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Chapter 6: Problem 35

Observations on the International Space Station indicate that if a small spherical liquid droplet suspended in space is deformed slightly, then its surface will oscillate with a frequency that depends on the diameter of the droplet and the density and viscosity of the liquid. Use dimensional analysis to estimate the functional relationship between the oscillation frequency of the droplet surface and the influencing variables.

Short Answer

Expert verified
The oscillation frequency is proportional to \(D^2\mu/\rho\).

Step by step solution

01

Identify Variables

Identify the primary variables that influence the oscillation: diameter of the droplet \(D\), density of the liquid \(\rho\), viscosity of the liquid \(\mu\), and the oscillation frequency \(f\).
02

Determine Dimensions of Variables

Express each variable in terms of their fundamental dimensions: \(D\) has dimensions \([L]\), \(\rho\) has dimensions \([M][L]^{-3}\), \(\mu\) has dimensions \([M][L]^{-1}[T]^{-1}\), and \(f\) has dimensions \([T]^{-1}\).
03

Apply Buckingham Pi Theorem

Use the Buckingham Pi theorem to find dimensionless groups. We have 3 primary quantities (\(M, L, T\)) and 4 variables, leading to a single dimensionless group. Assume \(f = C \times D^a \times \rho^b \times \mu^c\), where \(C\) is a dimensionless constant.
04

Set Up Dimensionless Equation

Balance dimensions: \([T]^{-1} = [L]^{a} [M]^{b} [L]^{-3b} [M]^{c} [L]^{-c} [T]^{-c}\). Simplify: \([T]^{-1} = [L]^{a - 3b - c} [M]^{b + c} [T]^{-c}\).
05

Solve for Exponents

Equate the exponents of dimensions from both sides. \(b + c = 0\), \(-c = -1\), and \(a - 3b - c = 0\). Solving gives \(c = 1\), \(b = -1\), and \(a = 2\).
06

Formulate Relationship

Substitute back the derived exponents to get: \(f \propto D^2 \times \rho^{-1} \times \mu\). Therefore, the frequency \(f\) is proportional to the square of the diameter, inversely proportional to the density, and directly proportional to the viscosity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Oscillation Frequency
The concept of oscillation frequency is crucial in understanding how often a surface, such as that of a liquid droplet in space, vibrates over a period of time.
Oscillation frequency is typically measured in Hertz (Hz), representing cycles per second.
For our experiment conducted in a microgravity environment like the International Space Station, a spherical droplet鈥檚 oscillation frequency can indicate the dynamic stability of its surface. When the droplet is slightly perturbed or deformed, it will tend to return to equilibrium, creating a wave-like motion on its surface.
During oscillation, various physical properties play a role in determining the frequency, such as:
  • Diameter of the droplet
  • Density of the liquid
  • Viscosity of the liquid
Ultimately, the oscillation frequency links directly to these variables, as dimensional analysis helps establish the proportional relationship between these quantities.
Fluid Mechanics
In the realm of physics, fluid mechanics plays a pivotal role in understanding the behavior of liquids and gases in motion.
On the International Space Station, the microgravity environment presents a unique setting for observing small spherical droplets suspended in space.
Here, fluid mechanics principles allow investigators to assess how liquids behave in the absence of strong gravitational forces.
Some key concepts in fluid mechanics that are particularly relevant to this scenario include:
  • Surface Tension: The force that causes the surface of a liquid to contract and behave like an elastic sheet, impacting droplet stability and deformation.
  • Viscosity: The internal friction within the fluid that resists flow and affects oscillation frequency.
  • Fluid Density: This determines how mass is distributed within a given volume, influencing the droplet鈥檚 oscillation and stability.
Dimensional analysis combines these attributes to better predict and understand the behavior of fluids, aiding in precise calculations and hypotheses about droplet oscillations under reduced gravity.
Dimensionless Groups
Dimensionless groups are powerful tools in engineering and physics used to simplify and analyze relationships between physical quantities.
Through Buckingham Pi Theorem, these groups help reduce complex equations into simpler, more generalized forms, devoid of constant units.
In our study of a droplet's oscillation in space, we found that only one dimensionless group was necessary.
This allows the problem to be analyzed in a more tractable form, where relationships between variables are clearer.
The steps to applying dimensional analysis often include:
  • Identifying relevant variables (e.g., droplet diameter, liquid density and viscosity, oscillation frequency).
  • Expressing each in terms of fundamental dimensions.
  • Using the Buckingham Pi theorem to establish a dimensionless group.
By leveraging dimensionless groups, it becomes easier to relate the droplet鈥檚 oscillation frequency to its physical characteristics, yielding a better understanding of this delicate process in fluid dynamics.

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Most popular questions from this chapter

The energy per unit mass (of fluid), \(e\), added by a pump of a given shape depends on the pump size, \(D,\) volume flow rate, \(Q,\) speed of the rotor, \(\omega,\) density of the fluid, \(\rho\), and dynamic viscosity of the fluid, \(\mu\). This functional relation can be stated as $$e=f(D, Q, \omega, \rho, \mu)$$ Express this as a relationship between dimensionless groups. What is gained by expressing the pump performance as an empirical relationship between dimensionless groups versus expressing the pump performance as a relationship between the given dimensional variables?

It is known from engineering analysis that for a liquid flowing upward in a vertical pipe, the gauge pressure, \(p\), in the a pipe at a height \(z\) above the liquid surface in the source reservoir is given by $$p=-\gamma\left(1+0.24 \frac{Q^{2}}{g D^{5}}\right) z$$ where \(\gamma\) is the specific weight of the liquid, \(Q\) is the volume flow rate \(\left(\mathrm{L}^{3} \mathrm{~T}^{-1}\right),\) and \(D\) is the diameter of the pipe. Show that Equation 6.25 is dimensionally homogeneous.

A prototype water pump with an impeller diameter of \(470 \mathrm{~mm}\) is designed to operate at a rotational speed of \(950 \mathrm{rpm},\) and at this speed, the pump delivers a volume flow rate of \(1.7 \mathrm{~m}^{3} / \mathrm{s}\). The temperature of the water in the prototype pump is \(20^{\circ} \mathrm{C}\). The performance of a 1: 6 scale model of the pump is tested using standard air as the fluid. The model pump has a rotational speed of \(1750 \mathrm{rpm},\) and the power required to drive the model pump is \(95 \mathrm{~W}\). The Reynolds number in the prototype and the model are both sufficiently high that Reynolds similarity is not a requirement. (a) Determine the volume flow rate of air in the model that corresponds to the design volume flow rate of water in the prototype. (b) Determine the power requirement of the prototype that corresponds to the measured power requirement in the model.

A 1: 12 model of a spillway is tested under a particular upstream condition. The measured velocity and flow rate over the model spillway are \(0.68 \mathrm{~m} / \mathrm{s}\) and \(0.12 \mathrm{~m}^{3} / \mathrm{s}\), respectively. What are the corresponding velocity and flow rate in the actual spillway?

An atomizer is a common term used to describe a spray nozzle that produces small droplets of a liquid that is drawn into the nozzle by the low pressure that exists in the nozzle. Consider a nozzle of diameter \(D\) that generates droplets of diameter \(d\) when the velocity at the nozzle exit is \(V\). The relevant liquid properties are the density, \(\rho\), the viscosity, \(\mu,\) and the surface tension, \(\sigma .\) With the objective of predicting the size of the droplets generated by an atomizer, express the relationship between the relevant variables in nondimensional form where, to the extent possible, dimensionless groups representing force ratios are used with each of the fluid properties.
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