with constant heat capacities. The latter equation implies that the product pυ
γ
remains constant as we follow the fluid element in its motion. If, for example, the
fluid element has pressure p 0 and specific volume υ 0 at some instant in time, then its
pressure p and specific volume υ at later times are related according to the equation,
pυ
γ
¼ p 0 υ
γ
0 :
Consequently, the quantity pυ
γ remains constant in the flow and, as such, it
assumes the status of a state variable which, we will see in the subsequent discussion,
is related to another state variable called the specific entropy s, where we show in
Sect. 1.6 that
s ¼ c V ln pυ
γ
ð Þþ constant:
1.5 Constancy of the Entropy with Time for a Fluid
Element
Provided the fluid motion experiences no abrupt changes in any of the quantities, u,p
or ρ etc., the conservation of energy implies the constancy of entropy [9, 10]. Accordingly, if the fluid element is in thermodynamic equilibrium during the motion the
entropy of a fluid element will remain constant with time. In order to see this we need
to consider reversible changes taking place in thermodynamic systems. For a
reversible thermodynamic change we have [11],
TdS ¼ dE þ pdV
where dS is the change in entropy, dE is the change in internal energy and pdV
represents the work done in the process. Writing the latter equation as
dE ¼ TdS À pdV
and dividing both sides of this latter equation by the mass m of a fluid element we
have
de ¼ Tds À pdυ
where e is the internal energy per unit mass, ds is the entropy change per unit mass
and υ is the specific volume. As υ ¼ 1/ρ, we have dυ ¼ À (1/ρ
2 )dρ, so that
de ¼ Tds þ
p
ρ 2 dρ:
ð1:55Þ
1.5 Constancy of the Entropy with Time for a Fluid Element
21
γ
remains constant as we follow the fluid element in its motion. If, for example, the
fluid element has pressure p 0 and specific volume υ 0 at some instant in time, then its
pressure p and specific volume υ at later times are related according to the equation,
pυ
γ
¼ p 0 υ
γ
0 :
Consequently, the quantity pυ
γ remains constant in the flow and, as such, it
assumes the status of a state variable which, we will see in the subsequent discussion,
is related to another state variable called the specific entropy s, where we show in
Sect. 1.6 that
s ¼ c V ln pυ
γ
ð Þþ constant:
1.5 Constancy of the Entropy with Time for a Fluid
Element
Provided the fluid motion experiences no abrupt changes in any of the quantities, u,p
or ρ etc., the conservation of energy implies the constancy of entropy [9, 10]. Accordingly, if the fluid element is in thermodynamic equilibrium during the motion the
entropy of a fluid element will remain constant with time. In order to see this we need
to consider reversible changes taking place in thermodynamic systems. For a
reversible thermodynamic change we have [11],
TdS ¼ dE þ pdV
where dS is the change in entropy, dE is the change in internal energy and pdV
represents the work done in the process. Writing the latter equation as
dE ¼ TdS À pdV
and dividing both sides of this latter equation by the mass m of a fluid element we
have
de ¼ Tds À pdυ
where e is the internal energy per unit mass, ds is the entropy change per unit mass
and υ is the specific volume. As υ ¼ 1/ρ, we have dυ ¼ À (1/ρ
2 )dρ, so that
de ¼ Tds þ
p
ρ 2 dρ:
ð1:55Þ
1.5 Constancy of the Entropy with Time for a Fluid Element
21