Exercise 1.36. The horizontal displacement Δ of the trajectory of a spinning ball depends on the
mass m and diameter d of the ball, the air density
ρ
and viscosity
µ
, the ball’s rotation rate
ω
, the
ball’s speed U, and the distance L traveled.
a) Use dimensional analysis to predict how Δ can depend on the other parameters.
b) Simplify your result from part a) for negligible viscous forces.
c) It is experimentally observed that Δ for a spinning sphere becomes essentially independent of
the rotation rate once the surface rotation speed,
ω
d/2, exceeds twice U. Simplify your result
from part b) for this high-spin regime.
d) Based on the result in part c), how does Δ depend on U?
Solution 1.36. a) Create the parameter matrix using the solution parameter is Δ, and the
boundary condition and material parameters are: Q,
ρ
, ΔP, and c.
Δ m d
ρ
µ
ω
U L
c) When the rotation rate is no longer a parameter, the fourth dimensionless group from part a)
U!L!
Δ“
Top view!spinning ball trajectory !
Exercise 1.37. A machine that fills peanut-butter jars must be reset to accommodate larger jars.
The new jars are twice as large as the old ones but they must be filled in the same amount of time
by the same machine. Fortunately, the viscosity of peanut butter decreases with increasing
temperature, and this property of peanut butter can be exploited to achieve the desired results
since the existing machine allows for temperature control.
a) Write a dimensionless law for the jar-filling time tf based on: the density of peanut butter
ρ
,
the jar volume V, the viscosity of peanut butter
µ
, the driving pressure that forces peanut butter
out of the machine P, and the diameter of the peanut butter-delivery tube d.
b) Assuming that the peanut butter flow is dominated by viscous forces, modify the relationship
you have written for part a) to eliminate the effects of fluid inertia.
c) Make a reasonable assumption concerning the relationship between tf and V when the other
variables are fixed so that you can determine the viscosity ratio
µ
new/
µ
old necessary for proper
operation of the old machine with the new jars.
d) Unfortunately, the auger mechanism that pumps the liquid peanut butter develops driving
pressure through viscous forces so that P is proportional to
µ
. Therefore, to meet the new jar-
filling requirement, what part of the machine should be changed and how much larger should it
be?
Solution 1.37. a) First create the parameter matrix. The solution parameter is tf. The boundary
condition and material parameters are: V,
ρ
, P, µ, and d.
tf V P
ρ
d µ
b) When fluid inertia is not important the fluid’s density is not a parameter. Therefore, drop ∏2
from the dimensional analysis formula: tf = (µ/P)Ψ(V/d3), where Ψ is yet another unknown
function.
Exercise 1.38. As an idealization of fuel injection in a Diesel engine, consider a stream of high-
speed fluid (called a jet) that emerges into a quiescent air reservoir at t = 0 from a small hole in
an infinite plate to form a plume where the fuel and air mix.
a) Develop a scaling law via dimensional analysis for the penetration distance D of the plume as
a function of: Δp the pressure difference across the orifice that drives the jet, do the diameter of
the jet orifice,
ρ
o the density of the fuel, µ∞ and
ρ
∞ the viscosity and density of the air, and t the
time since the jet was turned on.
b) Simplify this scaling law for turbulent flow where air viscosity is no longer a parameter.
c) For turbulent flow and D << do, do and
ρ
∞ are not parameters. Recreate the dimensionless law
for D.
d) For turbulent flow and D >> do, only the momentum flux of the jet matters, so Δp and do are
replaced by the single parameter Jo = jet momentum flux (Jo has the units of force and is
approximately equal to
Δpdo
2
). Recreate the dimensionless law for D using the new parameter Jo.
Solution 1.38. a) The parameters are: D, t, Δp,
ρ
o,
ρ
∞, µ∞, and do. With D as the solution
parameter, create the parameter matrix:
D t Δp
ρ
o
ρ
∞ µ∞ do
––––––––––––––––––––––––––––––––––––––––
do
ρ
∞
ρ
∞do
µ
∞
&
)
b) For high Reynolds number turbulent flow when the reservoir viscosity is no longer a
parameter, the above result becomes:
c) When do and
ρ
∞ are not parameters, there is only one dimensionless group:
Δpt2
ρ
∞D2
, so
the dimensionless law becomes:
D=const ⋅tΔp
ρ
o
.
Exercise 1.39. One of the simplest types of gasoline carburetors is a tube with small port for
transverse injection of fuel. It is desirable to have the fuel uniformly mixed in the passing air
stream as quickly as possible. A prediction of the mixing length L is sought. The parameters of
this problem are:
ρ
= density of the flowing air, d = diameter of the tube,
µ
= viscosity of the
flowing air, U = mean axial velocity of the flowing air, and J = momentum flux of the fuel
stream.
a) Write a dimensionless law for L.
b) Simplify your result from part a) for turbulent flow where
µ
must drop out of your
dimensional analysis.
c) When this flow is turbulent, it is observed that mixing is essentially complete after one
rotation of the counter rotating vortices driven by the injected-fuel momentum (see downstream-
view of the drawing for this problem), and that the vortex rotation rate is directly proportional to
J. Based on this information, assume that L ∝ (rotation time)(U) to eliminate the arbitrary
function in the result of part b). The final formula for L should contain an undetermined
dimensionless constant.
Solution 1.39. a) The parameters are: L, J, d,
µ
,
ρ
, and U. Use these to create the parameter
matrix with L as the solution parameter:
L J d
µ
ρ
U
–––––––––––––––––––––––––––––––––––
Exercise 1.40. Consider dune formation in a large horizontal desert of deep sand.
a) Develop a scaling relationship that describes how the height h of the dunes depends on the
average wind speed U, the length of time the wind has been blowing Δt, the average weight and
diameter of a sand grain w and d, and the air’s density
ρ
and kinematic viscosity
ν
.
b) Simplify the result of part a) when the sand-air interface is fully rough and
ν
is no longer a
parameter.
c) If the sand dune height is determined to be proportional to the density of the air, how do you
expect it to depend on the weight of a sand grain?
Solution 1.40. a) The solution parameter is h. The boundary condition and material parameters
are: U, Δt, w, d,
ρ
, and
ν
. First create the parameter matrix:
h U Δt w d
ρ ν
Exercise 1.41. The rim-to-rim diameter D of the impact crater produced by a vertically-falling
object depends on d = average diameter of the object, E = kinetic energy of the object lost on
impact,
ρ
= density of the ground at the impact site, Σ = yield stress of the ground at the impact
site, and g = acceleration of gravity.
a) Using dimensional analysis, determine a scaling law for D.
b) Simplify the result of part a) when D >> d, and d is no longer a parameter.
c) Further simplify the result of part b) when the ground plastically deforms to absorb the impact
energy and
ρ
is irrelevant. In this case, does gravity influence D? And, if E is doubled how much
bigger is D?
d) Alternatively, further simplify the result of part b) when the ground at the impact site is an
unconsolidated material like sand where Σ is irrelevant. In this case, does gravity influence D?
And, if E is doubled how much bigger is D?
e) Assume the relevant constant is unity and invert the algebraic relationship found in part d) to
estimate the impact energy that formed the 1.2-km-diameter Barringer Meteor Crater in Arizona
using the density of Coconino sandstone, 2.3 g/cm3, at the impact site. The impact energy that
formed this crater is likely between 1016 and 1017 J. How close to this range is your dimensional
analysis estimate?
Solution 1.41. The solution parameter is D. The boundary condition and material parameters are:
d, E,
θ
,
ρ
, Σ, and g. First create the parameter matrix:
D d E
ρ
Σ g
M 0 0 1 1 1 0
D!
g!
ρ
, Σ”
Ε#
d!
In this case, gravity does influence the crater diameter, and a doubling of the energy E increases
D a factor of
21 4 ≅1.19
.
Exercise 1.42. An isolated nominally spherical bubble with radius R undergoes shape
oscillations at frequency f. It is filled with air having density
ρ
a and resides in water with density
ρ
w and surface tension
σ
. What frequency ratio should be expected between two isolated bubbles
with 2 cm and 4 cm diameters undergoing geometrically similar shape oscillations? If a soluble
surfactant is added to the water that lowers
σ
by a factor of two, by what factor should air bubble
oscillation frequencies increase or decrease?
Solution 1.42. The boundary condition and material parameters are: R,
ρ
a,
ρ
w, and
σ
. The
solution parameter is f. First create the parameter matrix:
f R
ρ
a
ρ
w
σ
––––––––––––––––––––––––––––
Mass: 0 0 1 1 1
ρ
wR3Φ
ρ
a
&
)
where Φ is an unknown function. For a fixed density ratio, Φ(
ρ
w/
ρ
a) will be constant so f is
proportional to R–3/2 and to
σ
1/2. Thus, the required frequency ratio between different sizes
bubbles is:
Exercise 1.43. In general, boundary layer skin friction,
τ
w, depends on the fluid velocity U above
the boundary layer, the fluid density
ρ
, the fluid viscosity µ, the nominal boundary layer
thickness
δ
, and the surface roughness length scale
ε
.
a) Generate a dimensionless scaling law for boundary layer skin friction.
b) For laminar boundary layers, the skin friction is proportional to µ. When this is true, how must
τ
w depend on U and
ρ
?
c) For turbulent boundary layers, the dominant mechanisms for momentum exchange within the
flow do not directly involve the viscosity µ. Reformulate your dimensional analysis without it.
How must
τ
w depend on U and
ρ
when µ is not a parameter?
d) For turbulent boundary layers on smooth surfaces, the skin friction on a solid wall occurs in a
viscous sublayer that is very thin compared to
δ
. In fact, because the boundary layer provides a
buffer between the outer flow and this viscous sub-layer, the viscous sublayer thickness lv does
not depend directly on U or
δ
. Determine how lv depends on the remaining parameters.
e) Now consider nontrivial roughness. When
ε
is larger than lv a surface can no longer be
considered fluid-dynamically smooth. Thus, based on the results from parts a) through d) and
anything you may know about the relative friction levels in laminar and turbulent boundary
layers, are high- or low-speed boundary layer flows more likely to be influenced by surface
roughness?
Solution 1.43. a) Construct the parameter & units matrix and recognizing that
τ
w is a stress and
has units of pressure.
τ
w U
ρ
µ
δ
ε
–––––––––––––––––––––––––––––––
M 1 0 1 1 0 0
b) Use the result of part a) and set
τ
w∝
µ
. This involves requiring Π1 to be proportional to 1/ Π2
so the revised form of the dimensionless law in part a) is:
τ
w
ρ
U2=
µ
ρ
U
δ
g
ε
δ
&
‘
( )
*
+
, where g is an
Exercise 1.44. Turbulent boundary layer skin friction is one of the fluid phenomena that limit
the travel speed of aircraft and ships. One means for reducing the skin friction of liquid boundary
layers is to inject a gas (typically air) from the surface on which the boundary layer forms. The
shear stress,
τ
w, that is felt a distance L downstream of such an air injector depends on: the
volumetric gas flux per unit span q (in m2/s), the free stream flow speed U, the liquid density
ρ
,
the liquid viscosity
µ
, the surface tension
σ
, and gravitational acceleration g.
a) Formulate a dimensionless law for
τ
w in terms of the other parameters.
b) Experimental studies of air injection into liquid turbulent boundary layers on flat plates has
found that the bubbles may coalesce to form an air film that provides near perfect lubrication,
τ
w→0
for L > 0, when q is high enough and gravity tends to push the injected gas toward the
plate surface. Reformulate your answer to part a) by dropping
τ
w and L to determine a
dimensionless law for the minimum air injection rate, qc, necessary to form an air layer.
c) Simplify the result of part c) when surface tension can be neglected.
d) Experimental studies (Elbing et al. 2008) find that qc is proportional to U2. Using this
information, determine a scaling law for qc involving the other parameters. Would an increase in
g cause qc to increase or decrease?
Solution 1.44. a) Construct the parameter & units matrix and recognizing that
τ
w is a stress and
has units of pressure.
τ
w L q U
ρ
µ
σ
g
–––––––––––––––––––––––––––––––––––––––––––
=
Π4=
µ
U
σ
, and flux ratio =
Π5=
ρ
q
µ
. Thus the dimensionless law is:
Exercise 1.45. An industrial cooling system is in the design stage. The pumping requirements
are known and the drive motors have been selected. For maximum efficiency the pumps will be
directly driven (no gear boxes). The number Np and type of water pumps are to be determined
based on pump efficiency
η
(dimensionless), the total required volume flow rate Q, the required
pressure rise ΔP, the motor rotation rate Ω, and the power delivered by one motor W. Use
dimensional analysis and simple physical reasoning for the following items.
a) Determine a formula for the number of pumps.
b) Using Q, Np, ΔP, Ω, and the density (
ρ
) and viscosity (
µ
) of water, create the appropriate
number of dimensionless groups using ΔP as the dependent parameter.
c) Simplify the result of part b) by requiring the two extensive variables to appear as a ratio.
d) Simplify the result of part c) for high Reynolds number pumping where
µ
is no longer a
parameter.
e) Manipulate the remaining dimensionless group until Ω appears to the first power in the
numerator. This dimensionless group is known as the specific speed, and its value allows the
most efficient type of pump to be chosen (see Sabersky et al. 1999).
Solution 1.45. a) The total power that must be delivered to the fluid is QΔP. The power that one
pump delivers to the fluid will be
η
W. Thus, Np will be the next integer larger than QΔP/
η
W.
b) Construct the parameter & units matrix using ΔP as the solution parameter
ΔP Q Np Ω
ρ
µ
Exercise 1.46. Nearly all types of fluid filtration involve pressure driven flow through a porous
material.
a) For a given volume flow rate per unit area = Q/A, predict how the pressure difference across
the porous material = Δp, depends on the thickness of the filter material = L, the surface area per
unit volume of the filter material = Ψ, and other relevant parameters using dimensional analysis.
b) Often the Reynolds number of the flow in the filter pores is very much less than unity so fluid
inertia becomes unimportant. Redo the dimensional analysis for this situation.
c) To minimize pressure losses in heating, ventilating, and air-conditioning (HVAC) ductwork,
should hot or cold air be filtered?
d) If the filter material is changed and Ψ is lowered to one half its previous value, estimate the
change in Δp if all other parameters are constant. (Hint: make a reasonable assumption about the
dependence of Δp on L; they are both extensive variables in this situation).
Solution 1.46. This question can be answered with dimensional analysis. The parameters are
drawn from the problem statement and the two fluid properties
ρ
= density and
µ
= viscosity.
The solution parameter is Δp, and the unit matrix is:
Δp Q/A L Ψ
ρ
µ
––––––––––––––––––––––––––––––––––––––
b) Dropping the density reduces the number of dimensionless groups. The product of the first
and third group is independent of the density, thus the revised dimensional analysis result is:
L!
Gage pressure !
= Δp!
Q/A!Q/A!
Gage pressure !
= 0!
Porous Material!
Exercise 1.47. A new industrial process requires a volume V of hot air with initial density
ρ
to be
moved quickly from a spherical reaction chamber to a larger evacuated chamber using a single
pipe of length L and interior diameter of d. The vacuum chamber is also spherical and has a
volume of Vf. If the hot air cannot be transferred fast enough, the process fails. Thus, a prediction
of the transfer time t is needed based on these parameters, the air’s ratio of specific heats
γ
, and
initial values of the air’s speed of sound c and viscosity
µ
.
a) Formulate a dimensionless scaling law for t, involving six dimensionless groups.
b) Inexpensive small-scale tests of the air-transfer process are untaken before construction of the
commercial-scale reaction facility. Can all these dimensionless groups be matched if the target
size for the pipe diameter in the small-scale tests is d´ = d/10? Would lowering or raising the
initial air temperature in the small-scale experiments help match the dimensionless numbers?
Solution 1.47. a) This question can be answered with dimensional analysis. The parameters are
drawn from the problem statement. The solution parameter is t, and the unit matrix is:
t V Vf L d
γ
c
ρ
µ
–––––––––––––––––––––––––––––––––––––––––––––––––
M 0 0 0 0 0 0 0 1 1
b) The first dimensionless group will be matched if the other five are matched. So, let primes
denote the small-scale test parameter. Matching the five independent dimensionless groups
means:
Exercise 1.48. Create a small passive helicopter from ordinary photocopy-machine paper (as
shown) and drop it from a height of 2 m or so. Note the helicopter’s rotation and decent rates
once it’s rotating steadily. Repeat this simple experiment with different sizes of paper clips to
change the helicopter’s weight, and observe changes in the rotation and decent rates.
a) Using the helicopter’s weight W, blade length l, and blade width (chord) c, and the air’s
density
ρ
and viscosity
µ
as independent parameters, formulate two independent dimensionless
scaling laws for the helicopter’s rotation rate Ω, and decent rate dz/dt.
b) Simplify both scaling laws for the situation where
µ
is no longer a parameter.
c) Do the dimensionless scaling laws correctly predict the experimental trends?
d) If a new paper helicopter is made with all dimensions smaller by a factor of two. Use the
scaling laws found in part b) to predict changes in the rotation and decent rates. Make the new
smaller paper helicopter and see if the predictions are correct.
Solution 1.48. The experiments clearly show that Ω and dz/dt increase with increasing W.
a) This question can be answered with dimensional analysis. The parameters are drawn from the
problem statement. The first solution parameter is Ω (the rotation rate) and the units matrix is:
Ω W l c
ρ
µ
4 cm!
7 cm!
2 cm!
paper clip!
7 cm!
