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Solution Manual, Chapter 4 – Fluid Kinematics
Chapter 4
Fluid Kinematics
Introductory Problems
4-1C
Solution We are to define and explain kinematics and fluid kinematics.
Analysis Kinematics means the study of motion. Fluid kinematics is the study
of how fluids flow and how to describe fluid motion. Fluid kinematics deals with
describing the motion of fluids without considering (or even understanding) the forces
and moments that cause the motion.
Discussion Fluid kinematics deals with such things as describing how a fluid
particle translates, distorts, and rotates, and how to visualize flow fields.
4-2
Solution We are to write an equation for centerline speed through a nozzle,
given that the flow speed increases parabolically.
Assumptions 1 The flow is steady. 2 The flow is axisymmetric. 3 The water is
incompressible.
Analysis A general equation for a parabola in the x direction is
General parabolic equation:
()
2
uabxc=+ − (1)
We have two boundary conditions, namely at x = 0, u = uentrance and at x = L, u = uexit.
By inspection, Eq. 1 is satisfied by setting c = 0, a = uentrance and b = (uexit – uentrance)/L2.
Thus, Eq. 1 becomes
Parabolic speed:
(
)
exit entrance 2
entrance 2
uu
uu x
L
−
=+
(2)
Discussion You can verify Eq. 2 by plugging in x = 0 and x = L.
4-1
Solution Manual, Chapter 4 – Fluid Kinematics
4-3
Solution For a given velocity field we are to find out if there is a stagnation
point. If so, we are to calculate its location.
Assumptions 1 The flow is steady. 2 The flow is two-dimensional in the x-y plane.
Analysis The velocity field is
(
)
(
)( )
, 0.5 1.2 2.0 1.2Vuv xi y==+ +−−j
G
GG
(1)
At a stagnation point, both u and v must equal zero. At any point (x,y) in the flow
field, the velocity components u and v are obtained from Eq. 1,
Velocity components: 0.5 1.2 2.0 1.2uxv y
=
+=−− (2)
Setting these to zero yields
Stagnation point: 0 0.5 1.2 0.4167
0 2.0 1.2 1.667
xx
yy
=
+=−
=− − =− (3)
So, yes there is a stagnation point; its location is x = -0.417, y = -1.67 (to 3 digits).
Discussion If the flow were three-dimensional, we would have to set w = 0 as
well to determine the location of the stagnation point. In some flow fields there is
more than one stagnation point.
4-2
Solution Manual, Chapter 4 – Fluid Kinematics
4-4
Solution For a given velocity field we are to find out if there is a stagnation
point. If so, we are to calculate its location.
Assumptions 1 The flow is steady. 2 The flow is two-dimensional in the x-y plane.
Analysis The velocity field is
() ( )
(
)
()
2
22
,2V u v a b cx i cby c xy j==−− +−+2
G
GG
(1)
At a stagnation point, both u and v must equal zero. At any point (x,y) in the flow
field, the velocity components u and v are obtained from Eq. 1,
Velocity components:
()
2
22
2 2u a b cx v cby c xy=−− =− + (2)
Setting these to zero and solving simultaneously yields
Stagnation point:
()
2
2
2
0
2 2 0
ba
abcx x c
vcbycxyy
−
=−− =
=− + =
(3)
So, yes there is a stagnation point; its location is x = (b – a)/c, y = 0.
Discussion If the flow were three-dimensional, we would have to set w = 0 as
well to determine the location of the stagnation point. In some flow fields there is
more than one stagnation point.
Lagrangian and Eulerian Descriptions
4-5C
Solution We are to define the Lagrangian description of fluid motion.
Analysis In the Lagrangian description of fluid motion, individual fluid
particles (fluid elements composed of a fixed, identifiable mass of fluid) are followed.
Discussion The Lagrangian method of studying fluid motion is similar to that of
studying billiard balls and other solid objects in physics.
4-3
Solution Manual, Chapter 4 – Fluid Kinematics
4-6C
Solution We are to compare the Lagrangian method to the study of systems and
control volumes and determine to which of these it is most similar.
Analysis The Lagrangian method is more similar to system analysis. In both
cases, we follow a mass of fixed identity as it moves in a flow. In a control volume
analysis, on the other hand, mass moves into and out of the control volume, and we
don’t follow any particular chunk of fluid. Instead we analyze whatever fluid happens
to be inside the control volume at the time.
Discussion In fact, the Lagrangian analysis is the same as a system analysis in the
limit as the size of the system shrinks to a point.
4-7C
Solution We are to define the Eulerian description of fluid motion, and explain
how it differs from the Lagrangian description.
Analysis In the Eulerian description of fluid motion, we are concerned with
field variables, such as velocity, pressure, temperature, etc., as functions of space and
time within a flow domain or control volume. In contrast to the Lagrangian method,
fluid flows into and out of the Eulerian flow domain, and we do not keep track of the
motion of particular identifiable fluid particles.
Discussion The Eulerian method of studying fluid motion is not as “natural” as
the Lagrangian method since the fundamental conservation laws apply to moving
particles, not to fields.
4-8C
Solution We are to determine whether a measurement is Lagrangian or
Eulerian.
Analysis Since the probe is fixed in space and the fluid flows around it, we are
not following individual fluid particles as they move. Instead, we are measuring a
field variable at a particular location in space. Thus this is an Eulerian measurement.
Discussion If a neutrally buoyant probe were to move with the flow, its results
would be Lagrangian measurements – following fluid particles.
4-4
