Mechanics of Fluids

136 The momentum equation
=
=
� �
−
ABCD
� �
ABCD
� �
DCC ′ D ′
uxδm − �
uxδm
ABB ′ A ′
�
before δt
uxδm − �
ABB ′ A ′
uxδm + �
uxδm
�
DCC ′ D ′
after δt
uxδm
�
after δt
since, as the flow is assumed steady, �� uxδm �
is the same after δt as
ABCD
before δt. Thus, during the time intereval δt, the increase of x-momentum
of the batch of fluid considered is equal to the x-momentum leaving the
stream-tube in that time minus the x-momentum entering in that time:
� � � �
�
�
uxδm − uxδm
DCC ′ D ′
ABB ′ A ′
For a very small value of δt the distances AA ′ , BB ′ are very small, so
the values of ux, for all the particles in the space ABB ′ A ′ are substantially
the same. Similarly, all particles in the space DCC ′ D ′ have substantially the
same value of ux, although this may differ considerably from the value for
particles in ABB ′ A ′ . The ux terms may consequently be taken outside the
summations.
Therefore the increase of x-momentum during the interval δt is
� �
� �
δm
δm
�
ux
DCC ′ −
D ′
�
ux
ABB ′ A ′
(4.1)
Now �� δm �
DCC ′ D ′ is the mass of fluid which has crossed the plane CD
during the interval δt and so is expressed by ˙mδt, where ˙m denotes the rate
of mass flow. Since the flow is steady, �� δm �
ABB ′ A ′ also equals ˙mδt. Thus
expression 4.1 may be written ˙m(ux2 − ux1 )δt, where suffix 1 refers to the
inlet section of the stream-tube, suffix 2 to the outlet section. The rate of
increase of x-momentum is obtained by dividing by δt, and the result, by
Newton’s Second Law, equals the net force Fx on the fluid in the x direction
Fx = ˙m(ux2 − ux1 ) (4.2)
The corresponding force in the x direction exerted by the fluid on its
surroundings is, by Newton’s Third Law, −Fx.
A similar analysis for the relation between force and rate of increase of
momentum in the y direction gives
Fy = ˙m(uy2 − uy1 ) (4.3)
In steady flow ˙m is constant and so ˙m = ϱ1A1u1 = ϱ2A2u2 where ϱ represents
the density of the fluid and A the cross-sectional area of the stream-tube
(A being perpendicular to u).
We have so far considered only a single stream-tube with a cross-sectional
area so small that the velocity over each end face (AB, CD) may be considered
uniform. Let us now consider a bundle of adjacent stream-tubes, each of
cross-sectional area δA, which together carry all the flow being examined.