DERIVATION OF BASIC

EQUATIONS

Dr. Shahid Ali

Adhémar Jean Claude Barré de Saint-Venant

(23 August 1797 – January 1886)

He was a mechanician and mathematician

who contributed to early stress analysis and

also developed the unsteady open channel

flow shallow water equations, also known as

the Saint-Venant equations that are a

fundamental set of equations used in

modern hydraulic engineering

École Polytechnique (Paris)

In 1843 he published the correct derivation of the Navier–Stokes

equations for a viscous flow and was the first to "properly identify

the coefficient of viscosity and its role as a multiplying factor for the

velocity gradients in the flow". Even though he published before

Stokes, the equations do not bear his name.

He went on to teach mathematics at the École des Ponts et

Chaussées (National school of Bridges and Roads) where he

succeeded Coriolis.

(18 July 1853 – 4 February 1928)

Hendrik Antoon Lorentz was a

Dutch physicist who shared the 1902 Nobel

Prize in Physics withPieter Zeeman for the

discovery and theoretical explanation of

the Zeeman effect. He also derived

the transformation equations subsequently

used by Albert Einstein to describe space

and time.

Lorentz was also asked by the Dutch government to chair a committee to

calculate some of the effects of the proposed Afsluitdijk (Enclosure Dam) flood

control dam on water levels in the Waddenzee. Hydraulic engineering was mainly

an empirical science at that time, but the disturbance of the tidal flow caused by

the Afsluitdijk was so unprecedented that the empirical rules could not be trusted.

Originally Lorentz was only supposed to have a coordinating role in the

committee, but it quickly became apparent that Lorentz was the only physicist to

have any fundamental traction on the problem. In the period 1918 till 1926,

Lorentz invested a large portion of his time in the problem. Lorentz proposed to

start from the basic hydrodynamic equations of motion and solve the problem

numerically. This was feasible for a "human computer", because of the quasi-one-

dimensional nature of the water flow in the Waddenzee. The Afsluitdijk was

completed in 1932 and the predictions of Lorentz and his committee turned out to

be remarkably accurate. One of the two sets of locks in the Afsluitdijk was named

after him.

Afsluitdijk in the Waddenzee

Momentum Equation

....

.scvc

dAVVdVdt

dF

Sum of forces on

the C.V.

Momentum stored

within the C.V

Momentum flow

across the C. S.

0)(11 2

fo SSg

x

yg

A

Q

xAt

Q

A

Momentum Equation

0)(11 2

fo SSg

x

yg

A

Q

xAt

Q

A

0)(

fo SSg

x

yg

x

VV

t

V

Local

acceleration

term

Convective

acceleration

term

Pressure

force

term

Gravity

force

term

Friction

force

term

Kinematic Wave

Diffusion Wave

Dynamic Wave

Momentum Equation

fo SSx

y

x

V

g

V

t

V

g

1

Steady, uniform flow

Steady, non-uniform flow

Unsteady, non-uniform flow

Thanks