Mechanics of Fluids

576 Unsteady flow
The continuity equation
In the fourth term τ0 represents the mean frictional shear stress at the pipe
wall and P the mean perimeter. We assume that τ0 has the same value as
in steady flow at the same velocity and, from eqn 7.3, we can substitute
τ0 = 1
2 f ϱu2 . However, the friction force always opposes the motion, so to
ensure that the term changes sign as u changes sign, we put u|u| in place
of u 2 (where |u| means the magnitude of u regardless of its sign). A variation
of f with u can be incorporated into the equation if desired and so can
additional losses.
On the right-hand side of the equation, the expression for acceleration is
taken from eqn 3.1. Neglecting higher orders of small quantities, we write
the mass m as ϱAδx, and so, after division by −ϱAδx, eqn 12.14 reduces to
1 ∂p
ϱ ∂x
1 u|u|
∂u
+ f + g sin α + u∂u + = 0 (12.15)
2 A/P ∂x ∂t
For the unsteady conditions being studied, the rate at which mass enters the
volume of length δx is equal to the rate at which mass leaves the volume
plus the rate of increase of mass within the volume. This may be written in
mathematical form as
�
ϱAu = ϱAu + ∂
∂x (ϱAu)δx
�
+ ∂
(ϱA δx)
∂t
∴ 0 = ∂ ∂
(ϱAu) +
∂x ∂t (ϱA)
= u ∂
∂
(ϱA) + ϱA∂u +
∂x ∂x ∂t (ϱA)
= u ∂p d
∂p d
(ϱA) + ϱA∂u + (ϱA) (12.16)
∂x dp ∂x ∂t dp
Dividing eqn 12.16 by d (ϱA)/dp and putting
we obtain
A
= c2
(d/dp)(ϱA)
The left-hand side of eqn 12.17 equals
(12.17)
u ∂p ∂u ∂p
+ ϱc2 + = 0 (12.18)
∂x ∂x ∂t
A
A(dϱ/dp) + ϱ(dA/dp) =
A
(Aϱ/K) + ϱ(dA/dp)
(from the definition of bulk modulus, eqn 1.8), so the result corresponds
to eqn 12.7. That is, c represents the celerity of a small wave for which
�p/K and δA/A are both small compared with unity (as assumed in the
derivation of eqn 12.7). However, eqn 12.17 is no more than a convenient
mathematical substitution and the method is not restricted to waves of small
amplitude.