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Fluid Mechanics 4th

Frank M. White

Chapter 5

Dimensional Analysis and Similarity

Educators


Problem 1

For axial flow through a circular tube, the Reynolds number for transition to turbulence is approximately 2300 [see Eq. $(6.2)]$, based upon the diameter and average velocity. If $d=5 \mathrm{cm}$ and the fluid is kerosine at $20^{\circ} \mathrm{C}$, find the volume flow rate in $\mathrm{m}^{3} / \mathrm{h}$ which causes transition.

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Problem 2

In flow past a thin flat body such as an airfoil, transition to turbulence occurs at about $\mathrm{Re}=1 \mathrm{E} 6$, based on the distance $x$ from the leading edge of the wing. If an airplane flies at $450 \mathrm{mi} / \mathrm{h}$ at $8-\mathrm{km}$ standard altitude and undergoes transition at the 12 percent chord position, how long is its chord (wing length from leading to trailing edge)?

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Problem 3

An airplane has a chord length $L=1.2 \mathrm{m}$ and flies at a Mach number of 0.7 in the standard atmosphere. If its Reynolds number, based on chord length, is 7 E6, how high is it flying?

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Problem 4

When tested in water at $20^{\circ} \mathrm{C}$ flowing at $2 \mathrm{m} / \mathrm{s},$ an $8-\mathrm{cm}-\mathrm{di}$ ameter sphere has a measured drag of $5 \mathrm{N}$. 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 çonditions?

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Problem 5

An automobile has a characteristic length and area of $8 \mathrm{ft}$ and $60 \mathrm{ft}^{2},$ respectively. When tested in sea-level standard air, it has the following measured drag force versus speed:
$$\begin{array}{l|c|c|c}V, \mathrm{mi} / \mathrm{h} & 20 & 40 & 60 \\\hline \text { Drag, lbf } & 31 & 115 & 249\end{array}$$
The same car travels in Colorado at $65 \mathrm{mi} / \mathrm{h}$ at an altitude of $3500 \mathrm{m} .$ Using dimensional analysis, estimate $(a)$ its drag force and ( $b$ ) the horsepower required to overcome air drag.

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Problem 6

SAE 10 oil at $20^{\circ} \mathrm{C}$ flows past an 8 -cm-diameter sphere. At flow velocities of $1,2,$ and $3 \mathrm{m} / \mathrm{s},$ the measured sphere drag forces are $1.5,5.3,$ and $11.2 \mathrm{N},$ respectively. Estimate the drag force if the same sphere is tested at a velocity of 15 $\mathrm{m} / \mathrm{s}$ in glycerin at $20^{\circ} \mathrm{C}$

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Problem 7

A body is dropped on the moon $\left(g=1.62 \mathrm{m} / \mathrm{s}^{2}\right)$ with an initial velocity of $12 \mathrm{m} / \mathrm{s}$. By using option 2 variables, Eq. $(5.11),$ the ground impact occurs at $t^{* *}=0.34$ and $S^{* *}=0.84 .$ Estimate $(a)$ the initial displacement, $(b)$ the final displacement, and ( $c$ ) the time of impact.

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Problem 8

The Bernoulli equation ( 5.6 ) can be written in the form
$$p=p_{0}-\frac{1}{2} \rho V^{2}-\rho g z$$
where $p_{0}$ is the "stagnation" pressure at zero velocity and elevation. (a) State how many scaling variables are needed to nondimensionalize this equation. (b) Suppose that we wish to nondimensionalize Eq. (1) in order to plot dimensionless pressure versus velocity, with elevation as a parameter. Select the proper scaling variables and carry out and plot the resulting dimensionless relation.

Rashmi S.
Numerade Educator

Problem 9

Modify Prob. 5.8 as follows. Suppose that we wish to nondimensionalize Eq. (1) in order to plot dimensionless pressure versus gravity, with velocity as a parameter. Select the proper scaling variables and carry out and plot the resulting dimensionless relation.

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Problem 10

Determine the dimension $\{M L T \Theta\}$ of the following quantities:
(a) $\rho u \frac{\partial u}{\partial x}$
(b) $\int_{1}^{2}\left(p-p_{0}\right) d A$
$(c) \rho c_{p} \frac{\partial^{2} T}{\partial x \partial y}$
$(d) \iiint \rho \frac{\partial u}{\partial t} d x d y d z$
All quantities have their standard meanings; for example, $\rho$ is density.

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Problem 11

For a particle moving in a circle, its centripetal acceleration takes the form $a=\operatorname{fcn}(V, R),$ where $V$ is its velocity and $R$ the radius of its path. By pure dimensional reasoning, rewrite this function in algebraic form.

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Problem 12

The velocity of sound $a$ of a gas varies with pressure $p$ and density $\rho .$ Show by dimensional reasoning that the proper form must be $a=(\mathrm{const})(p / \rho)^{1 / 2}$

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Problem 13

The speed of propagation $C$ of a capillary wave in deep water is known to be a function only of density $\rho,$ wavelength
$\lambda,$ and surface tension $Y$. Find the proper functional relationship, completing it with a dimensionless constant. For a given density and wavelength, how does the propagation speed change if the surface tension is doubled?

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Problem 14

Consider laminar flow over a flat plate. The boundary layer thickness $\delta$ grows with distance $x$ down the plate and is also a function of free-stream velocity $U$, fluid viscosity $\mu$, and fluid density $\rho .$ Find the dimensionless parameters for this problem, being sure to rearrange if neessary to agree with the standard dimensionless groups in fluid mechanics, as given in Table 5.2

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Problem 15

It is desired to measure the drag on an airplane whose velocity is $300 \mathrm{mi} / \mathrm{h}$. Is it feasible to test a one-twentieth-scale model of the plane in a wind tunnel at the same pressure and temperature to determine the prototype drag coefficient?

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Problem 16

Convection heat-transfer data are often reported as a heattransfer coefficient $h,$ defined by
$$\dot{Q}=h A \Delta T$$
where $\dot{Q}=$ heat flow, $\mathrm{J} / \mathrm{s}$
\[
\begin{aligned}
A &=\text { surface area, } \mathrm{m}^{2} \\
\Delta T &=\text { temperature difference, } \mathrm{K}
\end{aligned}
\]
The dimensionless form of $h,$ called the Stanton number, is a combination of $h,$ fluid density $\rho,$ specific heat $c_{p},$ and flow velocity $V$. Derive the Stanton number if it is proportional to $h$

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Problem 17

In some heat-transfer textbooks, e.g., J. P. Holman, Heat Transfer, 5th ed., McGraw-Hill, $1981,$ p. $285,$ simplified formulas are given for the heat-transfer coefficient from Prob. 5.16 for buoyant or natural convection over hot surfaces. An example formula is
$$h=1.42\left(\frac{\Delta T}{L}\right)^{1 / 4}$$
where $L$ is the length of the hot surface. Comment on the dimensional homogeneity of this formula. What might be the SI units of constants 1.42 and $\frac{1}{4} ?$ What parameters might be missing or hidden?

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Problem 18

Under laminar conditions, the volume flow $Q$ through a small triangular-section pore of side length $b$ and length $L$
is a function of viscosity $\mu,$ pressure drop per unit length $\Delta p / L,$ and $b .$ Using the pi theorem, rewrite this relation in dimensionless form. How does the volume flow change if the pore size $b$ is doubled?

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Problem 19

The period of oscillation $T$ of a water surface wave is assumed to be a function of density $\rho,$ wavelength $\lambda,$ depth $h$ gravity $g,$ and surface tension $Y$. Rewrite this relationship in dimensionless form. What results if $Y$ is negligible? Hint Take $\lambda, \rho,$ and $g$ as repeating variables.

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Problem 20

The power input $P$ to a centrifugal pump is assumed to be a function of the volume flow $Q,$ impeller diameter $D, \mathrm{ro}$ tational rate $\Omega,$ and the density $\rho$ and viscosity $\mu$ of the fluid. Rewrite this as a dimensionless relationship. Hint:
Take $\Omega, \rho,$ and $D$ as repeating variables.

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Problem 21

In Example 5.1 we used the pi theorem to develop Eq. (5.2) from Eq. (5.1). Instead of merely listing the primary dimensions of each variable, some workers list the powers of each primary dimension for each variable in an array:
This array of exponents is called the dimensional matrix for the given function. Show that the rank of this matrix (the size of the largest nonzero determinant) is equal to $j=n-$ $k,$ the desired reduction between original variables and the pi groups. This is a general property of dimensional matrices, as noted by Buckingham [29].

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Problem 22

When freewheeling, the angular velocity $\Omega$ of a windmill is found to be a function of the windmill diameter $D$, the wind velocity $V$, the air density $\rho$, the windmill height $H$ as compared to the atmospheric boundary layer height $L$, and the number of blades $N$
$$\Omega=\operatorname{fcn}\left(D, V, \rho, \frac{H}{L}, N\right)$$
Viscosity effects are negligible. Find appropriate pi groups for this problem and rewrite the function above in dimensionless form.

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Problem 23

The period $T$ of vibration of a beam is a function of its length $L,$ area moment of inertia $I,$ modulus of elasticity $E,$ density $\rho,$ and Poisson's ratio $\sigma .$ Rewrite this relation in dimensionless form. What further reduction can we make if $E$ and $I$ can occur only in the product form $E I$ ? Hint: Take $L, \rho,$ and $E$ as repeating variables

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Problem 24

The lift force $F$ on a missile is a function of its length $L$, velocity $V,$ diameter $D,$ angle of attack $\alpha,$ density $\rho,$ viscosity $\mu,$ and speed of sound $a$ of the air. Write out the dimensional matrix of this function and determine its rank.
(See Prob. 5.21 for an explanation of this concept.) Rewrite the function in terms of pi groups.

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Problem 25

When a viscous fluid is confined between two long concentric cylinders as in Fig. 4.17 , the torque per unit length $T^{\prime}$ required to turn the inner cylinder at angular velocity $\Omega$ is a function of $\Omega,$ cylinder radii $a$ and $b,$ and viscosity $\mu$ Find the equivalent dimensionless function. What happens to the torque if both $a$ and $b$ are doubled?

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Problem 26

A pendulum has an oscillation period $T$ which is assumed to depend upon its length $L,$ bob mass $m,$ angle of swing $\theta$ and the acceleration of gravity. A pendulum $1 \mathrm{m}$ long, with a bob mass of $200 \mathrm{g}$, is tested on earth and found to have a period of 2.04 s when swinging at $20^{\circ} .(a)$ What is its period when it swings at $45^{\circ} ?$ A similarly constructed pendulum, with $L=30 \mathrm{cm}$ and $m=100 \mathrm{g},$ is to swing on the moon $\left(g=1.62 \mathrm{m} / \mathrm{s}^{2}\right)$ at $\theta=20^{\circ} .(b)$ What will be its period?

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Problem 27

In studying sand transport by ocean waves, A. Shields in 1936 postulated that the threshold wave-induced bottom shear stress $\tau$ required to move particles depends upon gravity $g,$ particle size $d$ and density $\rho_{p},$ and water density $\rho$ and viscosity $\mu .$ Find suitable dimensionless groups of this problem, which resulted in 1936 in the celebrated Shields sandtransport diagram.

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Problem 28

A simply supported beam of diameter $D$, length $L$, and modulus of elasticity $E$ is subjected to a fluid crossflow of velocity $V,$ density $\rho,$ and viscosity $\mu .$ Its center deflection $\delta$ is assumed to be a function of all these variables. (a) Rewrite this proposed function in dimensionless form. $(b)$ Suppose it is known that $\delta$ is independent of $\mu,$ inversely proportional to $E,$ and dependent only on $\rho V^{2},$ not $\rho$ and $V$ separately. Simplify the dimensionless function accordingly. Hint. Take $L, \rho,$ and $V$ as repeating variables.

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Problem 29

When fluid in a pipe is accelerated linearly from rest, it begins as laminar flow and then undergoes transition to turbulence at a time $t_{\mathrm{tr}}$ which depends upon the pipe diameter $D,$ fluid acceleration $a,$ density $\rho,$ and viscosity $\mu .$ Arrange this into a dimensionless relation between $t_{\mathrm{tr}}$ and $D$

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Problem 30

In forced convection, the heat-transfer coefficient $h,$ as defined in Prob. $5.16,$ is known to be a function of stream velocity $U,$ body size $L,$ and fluid properties $\rho, \mu, c_{p},$ and $k$ Rewrite this function in dimensionless form, and note by name any parameters you recognize. Hint: Take $L, \rho, k,$ and $\mu$ as repeating variables.

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Problem 31

The heat-transfer rate per unit area $q$ to a body from a fluid in natural or gravitational convection is a function of the temperature difference $\Delta T,$ gravity $g,$ body length $L,$ and three fluid properties: kinematic viscosity $\nu$, conductivity $k$ and thermal expansion coefficient $\beta .$ Rewrite in dimensionless form if it is known that $g$ and $\beta$ appear only as the product $g \beta$

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Problem 32

A weir is an obstruction in a channel flow which 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 $Q(g, b, H)$

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Problem 33

A spar buoy (see Prob. 2.113) has a period $T$ of vertical (heave) oscillation which depends upon the waterline cross-sectional area $A,$ buoy mass $m,$ and fluid specific weight $\gamma$. How does the period change due to doubling of $(a)$ the mass and $(b)$ the area? Instrument buoys should have long periods to avoid wave resonance. Sketch a possible long-period buoy design.

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Problem 34

To good approximation, the thermal conductivity $k$ of a gas (see Ref. 8 of Chap. 1 ) depends only upon the density $\rho$ mean free path $\ell,$ gas constant $R,$ and absolute temperature $T .$ For air at $20^{\circ} \mathrm{C}$ and $1 \mathrm{atm}, k \approx 0.026 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K})$ and $\ell \approx$
$6.5 \mathrm{E}-8 \mathrm{m} .$ Use this information to determine $k$ for hydro-

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Problem 35

The torque $M$ required to turn the cone-plate viscometer in Fig. P5.35 depends upon the radius $R$, rotation rate $\Omega$, fluid viscosity $\mu,$ and cone angle $\theta .$ Rewrite this relation in dimensionless form. How does the relation simplify it if it is known that $M$ is proportional to $\theta ?$

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Problem 36

The rate of heat loss, $\dot{Q}_{\text {loss }}$ through a window or wall is a function of the temperature difference between inside and outside $\Delta T,$ the window surface area $A,$ and the $R$ value of the window which has units of $\left(\mathrm{ft}^{2} \cdot \mathrm{h} \cdot^{\circ} \mathrm{F}\right) / \mathrm{Btu} .(a)$ Using Buckingham pi theorem, find an expression for rate of heat loss as a function of the other three parameters in the problem. ( $b$ ) If the temperature difference $\Delta T$ doubles, by what factor does the rate of heat loss increase?

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Problem 37

The pressure difference $\Delta p$ across an explosion or blast wave is a function of the distance $r$ from the blast center, time $t$ speed of sound $a$ of the medium, and total energy $E$ in the blast. Rewrite this relation in dimensionless form (see Ref. 18, chap. 4, for further details of blast-wave scaling). How does $\Delta p$ change if $E$ is doubled?

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Problem 38

The size $d$ of droplets produced by a liquid spray nozzle is thought to depend upon the nozzle diameter $D$, jet velocity $U,$ and the properties of the liquid $\rho, \mu,$ and $Y .$ Rewrite this relation in dimensionless form. Hint: Take $D, \rho,$ and $U$ as repeating variables.

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Problem 39

In turbulent flow past a flat surface, the velocity $u$ near the wall varies approximately logarithmically with distance $y$ from the wall and also depends upon viscosity $\mu,$ density $\rho,$ and wall shear stress $\tau_{w} .$ For a certain airflow at $20^{\circ} \mathrm{C}$ and 1 atm, $\tau_{w}=0.8 \mathrm{Pa}$ and $u=15 \mathrm{m} / \mathrm{s}$ at $y=3.6 \mathrm{mm} .$ Use this information to estimate the velocity $u$ at $y=6 \mathrm{mm}$

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Problem 40

Reconsider the slanted-plate surface tension problem (see Fig. $\mathrm{C} 1.1$ ) as an exercise in dimensional analysis. Let the capillary rise $h$ be a function only of fluid properties, gravity, bottom width, and the two angles in Fig. C1.1. That is, $h=\operatorname{fcn}(\rho, Y, g, L, \alpha, \theta) .(a)$ Use the pi theorem to rewrite this function in terms of dimensionless parameters. ( $b$ ) Verify that the exact solution from Prob. C1.1 is consistent with your result in part $(a)$

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Problem 41

A certain axial-flow turbine has an output torque $M$ which is proportional to the volume flow rate $Q$ and also depends upon the density $\rho,$ rotor diameter $D,$ and rotation rate $\Omega .$ How does the torque change due to a doubling of $(a) D$ and $(b) \Omega ?$

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Problem 42

Nondimensionalize the energy equation (4.75) and its boundary conditions $(4.62),(4.63),$ and (4.70) by defining $T^{*}=T / T_{0},$ where $T_{0}$ is the inlet temperature, assumed constant. Use other dimensionless variables as needed from Eqs. (5.23). Isolate all dimensionless parameters you find, and relate them to the list given in Table 5.2

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Problem 43

The differential equation of salt conservation for flowing seawater is
$$\frac{\partial S}{\partial t}+u \frac{\partial S}{\partial x}+v \frac{\partial S}{\partial y}+w \frac{\partial S}{\partial z}=\kappa\left(\frac{\partial^{2} S}{\partial x^{2}}+\frac{\partial^{2} S}{\partial y^{2}}+\frac{\partial^{2} S}{\partial z^{2}}\right)$$
where $\kappa$ is a (constant) coefficient of diffusion, with typical units of square meters per second, and $S$ is the salinity in parts per thousand. Nondimensionalize this equation and discuss any parameters which appear.

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Problem 44

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 which arise.

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Problem 45

In natural-convection problems, the variation of density due to the temperature difference $\Delta T$ creates an important buoyancy term in the momentum equation $(5.30) .$ To first-order accuracy, the density variation would be $\rho \approx \rho_{0}(1-\beta \Delta T)$ where $\beta$ is the thermal-expansion coefficient. The momentum equation thus becomes
$$\rho_{0} \frac{d \mathbf{V}}{d t}=-\boldsymbol{\nabla}\left(p+\rho_{0} g z\right)+\rho_{0} \beta \Delta T g \mathbf{k}+\mu \nabla^{2} \mathbf{V}$$
where we have assumed that $z$ is up. Nondimensionalize this equation, using Eqs. $(5.23),$ and relate the parameters you find to the list in Table 5.2

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Problem 46

The differential equation for compressible inviscid flow of a gas in the $x y$ plane is
$$\begin{aligned}
\frac{\partial^{2} \phi}{\partial t^{2}}+\frac{\partial}{\partial t}\left(u^{2}+v^{2}\right)+\left(u^{2}-a^{2}\right) \frac{\partial^{2} \phi}{\partial x^{2}} & \\
&+\left(v^{2}-a^{2}\right) \frac{\partial^{2} \phi}{\partial y^{2}}+2 u v \frac{\partial^{2} \phi}{\partial x \partial y}=0
\end{aligned}$$
where $\phi$ is the velocity potential and $a$ is the (variable) speed of sound of the gas. Nondimensionalize this relation, using a reference length $L$ and the inlet speed of sound $a_{0}$ as parameters for defining dimensionless variables

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Problem 47

The differential equation for small-amplitude vibrations $y(x, t)$ of a simple beam is given by
$$\rho A \frac{\partial^{2} y}{\partial t^{2}}+E I \frac{\partial^{4} y}{\partial x^{4}}=0$$
Use only the quantities $\rho, E,$ and $A$ to nondimensionalize $y$ $x,$ and $t,$ and rewrite the differential equation in dimensionless form. Do any parameters remain? Could they be removed by further manipulation of the variables?

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Problem 48

A smooth steel $(\mathrm{SG}=7.86)$ sphere is immersed in a stream of ethanol at $20^{\circ} \mathrm{C}$ moving at $1.5 \mathrm{m} / \mathrm{s}$. Estimate its drag in N from Fig. 5.3 $a$. What stream velocity would quadruple its drag? Take $D=2.5 \mathrm{cm}$

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Problem 49

The sphere in Prob. 5.48 is dropped in gasoline at $20^{\circ} \mathrm{C} .$ Ignoring its acceleration phase, what will its terminal (constant fall velocity be, from Fig. $5.3 a ?$

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Problem 50

When a micro-organism moves in a viscous fluid, it turns out that fluid density has nearly negligible influence on the drag force felt by the micro-organism. Such flows are called creeping flows. The only important parameters in the problem are the velocity of motion $U$, the viscosity of the fluid $\mu,$ and the length scale of the body. Here assume the $\mathrm{mi}$ cro-organism's body diameter $d$ as the appropriate length scale. ( $a$ ) Using the Buckingham pi theorem, generate an expression for the drag force $D$ as a function of the other parameters in the problem. ( $b$ ) The drag coefficient discussed in this chapter $C_{D}=D /\left(\frac{1}{2} \rho U^{2} A\right)$ is not appropriate for this kind of flow. Define instead a more appropriate drag coefficient, and call it $C_{c}$ (for creeping flow). (c) For a spherically shaped micro-organism, the drag force can be calculated exactly from the equations of motion for creeping flow. The result is $D=3 \pi \mu U d .$ Write expressions for both forms of the drag coefficient, $C_{c}$ and $C_{D},$ for a sphere under conditions of creeping flow.

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Problem 51

A ship is towing a sonar array which approximates a submerged cylinder $1 \mathrm{ft}$ in diameter and $30 \mathrm{ft}$ long with its axis normal to the direction of tow. If the tow speed is 12 kn $(1 \mathrm{kn}=1.69 \mathrm{ft} / \mathrm{s})$, estimate the horsepower required to tow this cylinder. What will be the frequency of vortices shed from the cylinder? Use Figs. 5.2 and 5.3.

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Problem 52

A 1-in-diameter telephone wire is mounted in air at $20^{\circ} \mathrm{C}$ and has a natural vibration frequency of $12 \mathrm{Hz}$. What wind velocity in ft/s will cause the wire to sing? At this condition what will the average drag force per unit wire length be?

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Problem 53

Vortex shedding can be used to design a vortex flowmeter (Fig. 6.32 ). A blunt rod stretched across the pipe sheds vortices whose frequency is read by the sensor downstream. Suppose the pipe diameter is $5 \mathrm{cm}$ and the rod is a cylinder of diameter $8 \mathrm{mm}$. If the sensor reads 5400 counts per minute, estimate the volume flow rate of water in $\mathrm{m}^{3} / \mathrm{h}$. How might the meter react to other liquids?

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Problem 54

A fishnet is made of 1 -mm-diameter strings knotted into $2 \times 2 \mathrm{cm}$ squares. Estimate the horsepower required to tow $300 \mathrm{ft}^{2}$ of this netting at $3 \mathrm{kn}$ in seawater at $20^{\circ} \mathrm{C}$. The net plane is normal to the flow direction.

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Problem 55

A fishnet is made of 1 -mm-diameter strings knotted into $2 \times 2 \mathrm{cm}$ squares. Estimate the horsepower required to tow $300 \mathrm{ft}^{2}$ of this netting at $3 \mathrm{kn}$ in seawater at $20^{\circ} \mathrm{C}$. The net plane is normal to the flow direction

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Problem 56

A wooden flagpole, of diameter 5 in and height $30 \mathrm{ft}$, fractures at its base in hurricane winds at sea level. If the fracture stress is $3500 \mathrm{lbf} / \mathrm{in}^{2},$ estimate the wind velocity in $\mathrm{mi} / \mathrm{h}$

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Problem 57

The simply supported 1040 carbon-steel rod of Fig. P5.57 is subjected to a crossflow stream of air at $20^{\circ} \mathrm{C}$ and 1 atm. For what stream velocity $U$ will the rod center deflection be approximately $1 \mathrm{cm} ?$

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Problem 58

For the steel rod of Prob. $5.57,$ at what airstream velocity $U$ will the rod begin to vibrate laterally in resonance in its first mode (a half sine wave)? Hint: Consult a vibration text under "lateral beam vibration."

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Problem 59

We wish to know the drag of a blimp which will move in $20^{\circ} \mathrm{C}$ air at $6 \mathrm{m} / \mathrm{s}$. If a one-thirtieth-scale model is tested in water at $20^{\circ} \mathrm{C}$, what should the water velocity be? At this velocity, if the measured water drag on the model is $2700 \mathrm{N}$ what is the drag on the prototype blimp and the power required to propel it?

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Problem 60

A prototype water pump has an impeller diameter of $2 \mathrm{ft}$ and is designed to pump $12 \mathrm{ft}^{3} / \mathrm{s}$ at $750 \mathrm{r} / \mathrm{min}$. A $1-\mathrm{ft}$ -diameter model pump is tested in $20^{\circ} \mathrm{C}$ air at $1800 \mathrm{r} / \mathrm{min}$, and Reynolds-number effects are found to be negligible. For similar conditions, what will the volume flow of the model be in $\mathrm{ft}^{3} / \mathrm{s} ?$ If the model pump requires 0.082 hp to drive it, what horsepower is required for the prototype?

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Problem 61

If viscosity is neglected, typical pump-flow results from Prob. 5.20 are shown in Fig. $\mathrm{P} 5.61$ for a model pump tested in water. The pressure rise decreases and the power required increases with the dimensionless flow coefficient. Curve-fit expressions are given for the data. Suppose a similar pump of 12 -cm diameter is built to move gasoline at $20^{\circ} \mathrm{C}$ and a flow rate of $25 \mathrm{m}^{3} / \mathrm{h}$. If the pump rotation speed is $30 \mathrm{r} / \mathrm{s}$ find $(a)$ the pressure rise and $(b)$ the power required.

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Problem 62

Modify Prob. 5.61 so that the rotation speed is unknown but $D=12 \mathrm{cm}$ and $Q=25 \mathrm{m}^{3} / \mathrm{h} .$ What is the maximum rotation speed for which the power will not exceed $300 \mathrm{W} ?$ What will the pressure rise be for this condition?

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Problem 63

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 flow 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 flow of kerosine at $20^{\circ} \mathrm{C}$ in a smooth pipe of diameter $5 \mathrm{cm}$ and length $200 \mathrm{m}$ if the flow rate is $50 \mathrm{m}^{3} / \mathrm{h}$

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Problem 64

The natural frequency $\omega$ of vibration of a mass $M$ attached to a rod, as in Fig. P5.64, depends only upon $M$ and the stiffness $E I$ and length $L$ of the rod. Tests with a 2 -kg mass attached to a 1040 carbon-steel rod of diameter $12 \mathrm{mm}$ and length $40 \mathrm{cm}$ reveal a natural frequency of $0.9 \mathrm{Hz}$. Use these data to predict the natural frequency of a 1 -kg mass attached to a 2024 aluminum-alloy rod of the same size.

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Problem 65

In turbulent flow near a flat wall, the local velocity $u$ varies only with distance $y$ from the wall, wall shear stress $\tau_{\mathrm{w}},$ and fluid properties $\rho$ and $\mu .$ The following data were taken in the University of Rhode Island wind tunnel for airflow, $\rho=$ 0.0023 slug $/ \mathrm{ft}^{3}, \mu=3.81 \mathrm{E}-7 \operatorname{slug} /(\mathrm{ft} \cdot \mathrm{s}),$ and $\tau_{\mathrm{w}}=0.029$
$\mathrm{Ibf} / \mathrm{ft}^{2}$

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Problem 66

A torpedo $8 \mathrm{m}$ below the surface in $20^{\circ} \mathrm{C}$ seawater cavitates at a speed of $21 \mathrm{m} / \mathrm{s}$ when atmospheric pressure is $101 \mathrm{kPa}$ If Reynolds-number and Froude-number effects are negligible, at what speed will it cavitate when running at a depth of $20 \mathrm{m}$ ? At what depth should it be to avoid cavitation at $30 \mathrm{m} / \mathrm{s} ?$

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Problem 67

A student needs to measure the drag on a prototype of characteristic dimension $d_{p}$ moving at velocity $U_{p}$ in air at standard atmospheric conditions. He constructs a model of characteristic dimension $d_{m},$ such that the ratio $d_{p} / d_{m}$ is some factor $f .$ He then measures the drag on the model at dynamically similar conditions (also with air at standard atmospheric conditions). The student claims that the drag force on the prototype will be identical to that measured on the model. Is this claim correct? Explain.

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Problem 68

Consider flow over a very small object in a viscous fluid. Analysis of the equations of motion shows that the inertial terms are much smaller than the viscous and pressure terms. It turns out, then, that fluid density drops out of the equations of motion. Such flows are called creeping flows. The only important parameters in the problem are
the velocity of motion $U$, the viscosity of the fluid $\mu$, and the length scale of the body. For three-dimensional bodies, like spheres, creeping flow analysis yields very good results. It is uncertain, however, if such analysis can be applied to two-dimensional bodies such as a circular cylinder, since even though the diameter may be very small, the length of the cylinder is infinite for a two-dimensional flow. Let us see if dimensional analysis can help. (a) Using the Buckingham pi theorem, generate an expression for the two-dimensional drag $D_{2-\mathrm{D}}$ as a function of the other parameters in the problem. Use cylinder diameter $d$ as the appropriate length scale. Be careful the two-dimensional drag has dimensions of force per unit length rather than simply force. ( $b$ ) Is your result physically plausible? If not, explain why not. $(c)$ It turns out that fluid density $\rho$ cannot be neglected in analysis of creeping flow over two-dimensional bodies. Repeat the dimensional analysis, this time with $\rho$ included as a parameter. Find the nondimensional relationship between the parameters in this problem.

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Problem 69

A one-sixteenth-scale model of a weir (see Fig. P5.32) has a measured flow rate $Q=2.1 \mathrm{ft}^{3} / \mathrm{s}$ when the upstream water height is $H=6.3$ in. If $Q$ is proportional to weir width
$b,$ predict the prototype flow rate when $H_{\mathrm{proto}}=3.2 \mathrm{ft}$

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Problem 70

A diamond-shaped body, of characteristic length 9 in, has the following measured drag forces when placed in a wind tunnel at sea-level standard conditions:
Use these data to predict the drag force of a similar 15 -in diamond placed at similar orientation in $20^{\circ} \mathrm{C}$ water flowing at $2.2 \mathrm{m} / \mathrm{s}$

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Problem 71

The pressure drop in a venturi meter (Fig. P3.165) varies only with the fluid density, pipe approach velocity, and diameter ratio of the meter. A model venturi meter tested in water at $20^{\circ} \mathrm{C}$ shows a 5 -kPa drop when the approach velocity is $4 \mathrm{m} / \mathrm{s}$. A geometrically similar prototype meter is used to measure gasoline at $20^{\circ} \mathrm{C}$ and a flow rate of $9 \mathrm{m}^{3} / \mathrm{min} .$ If the prototype pressure gage is most accurate at $15 \mathrm{kPa}$, what should the upstream pipe diameter be?

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Problem 72

A one-fifteenth-scale model of a parachute has a drag of 450 lbf when tested at $20 \mathrm{ft} / \mathrm{s}$ in a water tunnel. If Reynoldsnumber effects are negligible, estimate the terminal fall velocity at 5000 -ft standard altitude of a parachutist using the prototype if chute and chutist together weigh 160 lbf. Neglect the drag coefficient of the woman.

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Problem 73

The yawing moment on a torpedo control surface is tested on a one-eighth-scale model in a water tunnel at $20 \mathrm{m} / \mathrm{s}$, using Reynolds scaling. If the model measured moment is $14 \mathrm{N} \cdot \mathrm{m},$ what will the prototype moment be under similar conditions?

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Problem 74

A one-tenth-scale model of a supersonic wing tested at 700 $\mathrm{m} / \mathrm{s}$ in air at $20^{\circ} \mathrm{C}$ and 1 atm shows a pitching moment of $0.25 \mathrm{kN} \cdot \mathrm{m} .$ If Reynolds-number effects are negligible, what will the pitching moment of the prototype wing be if it is flying at the same Mach number at 8 -km standard altitude?

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Problem 75

A one-twelfth-scale model of an airplane is to be tested at $20^{\circ} \mathrm{C}$ in a pressurized wind tunnel. The prototype is to fly at $240 \mathrm{m} / \mathrm{s}$ at 10 -km standard altitude. What should the tunnel pressure be in atm to scale both the Mach number and the Reynolds number accurately?

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Problem 76

A 2 -ft-long model of a ship is tested in a freshwater tow tank. The measured drag may be split into "friction" drag (Reynolds scaling) and "wave" drag (Froude scaling). The model data are as follows:
The prototype ship is $150 \mathrm{ft}$ long. Estimate its total drag when cruising at $15 \mathrm{kn}$ in seawater at $20^{\circ} \mathrm{C}$

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Problem 77

A dam spillway is to be tested by using Froude scaling with a one-thirtieth-scale model. The model flow has an average velocity of $0.6 \mathrm{m} / \mathrm{s}$ and a volume flow of $0.05 \mathrm{m}^{3} / \mathrm{s}$. What will the velocity and flow of the prototype be? If the measured force on a certain part of the model is $1.5 \mathrm{N},$ what will the corresponding force on the prototype be?

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Problem 78

A prototype spillway has a characteristic velocity of $3 \mathrm{m} / \mathrm{s}$ and a characteristic length of $10 \mathrm{m}$. A small model is constructed by using Froude scaling. What is the minimum scale ratio of the model which will ensure that its minimum Weber number is $100 ?$ Both flows use water at $20^{\circ} \mathrm{C}$

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Problem 79

An East Coast estuary has a tidal period of $12.42 \mathrm{h}$ (the semidiurnal lunar tide) and tidal currents of approximately $80 \mathrm{cm} / \mathrm{s} .$ If a one-five-hundredth-scale model is constructed with tides driven by a pump and storage apparatus, what should the period of the model tides be and what model current speeds are expected?

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Problem 80

A prototype ship is $35 \mathrm{m}$ long and designed to cruise at $11 \mathrm{m} / \mathrm{s} \text { (about } 21 \mathrm{kn}) .$ Its drag is to be simulated by a $1-\mathrm{m}-$ long model pulled in a tow tank. For Froude scaling find
(a) the tow speed, ( $b$ ) the ratio of prototype to model drag, and $(c)$ the ratio of prototype to model power

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Problem 81

An airplane, of overall length $55 \mathrm{ft}$, is designed to fly at 680 $\mathrm{m} / \mathrm{s}$ at $8000-\mathrm{m}$ standard altitude. A one-thirtieth-scale model
is to be tested in a pressurized helium wind tunnel at $20^{\circ} \mathrm{C}$ What is the appropriate tunnel pressure in atm? Even at this (high) pressure, exact dynamic similarity is not achieved. Why?

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Problem 82

A prototype ship is $400 \mathrm{ft}$ long and has a wetted area of $30,000 \mathrm{ft}^{2} .$ A one-eightieth-scale model is tested in a tow tank according to Froude scaling at speeds of $1.3,2.0,$ and $2.7 \mathrm{kn}(1 \mathrm{kn}=1.689 \mathrm{ft} / \mathrm{s}) .$ The measured friction drag of
the model at these speeds is $0.11,0.24,$ and 0.41 lbf, respectively. What are the three prototype speeds? What is the estimated prototype friction drag at these speeds if we correct for Reynolds-number discrepancy by extrapolation?

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Problem 83

A one-fortieth-scale model of a ship's propeller is tested in a tow tank at $1200 \mathrm{r} / \mathrm{min}$ and exhibits a power output of 1.4
$\mathrm{ft} \cdot$ lbf/s. According to Froude scaling laws, what should the revolutions per minute and horsepower output of the prototype propeller be under dynamically similar conditions?

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Problem 84

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?

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Problem 85

Solve Prob. $5.49,$ using the modified sphere-drag plot of Fig. 5.11

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Problem 86

Solve Prob. 5.49 for glycerin at $20^{\circ} \mathrm{C},$ using the modified sphere-drag plot of Fig. 5.11

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Problem 87

In Prob. 5.62 it was difficult to solve for $\Omega$ because it appeared in both power and flow coefficients. Rescale the problem, using the data of Fig. $\mathrm{P} 5.61,$ to make a plot of dimensionless power versus dimensionless rotation speed. Enter this plot directly to solve Prob. 5.62 for $\Omega.$

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Problem 88

Modify Prob. 5.62 as follows: Let $\Omega=32 \mathrm{r} / \mathrm{s}$ and $Q=24$ $\mathrm{m}^{3} / \mathrm{h}$ for a geometrically similar pump. What is the maximum diameter if the power is not to exceed $340 \mathrm{W}$ ? Solve this problem by rescaling the data of Fig. P5.61 to make a plot of dimensionless power versus dimensionless diameter. Enter this plot directly to find the desired diameter.

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Problem 89

Knowing that $\Delta p$ is proportional to $L,$ rescale the data of Example 5.7 to plot dimensionless $\Delta p$ versus dimensionless diameter. Use this plot to find the diameter required in the first row of data in Example 5.7 if the pressure drop is increased to $10 \mathrm{kPa}$ for the same flow rate, length, and fluid.

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Problem 90

Knowing that $\Delta p$ is proportional to $L,$ rescale the data of Example 5.7 to plot dimensionless $\Delta p$ versus dimensionless viscosity. Use this plot to find the viscosity required in the first row of data in Example 5.7 if the pressure drop is increased to $10 \mathrm{kPa}$ for the same flow rate, length, and density.

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Problem 91

Develop a plot of dimensionless $\Delta p$ versus dimensionless viscosity, as described in Prob. $5.90 .$ Suppose that $L=200$ $\mathrm{m}, Q=60 \mathrm{m}^{3} / \mathrm{h},$ and the fluid is kerosine at $20^{\circ} \mathrm{C}$. Use your plot to determine the minimum pipe diameter for which the pressure drop is no more than $220 \mathrm{kPa}$

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