluid forces have on an object
Guiding Questions
- How does buoyant force contribute to an object or person staying afloat?
- Can I use the principles of forces to explain the concept of buoyancy?
- What is the difference between surface and form drag? Can I give examples of each in different sporting situations?
- Can I explain Bernoulli’s principle and the Magnus effect?
Buoyant Force
Two types of forces are exerted on an object by a fluid environment: a buoyant force due to the object’s immersion in the fluid and a dynamic force due to its relative motion in the fluid. The dynamic force is usually resolved into two components: drag and lift forces. The buoyant force, on the other hand, always acts vertically. A buoyant force acts upward on an object immersed in a fluid. You are probably most familiar with this principle in connection with objects in water.
Suppose you are in a pool of still water. As you dive deeper and deeper into the pool, the water exerts greater and greater pressure on you. The pressure the water exerts is due to the weight (force) of the water above you. But the water pressure doesn’t just act downward on you; the water below you pushes upward on you, and the water to either side pushes laterally on you. Water pressure acts in all directions with the same magnitude, as long as you stay at the same level. The deeper you go, the greater the pressure.
Specific Gravity and Density
Whether or not something floats is determined by the volume of the object immersed and the weight of the object compared to the weight of the same volume of water. Specific gravity is the ratio of the weight of an object to the weight of an equal volume of water. Something with a specific gravity of 1.0 or less will float. Another measure that can be used to determine if a material will float is density. Density is the ratio of mass to volume. The density of water is about 1000 kg/m3. The density of air is only about 1.2 kg/m3.
Buoyancy of the Human Body
Muscle and bone have densities greater than 1000 kg/m3 (specific gravities greater than 1.0), whereas fat has a density less than 1000 kg/m3 (specific gravity less than 1.0). These differences in density are the basis for the underwater weighing techniques used to determine body composition.
Someone who has low body fat can still float, because the lungs and other body cavities may be filled with air or other gases that are much less dense than water. A forced exhalation of the air out of the lungs may cause a lean person to sink, however. The volume of your chest increases when you inhale and decreases when you exhale, so you actually have some control over the total density of your body. To increase your buoyancy, increase your volume by inhaling forcibly. To decrease your buoyancy, decrease your volume by exhaling forcibly.
Dynamic Fluid Force: Force Relative to Motion
Buoyant force is the vertical force exerted on an object immersed in a fluid. It is present whether the object is at rest or is moving relative to the fluid. When an object moves within a fluid (or when a fluid moves past an object immersed in it), dynamic fluid forces are exerted on the object by the fluid. The dynamic fluid force is proportional to the density of the fluid, the surface area of the object immersed in the fluid, and the square of the relative velocity of the object to the fluid. The equation below summarises the relationship.
In the above equation, fluid density and object surface area are linear terms; an increase in either will cause a proportional increase in the dynamic fluid force. If the area doubles, the dynamic fluid force doubles. If the fluid becomes three times as dense, the dynamic fluid force becomes three times larger. The relative velocity term in equation is not linear, it is squared; so if the relative velocity doubles, the dynamic fluid force quadruples.
Relative Velocity
When we consider the dynamic fluid forces acting on an object, we must take into account the velocity of the object as well as the velocity of the fluid itself. Relative velocity is used to represent the effects of these two absolute velocities. Relative velocity is the difference between the object’s velocity and the fluid’s velocity. Suppose you were standing still on a running track and the wind was blowing in your face at 5 m/s. The relative velocity is the difference between your velocity and the wind’s velocity: 0 m/s − (−5 m/s) = 5 m/s.
Drag Force
Drag force, or drag, is the component of the resultant dynamic fluid force that acts in opposition to the relative motion of the object with respect to the fluid. A drag force will tend to slow down the relative velocity of an object through a fluid if it is the only force acting on the object. Drag force is defined by the equation:
Surface Drag
Surface drag is also called skin friction or viscous drag. As a fluid molecule slides past the surface of an object, the friction between the surface and the molecule slows down the molecule. On the opposite side of this molecule are fluid molecules that are now moving faster than this molecule, so these molecules are also slowed down as they slide past the molecules closest to the object. These molecules, in turn, slow down the molecules next to them. So the surface drag is proportional to the total mass of the molecules slowed down by the friction force and the average rate of change of velocity of these molecules.
Form Drag
Besides surface drag, form drag is the other means by which drag is produced. Form drag is also called shape drag, profile drag, or pressure drag. As a fluid molecule first strikes the surface of an object moving through it, it bounces off; but then the molecule strikes another fluid molecule and is pushed back toward the surface of the object. Thus the molecule will tend to follow the curvature of the object’s surface as the object moves past it. Because a change in direction is an acceleration, the object must exert a force on the molecule to change its direction. The molecule, in turn, exerts an equal but opposite force on the object as it changes direction. The larger the changes in direction, the larger the forces exerted.
Laminar and Turbulent Flow
On the trailing surfaces of the object, the forces exerted by the fluid molecules have components directed toward the front of the object. These forces
reduce the form drag; however, these forces are present only if the fluid molecules stay close to the surface of the object and press against it. This occurs during what is known as laminar flow as shown below. The lines and arrows in this figure represent the paths followed by molecules of the fluid as they pass by the object. But if the change in the surface curvature is too large or the relative velocity is too fast, the impact forces between fluid molecules are not large enough to deflect the molecules back toward the surface, and the molecules separate from the surface. Turbulent flow results, and the molecules no longer press against the surface of the object. If the fluid molecules are not pressing against the trailing surface of the object, then nothing is, and a vacuum may be created behind the object. Less force (or no force at all) is thus exerted against the trailing surfaces of the object. The difference between these forward- and backward directed forces is form drag. Form drag thus increases as the amount of turbulent flow increases.
Lift Force
Lift force is the dynamic fluid force component that acts perpendicular to the relative motion of the object with respect to the fluid. Rather than opposing the relative motion of the object through the fluid, the effect of lift force is to change the direction of the relative motion of the object through the fluid. The word lift implies that the lift force is directed upward, but this is not necessarily the case. A lift force can be directed upward, downward, or in any direction. The possible directions of the lift force are determined by the direction of flow of the fluid. The lift force must be perpendicular to this flow. Lift force is defined by equation below:
Qualitatively, lift force can be considered in the following manner. Lift is caused by the lateral deflection of fluid molecules as they pass the object. The object exerts a force on the molecules that causes this lateral deflection (an acceleration, because the molecules change direction).
According to Newton’s third law, an equal but opposite lateral force is exerted by the molecules on the object. This is the lift force. Lift force is thus proportional to the lateral acceleration of the fluid molecules and the mass of the molecules that are deflected.
Bernoulli’s Principle
In 1738, Daniel Bernoulli (1700-1782), a Swiss mathematician, discovered that faster-moving fluids exert less pressure laterally than do slower-moving fluids. This principle, called Bernoulli’s principle, may explain why some objects are able to generate lift forces even if they are not longer in one dimension than another or if their longest dimension is aligned with the flow.
Spin and Magnus Effect
In 1852, the German scientist Gustav Magnus noticed that lift forces are also generated by spinning balls. This effect is called the Magnus effect, and a lift force caused by a spin is called a Magnus force. But how can an object that doesn’t have any broad, relatively flat surface deflect the air laterally to cause a lift force?
Notice that on a ball with topspin, the upper surface of the ball has a forward velocity (to the right) relative to the center of the ball, and the lower surface has a backward velocity (to the left) relative to the center of the ball. The air molecules, on the other hand, all have a backward velocity (to the left) relative to the center of the ball. When molecules strike the lower surface of the ball, they don’t slow down as much because this surface is moving in the same direction as the molecules (backward or to the left) relative to the center of the ball. When
the molecules strike the top surface of the ball, they are slowed down more because this surface is moving in the opposite direction (forward or to the right) relative to the center of the ball. The velocity of the air over the top surface of the ball is lower than the velocity of the air over its bottom surface.
According to Bernoulli’s principle, then, less pressure will be exerted by the faster-moving
molecules on the bottom surface of the ball. This difference in pressure results in a lift force acting downward on the ball, as shown in the figure above.
Summary
- Fluid forces affect movements in a variety of sports.
- Buoyant force is an upward force equal in magnitude to the weight of water displaced by the immersed object.
- The dynamic fluid force is resolved into drag and life components and is proportional to the density of the fluid, frontal area of the object, and the square of the relative velocity.
- Ways to reduce drag include adopting a streamlined shape, keeping the surface of the object smooth and reducing frontal area.
- Lift can be controlled by the shape and orientation of an object.
- Spin on a ball can cause lift and has applications in many ball games.