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Chapter 5: Problem 11

During World War II, Sir Geoffrey Taylor, a British fluid dynamicist, used dimensional analysis to estimate the wave speed of an atomic bomb explosion. He assumed that the blast wave radius \(R\) was a function of energy released \(E,\) air density \(\rho,\) and time \(t .\) Use dimensional reasoning to show how wave radius must vary with time.

Short Answer

Expert verified
The wave radius \( R \) varies with time \( t^{2/5} \).

Step by step solution

01

Identify the Physical Quantities Involved

We are given that the blast wave radius \( R \) is a function of the energy released \( E \), the air density \( \rho \), and the time \( t \). We denote these relationships as \( R = f(E, \rho, t) \).
02

Determine the Dimensions of Each Quantity

Next, we need to determine the dimensional formula for each physical quantity. - Energy \( E \) has dimensions \( [M][L]^2[T]^{-2} \).- Air density \( \rho \) has dimensions \( [M][L]^{-3} \).- Time \( t \) has dimensions \( [T] \).- Radius \( R \) has dimensions \( [L] \).
03

Set Up Dimensional Equations

To find how \( R \) varies with \( t \), we assume a dimensionally consistent equation \( R = C E^a \rho^b t^c \), where \( C \) is a dimensionless constant, and \( a, b, c \) are the exponents we need to find.
04

Write and Solve the Dimensional Equation

Breaking down the assumed equation using dimensions:- Left-hand side: \([L]\)- Right-hand side: \(( [M][L]^2[T]^{-2} )^a( [M][L]^{-3} )^b( [T] )^c = [M]^{a+b}[L]^{2a-3b}[T]^{-2a+c}\)Equating dimensions:- For mass: \( a + b = 0 \)- For length: \( 2a - 3b = 1 \)- For time: \( -2a + c = 0 \).Solving these equations gives \( a = \frac{1}{5}, b = -\frac{1}{5}, c = \frac{2}{5} \).
05

Conclude the Dimensional Relationship

Now, substituting \( a, b, c \) into our assumed equation, we have \( R = C E^{1/5} \rho^{-1/5} t^{2/5} \). Thus, the wave radius \( R \) varies with time \( t \) as \( t^{2/5} \).

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Key Concepts

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

Wave Radius
The wave radius in the context of a blast wave refers to the distance from the center of an explosion to the outermost boundary of the effect caused by the blast. Understanding how this radius changes over time is crucial when studying explosions, such as those from atomic bombs.
Dimensional analysis helps us model the variation of this radius based on key physical parameters.
In our specific exercise, we found that the wave radius \( R \) depends on the energy released \( E \), the air's density \( \rho \), and the time \( t \) since the explosion occurred. By determining how these factors interact dimensionally, we are able to express the wave radius as a function of time in mathematical terms.
Fluid Dynamics
Fluid dynamics is the branch of physics that describes the behavior of fluids (liquids and gases) in motion. This is particularly relevant for understanding explosion waves in air, a gaseous fluid.
When a blast occurs, it generates a rapid expansion of gases, creating a shock wave that propagates through the medium, affecting the wave radius.
Key principles of fluid dynamics like pressure changes, energy transfer, and fluid flow play a significant role in predicting and analyzing these phenomena through the study of variables like velocity and density.
Blast Wave
A blast wave is the intense wave of pressure that is created after an explosion. It consists of highly compressed air moving outward rapidly.
  • The wave travels faster than the speed of sound initially and decreases in speed as it moves further out.
  • The ability to mathematically model how the radius of this wave changes over time is crucial in applications such as safety planning and understanding the impact of explosions.
The mathematical relation derived from dimensional analysis exemplifies how a blast's energy, air density, and time interact to affect the radius of a blast wave.
Dimensional Consistency
Dimensional consistency refers to ensuring that the dimensions on both sides of an equation match. In the context of our exercise, we created a model: \( R = C E^a \rho^b t^c \).
Here, dimensional consistency ensures that the combination of parameters \( E \), \( \rho \), and \( t \) result in the dimension of length, \([L]\), to match the wave radius \( R \). Applying this principle is essential for deriving physically accurate equations and ensuring logical interrelations among the parameters.
  • Mass dimensions have to balance out such that \( a + b = 0 \).
  • Length dimensions are balanced by \( 2a - 3b = 1 \).
  • Time dimensions require \( -2a + c = 0 \).
By solving these equations, we maintain dimensional consistency and derive meaningful results.

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Most popular questions from this chapter

A weir is an obstruction in a channel flow that can be calibrated to measure the flow rate, as in Fig. P5.32. The volume flow \(Q\) varies with gravity \(g,\) weir width \(b\) into the paper, and upstream water height \(H\) above the weir crest. If it is known that \(Q\) is proportional to \(b,\) use the pi theorem to find a unique functional relationship \(Q(g, b, H)\)

A prototype ocean platform piling is expected to encounter currents of \(150 \mathrm{cm} / \mathrm{s}\) and waves of 12 -s period and \(3-\mathrm{m}\) height. If a one-fifteenth-scale model is tested in a wave channel, what current speed, wave period, and wave height should be encountered by the model?

The pressure drop per unit length \(\Delta p / L\) in smooth pipe flow is known to be a function only of the average velocity \(V,\) diameter \(D,\) and fluid properties \(\rho\) and \(\mu .\) The following data were obtained for \(\mathrm{Aw}\) of water at \(20^{\circ} \mathrm{C}\) in an 8 -cm-diameter pipe \(50 \mathrm{m}\) long: $$\begin{array}{l|l|l|l|l} Q, \mathrm{m}^{3} / \mathrm{s} & 0.005 & 0.01 & 0.015 & 0.020 \\ \hline \Delta p, \mathrm{Pa} & 5800 & 20,300 & 42,100 & 70,800 \end{array}$$ Verify that these data are slightly outside the range of Fig. \(5.10 .\) What is a suitable power-law curve fit for the present data? Use these data to estimate the pressure drop for Aw of kerosene at \(20^{\circ} \mathrm{C}\) in a smooth pipe of diameter \(5 \mathrm{cm}\) and length \(200 \mathrm{m}\) if the flow rate is \(50 \mathrm{m}^{3} / \mathrm{h}\)

The differential energy equation for incompressible two dimensional flow through a "Darcy-type" porous medium is approximately $$\rho c_{p} \frac{\sigma}{\mu} \frac{\partial p}{\partial x} \frac{\partial T}{\partial x}+\rho c_{p} \frac{\sigma}{\mu} \frac{\partial p}{\partial y} \frac{\partial T}{\partial y}+k \frac{\partial^{2} T}{\partial y^{2}}=0$$ where \(\sigma\) is the permeability of the porous medium. All other symbols have their usual meanings. (a) What are the appropriate dimensions for \(\sigma ?\) (b) Nondimensionalize this equation, using \(\left(L, U, \rho, T_{0}\right)\) as scaling constants, and discuss any dimensionless parameters that arise.

When tested in water at \(20^{\circ} \mathrm{C}\) flwing at \(2 \mathrm{m} / \mathrm{s}\), an \(8-\mathrm{cm}\) diameter sphere has a measured drag of 5 N. What will be the velocity and drag force on a 1.5 -m-diameter weather balloon moored in sea-level standard air under dynamically similar conditions?
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