# MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS - PowerPoint PPT Presentation

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## MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS

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### MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS Instructor: Professor C. T. HSU – PowerPoint PPT presentation

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Title: MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS

1
MECH 221 FLUID MECHANICS(Fall 06/07)Chapter 4
FLUID KINETMATICS
• Instructor Professor C. T. HSU

2
4. FLUID KINETMATICS
• Fluid kinematics concerns the motion of fluid
element. As the fluid flows, a fluid particle
(element) can translate, rotate, and deform
linearly and angularly

Translation Rotation
Linear deformation Circulation
Dilation Viscous stress
Angular deformation
3
4.1. Translation
• The translation considers mainly the velocity and
acceleration along the trajectory of fluid
element in linear motion

4
4.1. Translation
• For the fluid element moving along the trajectory
r(t), the velocity is simply given by v dr/dt
(u,v,w). As the description is basically
Lagrangian, the acceleration a is given by
• which, for steady flows, reduces to

5
4.2. Linear Deformation (Strain)
• Deformation change of shape of fluid element
• For easily understanding, we illustrate here in
two-dimensions. The results then can be easily
extended to 3-dimensions. Consider the
rectangular fluid element at the initial time
instant given in the following picture

6
4.2. Linear Deformation (Strain)
• The initial distance between points A and B is ?x
and between A and C is ?y. After a short time of
?t, the distances then become ?x?Lx and ?y?Ly
due to different velocities at B and C from A

7
4.2. Linear Deformation (Strain)
• The linear strain rate in x and y directions are
then given by
• Similarly, for 3-D flows we have in the
z-direction,

8
4.3. Dilation
• Volumetric expansion contraction
• The fluid dilation is defined as the change of
volume per unit volume. We are more interested in
the rate of dilation that determines the
compressibility of fluids. For 2-D flows,

9
4.3. Dilation
• Then, the rate of dilatation becomes,
• It is easy to generalize this dilation rate for
3-D flows and to reach
• For incompressible flow, the rate of dilation is
zero,

for 2-D flows
10
4.4. Angular Deformation (Strain)
• Now consider the deformation between A and B
caused by the change in velocity v, and the
deformation between A and C by change in u

11
4.4. Angular Deformation (Strain)
• For , the counter clockwise rotation
of AB is equal to clockwise rotation of AC
therefore, the fluid element is in pure angular
strain without net rotation and the angular
strain is equal to either or .
However, if ? , the strain then is
equal to
• . The rate of angular strain
is then given by

12
4.4. Angular Deformation (Strain)
• Similarly, we can extend to other planes y-z and
z-x to obtain

13
4.5. Rotation
• If then the fluid element is under
rigid body rotation on the x-y plane. No angular
strain is experienced, i.e.,

14
4.5. Rotation
• When ? , the rotation of fluid element
in x-y plane is the average rotation of the two
mutually perpendicular lines AB and AC
therefore,
• where a counter clockwise rotation is chosen as
positive and the rotation axis is in the z
direction

15
4.5. Rotation
• Rotation is a vector quantity for fluid elements
in 3-D motion. A fluid particle moving in a
general 3-D flow field may rotate about all three
coordinate axes, thus

16
4.5. Rotation
• The vorticity of a flow field is defined as

17
4.5. Rotation
• Therefore,
• The flow vorticity is twice the rotation
• In 2-D flow, ?/?z0 and w0 (or const.), so there
is only one component of vorticity,
• Irrotational flow is defined as having

18
4.5. Rotation
• A fluid particle moving, without rotation, in a
flow field cannot develop a rotation under the
action of a body force or normal surface force.
If fluid is initially without rotation, the
development of rotation requires the action of
shear stresses. The presence of viscous forces
implies the flow is rotational
• The condition of irrotationality can be a valid
assumption only when the viscous forces are
negligible. (as example, for flow at very high
Reynolds number, Re, but not near a solid
boundary)

19
4.6. Circulation
• Consider the flow field as shown below
• The circulation, , is defined as the line
integral of the tangential velocity about a
closed curve fixed in the flow,

20
4.6. Circulation
• Where is the line-element vector tangent to the
closed loop C of the integral. It is possible to
decompose the integral loop C into the sum of
small sub-loops, i.e.,
• Without loss of generality, each sub-loop can be
a rectangular grid as illustrated below.

21
4.6. Circulation
• Therefore,
• As a result, we have

where A is the area enclosed the contour
22
4.6. Circulation
• Stokes' theorem in 2-D
• The circulation around a closed contour (loop)
is the sum of the vorticity (flux) passing
through the loop
• This is an expression to illustrate the Greens
Theorem. In fact, the surface A can be a curved
surface

23
4.6. Circulation
• Then for each sub-loop on the surface, we have
locally
• where is the vorticity normal to the surface
enclosed by the small increment loop C

24
4.7. Viscous Stresses
• The strain rate tensor S is a symmetric tensor
that measures the rate of linear and angular
deformations of fluid element. The strain rate
tensor is expressed as
• where the superscript T represents the
transpose

25
4.7. Viscous Stresses
• In term of a Cartesian coordinate system, they
are expressed as

and
26
4.7. Viscous Stresses
• Following the Stokes hypothesis, the viscous
stress tensor is linearly related to the rate
of dilation and the strain rate tensor by

where I represents the unit tensor, i.e.,
27
4.7. Viscous Stresses
• The proportional constants of the above
linear relation are the volume viscosity and
shear viscosity of the fluid respectively. It is
seen that the fluid viscosity leads to additional
normal stresses, as well as shear stresses. Note
that is a symmetric tensor, i.e.,
• Total stress is given by
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