Chapter 1, Problem 1.85P

To determine

i.

A class report on the achievements of Evangelista Torricelli $\left(1608-1647\right)$.

Explanation of Solution

• Torricelli $\left(1608-1647\right)$ was a famous mathematician and born in Faenza, Italy. He completed his earlier studies under the guidance of Benedetto Castelli in mathematics, philosophy and hydraulic engineering.
• He works as an assistant for Galileo. After the death of Galileo, Torricelli was appointed in his place as “mathematician and philosopher” by Duke Ferdinando II of Tuscany.
• In the mean time before his death he published a book name ‘Opera Geometrica’ which made him famous as a mathematician.
• Also, Torricelli was the first who showed that a parabolic curve is formed by a zero-drag projectile. He also made the finest telescope lenses and deduced his theorem which stated, ‘the velocity of efflux was equal to $\sqrt{2gh}$ ’.
• He also invented the barometer.

To determine

ii.

Report to the class on the achievements of Henri de Pitot.

Explanation of Solution

• Henri de Pitot $\left(1695-1771\right)$ was born in Aramon, France. Initially, he did not like studying much and joined the army instead.
• In Grenoble, he got a chance of reading a book and then he decided to study mathematics, astronomy, and became a civil engineer in $1740$ . After that he was appointed in Languedoc Province as the director of public works.
• He invented a glass tube, which is bent at right angles with its opening facing upstream was inserted into a moving stream. It was one of the basic instruments in Fluid Mechanics and used today as well.
• He was retired in 1756 and died in 1771.

To determine

iii.

Report to the class on the achievements of Antoine Chezy.

Explanation of Solution

• Antoine Chezy $\left(1718-1798\right)$ was born in Chalons-sur-Marne, France. In his academic life, he studied engineering at the Ecole des Ponts et Chaussees.
• He spent his whole life working for the progress of the school in which he studied.
• He contributed to study the flow in rivers and open channels and developed a formula, used even today, for calculating the average velocity:

$V\approx const\sqrt{AS/P}$.

To determine

iv.

A class report on the achievements of Gotthilf Heinrich Ludwig Hagen.

Explanation of Solution

• Hagen $\left(1884\right)$ was born in Konigsberg, East Prussia, and studied there, having famous mathematician Bessel as one of his teachers.
• He was an engineer, a teacher, and a writer and published a book in $\left(1841\right)$ on hydraulic engineering. He is best known for his study of flow of resistance in pipes which he carried out in 1839.
• He published a paper in 1854, in which he described that the difference between a laminar and turbulent flow was clearly visible in efflux jet and can be identified whether it is smooth and fluctuating.

To determine

v.

Report to the class on the achievements of Julius Weisbach.

Explanation of Solution

• Weisbach $\left(1806-1871\right)$ was born near Annaberg, Germany. After that he completed his studies by studying Mathematics, Physics and Mechanics.
• After studying, he worked as a mathematics instructor at Gymnasium. And in 1835, he was promoted as a professor in Freiberg.
• In the field of hydraulic engineering he published almost 59 papers with 15 textbooks.
• He was the first who defined the resistance head-loss formula in modern form like $h_{f\left(pipe\right)}=f\left(L/D\right)\left(V^2/2g\right)$ where f was the dimensionless ‘friction factor’
• He was also the first to derive the “weir equation” for volume flow rate Q over a dam of crest length L:

$Q\approx \frac{2}{3}{C}_w\sqrt{\left(2g\right)}\left[{\left(H+\frac{V^2}{2g}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}-{\left(\frac{V^2}{2g}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]\approx \frac{2}{3}{C}_w\sqrt{\left(2g\right)}{H}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Where H is the upstream water head level above the dam crest and $C_w$ is a dimensionless weir coefficient $\approx 0$.

To determine

vi.

Class report on the achievements of George Gabriel Stokes.

Explanation of Solution

• Stokes $\left(1819-1903\right)$ was born in Skreen, Ireland. He studied in Bristol College and Cambridge University.
• After completing his graduation, he became Professor at Cambridge.
• He works in the field of hydrodynamics, elasticity, wave mechanics, diffraction, gravity, acoustics, heat, meteorology, and chemistry.
• Stokes has several formulas and fields named after him like:

$\left(1\right)$ The equations of motion of a linear viscous fluid: the Navier-Stokes equations.

$\left(2\right)$ The motion of nonlinear deep-water surface waves: Stokes waves.

$\left(3\right)$ The drag on a sphere at low Reynolds number: Stokes’ formula, $F=3\pi \mu VD$.

$\left(4\right)$ Flow over immersed bodies for $\mathrm{Re}\ll 1$ : Stokes flow.

$\left(5\right)$ A metric (CGS) unit of kinematic viscosity, $V:1c{m}^2/s$

$=1$ stoke.

$\left(6\right)$ A relation between the $1$ st and $2$ nd coefficients of viscosity: Stokes’ hypothesis.

$\left(7\right)$ A stream function for axisymmetric flow: Stokes’ stream function.

To determine

vii.

Report to the class on the achievements of Moritz Weber.

Explanation of Solution

• Weber (1871-1951) was born in Germany. After completing his studies, he worked as a government architect.
• After completing his projects, he became a professor of mechanics at the Polytechnic Institute of Berlin.
• It was Weber who named invented the Froude and Reynolds number.
• Later, he published a paper, in which he developed a surface-tension (capillarity) parameter which is dimensionless in nature which was later known as the Weber number in his honor.

To determine

viii.

Report to the class on the achievements of Theodor Von Karman.

Explanation of Solution

• Karman $\left(1881-1963\right)$ was born in Budapest, Hungary. He did his fellowship from Gottingen and worked with Ludwig Prandtl for almost six years.
• After receiving his doctorate in 1912 he was then appointed as a director of aeronautics at the Polytechnic Institute of Aachen.
• Karman was skilled in physics, mathematics, and fluid mechanics.
• He was the one who write almost 171 articles and 5 textbooks on fluid mechanics.

To determine

ix.

Report to the class on the achievements of Paul Richard Heinrich Blasius.

Explanation of Solution

Blasius $\left(1883-1970\right)$ was first graduate student of Ludwig Prandtl’s. In his studies, he gave a theory that the flow of pipe resistance could be non-dimensionless in nature. He also developed ‘Moody-type’ chart.

He later worked on analytical solutions of boundary layers with variable pressure gradients.

To determine

x.

Class report on the achievements of Ludwig Prandtl.

Explanation of Solution

Ludwig Prandtl $\left(1875-1953\right)$ was considered as the father of modern fluid mechanics and was born in Munich. He completed his doctorate in the field of elasticity. But later during his work as engineer he found there is lack of proper correlation exist in theory and experiment in fluid mechanics. After that he made research and published a paper in 1904 explaining the concept of ‘boundary layer theory’. After working as a professor of mechanics he gave some aspects in fluid mechanics like:

Prandtl made important contributions to the following:

$\left(1\right)$ Wing theory.

$\left(2\right)$ Turbulence modeling.

$\left(3\right)$ Supersonic flow.

$\left(4\right)$ Dimensional analysis. And

$\left(5\right)$ Instability and transition of laminar flow.

To determine

xi.

Class report on the achievements of Osborne Reynolds.

Explanation of Solution

Osborne Reynolds $\left(1842-1912\right)$ was born in Belfast, Ireland, and completed his graduation in mathematics from Cambridge University. He was the person who firstly discovered the cavitation number which gives an idea of forming bubbles of vapor due to high velocity and low pressure.

He also gives a dimensionless stability parameter $\left(\frac{\rho VD}{\mu }\right)$ and was named as the Reynolds number. He also contributes in the studies of integral flow analysis.

To determine

xii.

Report to the class on the achievements of John William Strutt, Lord Rayleigh.

Explanation of Solution

John William Strutt $\left(1842-1919\right)$ was born in Essex, England. He studied at Cambridge University. He also served as the president of the Royal Society and works in the field of sounds. He is the one who discovered argon gas and won the Nobel Prize for the one.

He made five important contributions to hydrodynamics:

$\left(1\right)$ The equations of bubble dynamics in liquids, now known as Rayleigh-Plesset theory.

$\left(2\right)$ The theory of nonlinear surface waves.

$\left(3\right)$ The surface tension instability of jets

$\left(4\right)$ The “heat-transfer analogy” of laminar flow. And

$\left(5\right)$ Dimensional similarity especially related to viscosity data for argon gas and later generalized into group theory which previewed Buckingham’s Pi Theorem.

He ended his career as president, in $1909$, of the first British committee on aeronautics.

To determine

xiii.

Class report on the achievements of Daniel Bernoulli.

Explanation of Solution

Daniel Bernoulli $\left(1700-1782\right)$ was born in Groningen, Holland. He studied at the University of Basel, Switzerland. He is best known for his book ‘’Hydrodynamica’’which he wrote in 1738. Bernoulli used various concepts to establish proportional relationship between kinetic and potential energy. The famous equation that bears his name was not driven by him.

To determine

xiv.

Report to the class on the achievements of Leonhard Euler.

Explanation of Solution

Leonhard Euler $\left(1707-1783\right)$ was born in Basel, Switzerland, and studied mathematics. He initially joined as a professor of mathematics and later works as a faculty in Berlin University. After some time, he lost his sight but still works in the field if science and mathematics. His famous paper of $1755$ on fluid flow derived the full inviscid equations of fluid motion now called Euler’s equations. He also represents the correct form of Bernoulli’s equation. He also developed Euler’s turbine equation.