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Solution 1.80
First convert 1180 ft/s to 360 m/s. With T2 given as 223 K, evaluate the speed of sound at
Problem 1.81
Use Eq. (1.39) to find and sketch the streamlines of the following flow field:
; ; 0 , where is a constantu Kx v Ky w K= = − =
Solution 1.81*
The asterisk * denotes this as a relatively difficult problem. From Eq. (1.39),
Problem 1.82
A velocity field is given by u
=
V cos
, v
=
V sin
, and w = 0, where V and
are constants. Derive
a formula for the streamlines of this flow.
Solution 1.82
Equation (1.44) may be used to find the streamlines:
Problem 1.83*
Use Eq. (1.39) to find and sketch the streamlines of the following flow field:
22
( ) ; 2 ; 0 , where is a constantu K x y v K x y w K= − = − =
Hint: This is a first-order exact differential equation.
Solution 1.83
Equation (1.44) applies with time as a parameter:
Problem 1.84
In the early 1900’s, the British chemist Sir Cyril Hinshelwood quipped that fluid dynamics study
was divided into ”workers who observed things they could not explain and workers who explained
things they could not observe”. To what historic situation was he referring?
Solution 1.84
He was referring to the split between hydraulics engineers, who performed experiments but had
Problem 1.85
Do some reading and report to the class on the life and achievements, especially vis-à-vis fluid
mechanics, of
( a ) Evangelista Torricelli (1608–1647)
( b ) Henri de Pitot (1695–1771)
( c ) Antoine Chézy (1718–1798)
( d ) Gotthilf Heinrich Ludwig Hagen (1797–1884)
( e ) Julius Weisbach (1806–1871)
( f ) George Gabriel Stokes (1819–1903)
( g ) Moritz Weber (1871–1951)
( h ) Theodor von Kfirmfin (1881–1963)
( i ) Paul Richard Heinrich Blasius (1883–1970)
( j ) Ludwig Prandtl (1875–1953)
( k ) Osborne Reynolds (1842–1912)
( l ) John William Strutt, Lord Rayleigh (1842–1919)
( m ) Daniel Bernoulli (1700–1782)
( n ) Leonhard Euler (1707–1783)
Solution 1.85
(a) Evangelista Torricelli.
Torricelli’s biography is taken from a goldmine of information which I did not put in the
references, preferring to let the students find it themselves: C. C. Gillespie (ed.), Dictionary of
Pitot’s research was apparently mediocre, described as “competent solutions to minor problems
without lasting significance”⎯not a good recommendation for tenure nowadays! His lasting
contribution was the invention, in 1735, of the instrument which bears his name: a glass tube
bent at right angles and inserted into a moving stream with the opening facing upstream. The
water level in the tube rises a distance h above the surface, and Pitot correctly deduced that the
stream velocity (2gh). This is still a basic instrument in fluid mechanics.
V const AS/P
where A is the cross-section area, S the bottom slope, and P the wetted perimeter, i.e., the length
of the bottom and sides of the cross-section. The “constant” depends primarily on the roughness
of the channel bottom and sides. [See Chap. 10 for further details.]
Later, in an 1854 paper, Hagen noted that the difference between laminar and turbulent flow was
clearly visible in the efflux jet, which was either “smooth or fluctuating,” and in glass tubes, where
sawdust particles either “moved axially” or, at higher Q, “came into whirling motion.” Thus Hagen
was a true pioneer in fluid mechanics experimentation. Unfortunately, his achievements were
somewhat overshadowed by the more widely publicized 1840 tube-flow studies of J. L. M.
Poiseuille, the French physician.
(f) George Gabriel Stokes
The following notes are abstracted from the Dictionary of Scientific Biography (see Prob. 1.85-
a).
Stokes (1819–1903) was born in Skreen, County Sligo, Ireland, to a clergical family associated
for generations with the Church of Ireland. He attended Bristol College and Cambridge
In hydrodynamics, Stokes has several formulas and fields named after him:
(1) The equations of motion of a linear viscous fluid: the Navier-Stokes equations.
(2) The motion of nonlinear deep-water surface waves: Stokes waves.
(3) The drag on a sphere at low Reynolds number: Stokes’ formula, F = 3
VD.
Although Navier, Poisson, and Saint-Venant had made derivations of the equations of motion of
a viscous fluid in the 1820’s and 1830’s, Stokes was quite unfamiliar with the French literature.
He published a completely independent derivation in 1845 of the Navier-Stokes equations [see
Sect. 4.3], using a ‘continuum-calculus’ rather than a ‘molecular’ viewpoint, and showed that
these equations were directly analogous to the motion of elastic solids. Although not really new,
Stokes’ equations were notable for being the first to replace the mysterious French ‘molecular
coefficient’
by the coefficient of absolute viscosity,
.
(g) Moritz Weber.
The following notes are from Rouse and Ince [Ref. 12].
(h) Theodor von Kfirmfin.
The following notes are abstracted from the Dictionary of Scientific Biography (see Prob. 1.85-a).
Another good reference is his ghost-written (by Lee Edson) auto-biography, The Wind and
Beyond, Little-Brown, Boston, 1967.
Kfirmfin was uniquely skilled in integrating physics, mathematics, and fluid mechanics into a
variety of phenomena. His most famous paper was written in 1912 to explain the puzzling
alternating vortices shed behind cylinders in a steady-flow experiment conducted by K. Hiemenz,
(i) Paul Richard Heinrich Blasius.
The following notes are from Rouse and Ince [Ref. 12].
Blasius (1883–1970) was Ludwig Prandtl’s first graduate student at Göttingen. His 1908
(j) Ludwig Prandtl.
The following notes are from Rouse and Ince [Ref. 12].
Ludwig Prandtl (1875–1953) is described by Rouse and Ince [23] as the father of modern fluid
mechanics. Born in Munich, the son of a professor, Prandtl studied engineering and received a
doctorate in elasticity. But his first job as an engineer made him aware of the lack of correlation
(k) Osborne Reynolds.
The following notes are from Rouse and Ince [Ref. 12].
Osborne Reynolds (1842–1912) was born in Belfast, Ireland, to a clerical family and studied
mathematics at Cambridge University. In 1868 he was appointed chair of engineering at a
college which is now known as the University of Manchester Institute of Science and
(l) John William Strutt, Lord Rayleigh.
The following notes are from Rouse and Ince [Ref. 12].
(m) Daniel Bernoulli
The following notes are from Rouse and Ince [Ref. 12].
Daniel Bernoulli (1700–1782) was born in Groningen, Holland, his father, Johann, being a Dutch
professor. He studied at the University of Basel, Switzerland, and taught mathematics for a few
(n) Leonhard Euler
The following notes are from Rouse and Ince [Ref. 12].
Leonhard Euler (1707–1783) was born in Basel, Switzerland, and studied mathematics under
Johann Bernoulli, Daniel’s father. He succeeded Daniel Bernoulli as professor of mathematics at
the St. Petersburg Academy, leaving there in 1741 to join the faculty of Berlin University. He
Problem 1.86
A right circular cylinder volume
is to be calculated from the measured base radius R and height H.
If the uncertainty in R is 2 percent and the uncertainty in H is 3 percent, estimate the overall
uncertainty in the calculated volume. Hint: Read Appendix E.
Solution 1.86
The formula for volume is, of course,
= R2H. There are two terms to be calculated on the
right-hand side of Eq. (1.43):
