Question

Use dimensional analysis to derive a possible expression for the drag force \(F_{D}\) on a soccer ball of diameter \(D\) moving at speed \(v\) through air of density \(\rho\) and viscosity \(\mu .\) [Hint: Assuming viscosity has units \(\frac{[M]}{[L][T]}\), there are two possible dimensionless combinations: \(\pi_{1}=\mu D^{\alpha} \rho^{\beta} v^{\gamma}\) and \(\pi_{2}=F_{D} D^{a} \rho^{\beta} v^{\gamma} .\) Determine \(\alpha, \beta\), and \(\gamma\) for each case, and interpret your results.]

Short answer

The drag force \(F_D\) on the soccer ball can be represented proportionally as \(F_D \propto \mu v D + \rho v^2 D^2\). The powers indicate that drag force increases with viscosity, velocity and diameter of the ball.

Step-by-step solution

- Set up the dimensional analysis

Start by setting up the required dimensionless parameters. The hint states that there are two dimensionless combinations that have constant ratios, these are denoted as \(\pi_1\) and \(\pi_2\). Specifically, \(\pi_1 = \mu D^{\alpha} \rho^{\beta} v^{\gamma}\) and \(\pi_2 = F_{D} D^{a} \rho^{b} v^{c}\). Have a close look at the units involved, the units of \(\mu\) are given as [M]/[L][T]}

- Determine the dimensions of other quantities

Next, we will determine the dimensions of the other quantities in the dimensions, length [L], mass [M] and time [T]. Diameter \(D\) is a linear dimension and its dimensional unit is [L]. Velocity \(v\) is length per unit time which gives it dimensional units of [L]/[T]. Air density \(\rho\) which is mass per unit volume, has dimensional units of [M]/[L]^3. Drag force \(F_D\) which is basically force has dimensional units of [M][L]/[T]^2 as it is mass times acceleration.

- Apply the Principle of Dimensional Homogeneity

The Principle of Dimensional Homogeneity states that equations must be dimensionally consistent. The dimensions on the left-hand side must equal the dimensions on the right-hand side. Using the principle on our \(\pi\) terms, \(\pi_1 = \mu D^{\alpha} \rho^{\beta} v^{\gamma}\) and rearrange to get the individual coefficients, we get \(\alpha = -1, \beta = -1, \gamma = 1\). For \(\pi_2 = F_{D} D^{a} \rho^{b} v^{c}\), we get \(a=2, b=-1, c=-1\).

- Interpret the results

Using these coefficients \(\alpha, \beta, \gamma, a, b, c\), we can formulate a proportionality expression for \(F_D\) as follows: \(F_D \propto \mu v D + \rho v^2 D^2\)

Key concepts

Understanding Drag Force

Drag force is the resistance an object encounters when moving through a fluid, such as air or water. It acts in the opposite direction to the motion of the object. Every object moving through a fluid experiences drag force, whether it's a soccer ball flying through the air or a boat rafting on a river.
Here's how drag force can be visualized:

• It's similar to trying to walk against a strong wind. The wind pushes back, making it harder to move forward.
• The larger the surface area, like the sail on a sailboat, the more drag force there is to overcome.

Drag force, denoted as \( F_D \), is crucial in understanding how objects move through different mediums. It influences the motion by slowing objects down, and it is an essential factor in aerodynamic and hydrodynamic design.

Exploring Drag Coefficient

The drag coefficient is a dimensionless number that represents the drag force acting on an object relative to its size and speed. It is a critical factor in the drag equation, which determines how much drag an object experiences.
The drag coefficient helps predict how smoothly an object can pass through the air or water. It depends on various factors:

• The shape of the object: Sleek, streamlined shapes have lower drag coefficients.
• The surface texture: Smooth surfaces allow for less resistance.
• The flow conditions: Turbulent versus laminar flow impacts the drag coefficient.

A low drag coefficient implies that the object will face less resistance, making it more efficient in terms of energy consumption. Engineers use it in designing vehicles, aircraft, and sports equipment to optimize performance.

Reynolds Number and Its Significance

Reynolds number is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It helps determine whether the fluid flow is laminar or turbulent.
The Reynolds number is calculated using the formula:\[Re = \frac{\rho v D}{\mu}\]where:

• \( \rho \) is the fluid density
• \( v \) is the velocity of the fluid or object
• \( D \) is the characteristic length (such as diameter)
• \( \mu \) is the fluid's viscosity

A low Reynolds number indicates laminar flow, which is smooth and orderly, whereas a high Reynolds number indicates turbulent flow, which is chaotic and mixed. Understanding whether the flow is laminar or turbulent helps in predicting how the drag force will behave, impacting designs in engineering and natural environments.

What is Viscosity?

Viscosity is a measure of a fluid's resistance to flow. It's like the internal friction in a fluid, determining how easily it flows.
Think of it this way:

• Honey has high viscosity; it flows slowly because its molecules resist motion.
• Water, on the other hand, has low viscosity and flows easily.

In mathematical terms, viscosity is represented as \( \mu \). It is critical for calculating the Reynolds number and assessing drag force because it influences the fluid dynamics around an object.
Viscosity is crucial in many real-world applications, such as in automotive engineering, where oil viscosity affects engine performance, or in meteorology, where air viscosity impacts weather patterns. Understanding viscosity enables engineers and scientists to create more efficient systems and predict natural behaviors in fluid contexts.