acceleration (following the particle). However, an observer on the bank of the river
who concentrates on the velocity at fixed points in the flow will not see any
acceleration as a succession of logs that pass the same point will do so at the same
velocity since the river as a whole is not accelerating.
In mathematical terms, Du/Dt is the Lagrangian acceleration, while ∂u/∂t or more
specifically, (∂u/∂t) x is the Eulerian acceleration, where the subscript indicates that
the acceleration is measured at a particular point x. However, u u(x, t), so that
du
dt
Du
Dt
¼
∂u
∂t
þ u
∂u
∂x
:
In relation to the log entering the rapids we have, ∂u/∂t ¼ 0, so that Du/Dt ¼ u
(∂u/∂x); implying that the river exhibits a spatial variation in velocity, thereby
accounting for the Lagrangian acceleration.
1.3 Some Elements of Thermodynamics
A concise review of some elements of thermodynamics is provided in this section.
Only those aspects of thermodynamics that are relevant to gas dynamics are considered and, in particular, to air which is assumed in this text to behave as an ideal gas.
1.3.1 Ideal Gas Equation
The equation of state for an ideal gas is given by the equation
pV ¼
m
M
ℜT,
ð1:1Þ
where p is the pressure, V is the volume of gas of mass m, M is the molecular weight,
ℜ is the universal gas constant (ℜ ¼ 8.31JK
À1 mole
À1 ) and T is the temperature. In
this text we will assume that air behaves as an ideal gas. One mole of air has an
approximate molecular weight (based on its composition which contains largely
nitrogen and oxygen) of 28.97 Â 10
À3 kg, so that ℜ/M ¼ 287Jkg
À1 K
À1 . Accordingly, Eq. (1.1) for air can be written as
pV ¼ mRT,
ð1:2Þ
where R ¼ 287JK
À1 kg
À1 , or we can also write it as
p ¼ ρRT or pυ ¼ RT
ð1:3Þ
1.3 Some Elements of Thermodynamics
3