fluid_mechanics

376 Chapter 7 ■ Dimensional Analysis, Similitude, and Modeling
both the FLT and MLT systems for basic dimensions. Note: the results
should obviously be the same regardless of the system of dimensions
used. If your analysis indicates otherwise, go back and
check your work, giving particular attention to the required number
of reference dimensions.
7.21 A cone and plate viscometer consists of a cone with a very
small angle a which rotates above a flat surface as shown in
Fig. P7.21. The torque, , required to rotate the cone at an angular
velocity v is a function of the radius, R, the cone angle, a, and the
fluid viscosity, m, in addition to v. With the aid of dimensional
analysis, determine how the torque will change if both the viscosity
and angular velocity are doubled.
R
between the pulse-wave velocity and the variables listed. Form the
nondimensional parameters by inspection.
7.26 As shown in Fig. P7.26 and Video V5.6, a jet of liquid directed
against a block can tip over the block. Assume that the velocity,
V, needed to tip over the block is a function of the fluid density,
r, the diameter of the jet, D, the weight of the block, w, the width
of the block, b, and the distance, d, between the jet and the bottom
of the block. (a) Determine a set of dimensionless parameters for
this problem. Form the dimensionless parameters by inspection. (b)
Use the momentum equation to determine an equation for V in terms
of the other variables. (c) Compare the results of parts (a) and (b).
b
ω
Fluid
α
d
ρ
D
V
F I G U R E P7.21
7.22 The pressure drop, ¢p, along a straight pipe of diameter D has
been experimentally studied, and it is observed that for laminar flow
of a given fluid and pipe, the pressure drop varies directly with the
distance, /, between pressure taps. Assume that ¢p is a function of
D and /, the velocity, V, and the fluid viscosity, m. Use dimensional
analysis to deduce how the pressure drop varies with pipe diameter.
7.23 A cylinder with a diameter D floats upright in a liquid as
shown in Fig. P7.23. When the cylinder is displaced slightly along
its vertical axis it will oscillate about its equilibrium position with a
frequency, v. Assume that this frequency is a function of the diameter,
D, the mass of the cylinder, m, and the specific weight, g, of
the liquid. Determine, with the aid of dimensional analysis, how the
frequency is related to these variables. If the mass of the cylinder
were increased, would the frequency increase or decrease?
F I G U R E P7.23
Cylinder
diameter = D
Section 7.5 Determination of Pi Terms by Inspection
7.24 A liquid spray nozzle is designed to produce a specific size
droplet with diameter, d. The droplet size depends on the nozzle diameter,
D, nozzle velocity, V, and the liquid properties r, m, s. Using
the common dimensionless terms found in Table 7.1, determine
the functional relationship for the dependent diameter ratio of dD.
7.25 The velocity, c, at which pressure pulses travel through arteries
(pulse-wave velocity) is a function of the artery diameter, D, and
wall thickness, h, the density of blood, r, and the modulus of elasticity,
E, of the arterial wall. Determine a set of nondimensional parameters
that can be used to study experimentally the relationship
F I G U R E P7.26
7.27 Assume that the drag, d, on an aircraft flying at supersonic
speeds is a function of its velocity, V, fluid density, r, speed of
sound, c, and a series of lengths, / 1 , . . . , / i , which describe the
geometry of the aircraft. Develop a set of pi terms that could
be used to investigate experimentally how the drag is affected by
the various factors listed. Form the pi terms by inspection.
Section 7.7 Correlation of Experimental Data (Also See
Lab Problems 7.82, 7.83, 7.84, and 7.85)
7.28 The measurement of pressure is typically an important task in
fluids experiments. Obtain a photograph/image of a pressure measurement
device. Print this photo and write a brief paragraph that
describes its use.
*7.29 The pressure drop, ¢p, over a certain length of horizontal
pipe is assumed to be a function of the velocity, V, of the fluid in the
pipe, the pipe diameter, D, and the fluid density and viscosity, r and
m. (a) Show that this flow can be described in dimensionless form as
a “pressure coefficient,” C p ¢p10.5 rV 2 2 that depends on the
Reynolds number, Re rVDm. (b) The following data were obtained
in an experiment involving a fluid with r 2 slugs/ft 3 ,
m 2 10 3 lb # sft 2 , and D 0.1 ft. Plot a dimensionless graph
and use a power law equation to determine the functional relationship
between the pressure coefficient and the Reynolds number. (c) What
are the limitations on the applicability of your equation obtained in
part (b)?
V, fts
¢p, lbft 2
3 192
11 704
17 1088
20 1280
*7.30 The pressure drop across a short hollowed plug placed in a
circular tube through which a liquid is flowing (see Fig. P7.30) can
be expressed as
¢p f 1r, V, D, d2