1) The document discusses fluid kinematics, which deals with the motion of fluids without considering the forces that create motion. It covers topics like velocity fields, acceleration fields, control volumes, and flow visualization techniques.
2) There are two main descriptions of fluid motion - Lagrangian, which follows individual particles, and Eulerian, which observes the flow at fixed points in space. Most practical analysis uses the Eulerian description.
3) The Reynolds Transport Theorem allows equations written for a fluid system to be applied to a fixed control volume, which is useful for analyzing forces on objects in a flow. It relates the time rate of change of an extensive property within the control volume to surface fluxes and the property accumulation.
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
Reynolds number and geometry concept, Momentum integral equations, Boundary layer equations, Flow over a flat plate, Flow over cylinder, Pipe flow, fully developed laminar pipe flow, turbulent pipe flow, Losses in pipe flow
1. Introduction to Kinematics
2. Methods of Describing Fluid Motion
a). Lagrangian Method
b). Eulerian Method
3. Flow Patterns
- Stream Line
- Path Line
- Streak Line
- Streak Tube
4. Classification of Fluid Flow
a). Steady and Unsteady Flow
b). Uniform and Non-Uniform Flow
c). Laminar and Turbulent Flow
d). Rotational and Irrotational Flow
e). Compressible and Incompressible Flow
f). Ideal and Real Flow
g). One, Two and Three Dimensional Flow
5. Rate of Flow (Discharge) and Continuity Equation
6. Continuity Equation in Three Dimensions
7. Velocity and Acceleration
8. Stream and Velocity Potential Functions
A fluid is a state of matter in which its molecules move freely and do not bear a constant relationship in space to other molecules.
In physics, fluid flow has all kinds of aspects: steady or unsteady, compressible or incompressible, viscous or non-viscous, and rotational or irrotational to name a few. Some of these characteristics reflect properties of the liquid itself, and others focus on how the fluid is moving.
Fluids are :-
Liquid : blood, i.v. infusions)
Gas : O2 , N2O)
Vapour (transition from liquid to gas) : N2O (under compression in cylinder), volatile inhalational agents (halothane, isoflurane, etc)
Sublimate (transition from solid to gas bypassing liquid state) : Dry ice (solid CO2), iodine
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
A fluid is a state of matter in which its molecules move freely and do not bear a constant relationship in space to other molecules.
In physics, fluid flow has all kinds of aspects: steady or unsteady, compressible or incompressible, viscous or non-viscous, and rotational or irrotational to name a few. Some of these characteristics reflect properties of the liquid itself, and others focus on how the fluid is moving.
Fluids are :-
Liquid : blood, i.v. infusions)
Gas : O2 , N2O)
Vapour (transition from liquid to gas) : N2O (under compression in cylinder), volatile inhalational agents (halothane, isoflurane, etc)
Sublimate (transition from solid to gas bypassing liquid state) : Dry ice (solid CO2), iodine
Fluid Mechanics Chapter 4. Differential relations for a fluid flowAddisu Dagne Zegeye
Introduction, Acceleration field, Conservation of mass equation, Linear momentum equation, Energy equation, Boundary condition, Stream function, Vorticity and Irrotationality
Report on Types of fluid flow
fluid dynamics
Introduction
In physics, fluid flow has all kinds of aspects: steady or unsteady, compressible or incompressible, viscous or non-viscous, and rotational or irrotational to name a few. Some of these characteristics reflect properties of the liquid itself, and others focus on how the fluid is moving. Note that fluid flow can get very complex when it becomes turbulent. Physicists haven’t developed any elegant equations to describe turbulence because how turbulence works depends on the individual system whether you have water cascading through a pipe or air streaming out of a jet engine. Usually, you have to resort to computers to handle problems that involve fluid turbulence. Types of fluid flow:
Aerodynamic force
Cavitation
Compressible flow
Couette flow
Free molecular flow
Incompressible flow
Fluid Mechanics Chapter 3. Integral relations for a control volumeAddisu Dagne Zegeye
Introduction, physical laws of fluid mechanics, the Reynolds transport theorem, Conservation of mass equation, Linear momentum equation, Angular momentum equation, Energy equation, Bernoulli equation
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
2. Introduction
Fluid Kinematics deals with the motion of fluids
without considering the forces and moments which
create the motion.
Items discussed in this Chapter.
Introduction
Velocity Field
Acceleration Field
Control Volume and System Representation
Reynolds Transport Theorem
Examples
3. Introduction
• If a Fluids is subjected to shear force then it tends to flow
• Fluids that are subjected to pressure imbalance resulted in
fluid flow
• In kinematics we are not concerned with the force, but the
motion, thus, we are interested in flow visualization.
• We can learn a lot about flows from watching.
4. Velocity field-Continuum Hypothesis
Continuum Hypothesis: The flow is made of tightly packed fluid
particles that interact with each other. Each particle consists of
numerous molecules, and we can describe the field variables
velocity, acceleration, pressure, and density of these particles at a
given time.
5. Velocity field
Different particles in fluid flow, which move at different
velocities and may be subjected to different accelerations.
The velocity and acceleration of a fluid particle may change
both with respect to time and space.
In the study of fluid flow it is necessary to observe the motion
of the fluid particles at various points in space and at a
successive instants of time.
6. Velocity field - Lagarangian Vs Eulerian method
Generally there are two methods by which the motion of a fluid
may be described, Lagarangian and Eulerian method.
In the Lagarangian method any individual fluid particle is selected ,
and observation is made about the behavior of this particle during
its course of motion through space (Control mass approach).
In the Eulerian method any point in the space occupied by the fluid
is selected and observation is made of whatever changes of
velocity, density and pressure which take place at that point (control
volume approach).
7. Lagrangian Description
Lagrangian description of fluid flow tracks the position and velocity
of individual particles.
Based upon Newton's laws of motion, it is difficult to use Lagrangian
description for practical flow analysis as.
Fluids are composed of billions of molecules.
Interaction between molecules hard to describe/model.
However, useful for specialized applications in
Sprays, particles, bubble dynamics, rarefied gases.
Coupled Eulerian-Lagrangian methods.
Named after Italian mathematician Joseph Louis Lagrange (1736-
1813).
8. Eulerian Description
In a Eulerian description of fluid flow a flow domain or control volume is
defined by which fluid flows in and out.
We define field variables which are functions of space and time.
Pressure field, P = P(x, y, z, t)
Velocity field,
Acceleration field,
These (and other) field variables define the flow field.
Well suited for formulation of initial boundary-value problems (PDE's).
Named after Swiss mathematician Leonhard Euler (1707-1783).
( ) ( ) ( ), , , , , , , , ,V u x y z t i v x y z t j w x y z t k= + +
( ) ( ) ( ), , , , , , , , ,x y za a x y z t i a x y z t j a x y z t k= + +
( ), , ,a a x y z t=
( ), , ,V V x y z t=
9. Lagarangian Vs Eulerian method
Lagrangian
Measurement of fluid temperature
Eulerian
Eulerian methods are commonly
used in fluid experiments or
analysis—a probe placed in a
flow.
10. Example: Coupled Eulerian-Lagrangian Method
Simulation of micron-scale
airborne probes. The probe
positions are tracked using a
Lagrangian particle model
embedded within a flow field
computed using an Eulerian
CFD code.
http://www.ensco.com/products/atmospheric/gem/gem_ovr.htm
11. Velocity field- Uniform Vs Non-Uniform
Uniform flow: If the flow velocity is the same magnitude and direction
at every point in the fluid at the given instant of time, it is said to be
uniform.
Non-uniform: If at a given instant, the velocity is not the same at
every point the flow is non-uniform. (In practice, fluid that flows near
a solid boundary will be non-uniform – as the fluid at the boundary
must take the speed of the boundary, usually zero.
0=
∂
∂
s
V
0≠
∂
∂
s
V
12. Velocity field- Steady Vs Unsteady
Steady Flow: The velocity at a given point in space does not vary
with time.
Very often, we assume steady flow conditions for cases where there
is only a slight time dependence, since the analysis is “easier”
Unsteady Flow: The velocity at a given point in space does vary with
time.
Almost all flows have some unsteadiness. In addition, there are
periodic flows, non-periodic flows, and completely random flows.
13. Velocity field- Steady Vs Unsteady
Periodic flow: “fuel injectors” creating a periodic swirling in the
combustion chamber. Effect occurs time after time.
Random flow: “Turbulent”, gusts of wind, splashing of water in the
sink
Steady or Unsteady only pertains to fixed measurements, i.e.
exhaust temperature from a tail pipe is relatively constant “steady”;
however, if we followed individual particles of exhaust they cool!
14. Velocity field- Steady Vs Uniform
Combining the above we can classify any flow into one of four type:
Steady uniform flow. Conditions do not change with position in the
stream and with time at a point.
An example is the flow of water in a pipe of constant diameter
at constant velocity.
Steady non-uniform flow. Conditions change from point to point in the
stream but do not change with time at a point.
An example is flow in a tapering pipe with constant velocity at
the inlet - velocity will change as you move along the length of
the pipe toward the exit.
15. Velocity field- Steady Vs Uniform
Unsteady uniform flow. At a given instant in time the conditions at
every point are the same, but will change with time.
Example : An example is a pipe of constant diameter connected
to a pump pumping at a constant rate which is then switched off.
4. Unsteady non-uniform flow. Every condition of the flow may
change from point to point and with time at every point.
For example waves in a channel.
16. Velocity Field- 1D, 2D, and 3D Flows
Most fluid flows are complex three dimensional, time-dependent
phenomenon, however we can make simplifying assumptions allowing
an easier analysis or understanding without sacrificing accuracy. In
many cases we can treat the flow as 1D or 2D flow.
Three-Dimensional Flow: All three velocity components are
important and of equal magnitude. Flow past a wing is complex 3D
flow, and simplifying by eliminating any of the three velocities would
lead to severe errors.
17. Velocity Field- 1D, 2D, and 3D Flows
Two-Dimensional Flow : In many situations one of the velocity
components may be small relative to the other two, thus it is
reasonable in this case to assume 2D flow.
One-Dimensional Flow: In some situations two of the velocity
components may be small relative to the other one, thus it is
reasonable in this case to assume 1D flow. There are very few
flows that are truly 1D, but there are a number where it is a
reasonable approximation.
18. Flow Visualization
Flow visualization is the visual examination of flow-field
features.
Important for both physical experiments and numerical (CFD)
solutions.
Numerous methods
Streamlines and streamtubes
Pathlines
Streaklines
Timelines
Refractive techniques
Surface flow techniques
19. Velocity field - Streamlines
A Streamline is an imaginary curve
drawn through the flowing fluid in such a
way that the tangent to it at any point
gives the direction of the velocity at that
point.
Because the fluid is moving in the
same direction as the streamlines,
fluid can not cross a streamline.
Streamlines can not cross each other.
If they were to cross this would
indicate two different velocities at the
same point. This is not physically
possible.
The above point implies that any
particles of fluid starting on one
streamline will stay on that same
streamline throughout the fluid.
20. Streamlines
A Streamline is a curve that is
everywhere tangent to the
instantaneous local velocity vector.
Consider an arc length
must be parallel to the local
velocity vector
Geometric arguments results in the
equation for a streamline
dr dxi dyj dzk= + +
dr
V ui vj wk= + +
dr dx dy dz
V u v w
= = =
21. Velocity field - Streamlines
NASCAR surface pressure contours
and streamlines
Airplane surface pressure contours,
volume streamlines, and surface
streamlines
22. Velocity field - Pathlines
( ) ( ) ( )( ), ,particle particle particlex t y t z t
A Pathline is the actual path
traveled by an individual fluid
particle over some time period.
Same as the fluid particle's
material position vector
Particle location at time t:
Particle Image Velocimetry (PIV) is
a modern experimental technique
to measure velocity field over a
plane in the flow field.
start
t
start
t
x x Vdt= + ∫
23. Velocity field - Streakline
A Streakline is the locus of
fluid particles that have passed
sequentially through a
prescribed point in the flow.
Easy to generate in
experiments: dye in a water
flow, or smoke in an airflow.
24. Comparisons
For steady flow, streamlines, pathlines, and streaklines are
identical.
For unsteady flow, they can be very different.
Streamlines are an instantaneous picture of the flow field
Pathlines and Streaklines are flow patterns that have a time
history associated with them.
Streakline: instantaneous snapshot of a time-integrated flow
pattern.
Pathline: time-exposed flow path of an individual particle.
25. Plots of Data
A Profile plot indicates how the value of a scalar property varies
along some desired direction in the flow field.
A Vector plot is an array of arrows indicating the magnitude and
direction of a vector property at an instant in time.
A Contour plot shows curves of constant values of a scalar property
for magnitude of a vector property at an instant in time.
26. Acceleration Field
Lagrangian Frame:
Eulerian Frame: we describe the acceleration in terms of position and time
without following an individual particle. This is analogous to describing the
velocity field in terms of space and time.
A fluid particle can accelerate due to a change in velocity in time (“unsteady”)
or in space (moving to a place with a greater velocity).
27. Acceleration Field: Material (Substantial) Derivative
time dependence
spatial dependence
We note:
Then, substituting:
The above is good for any fluid particle, so we drop “A”:
28. Acceleration Field: Material (Substantial) Derivative
Writing out these terms in vector components:
x-direction:
y-direction:
z-direction:
Writing these results in “short-hand”:
where,
k
z
j
y
i
x
ˆˆˆ()
∂
∂
+
∂
∂
+
∂
∂
=∇ ,
Fluid flows experience fairly
large accelerations or
decelerations, especially
approaching stagnation
points.
29. Acceleration field
The time dependant term in the acceleration field is called the local
acceleration and is nonzero only for unsteady flows.
The spacial dependant term in the acceleration field is called the
advective acceleration and accounts for the effect of the fluid
particle moving to a new location in the flow, where the velocity is
different.
The total derivative operator d/dt is called the material derivative
and is often given special notation, D/Dt.
Advective acceleration is nonlinear: source of many phenomenon
and primary challenge in solving fluid flow problems.
Advective acceleration provides ``transformation'' between
Lagrangian and Eulerian frames.
Other names for the material derivative include: total, particle,
Lagrangian, Eulerian, and substantial derivative.
30. Acceleration Field: Material (Substantial) Derivative
Applied to the Temperature Field in a Flow:
The material derivative of any variable is the rate at which that variable changes
with time for a given particle (as seen by one moving along with the fluid—
Lagrangian description).
31. Acceleration Field: Unsteady Effects
If the flow is unsteady, its paramater values at any location may change with
time (velocity, temperature, density, etc.)
The local derivative represents the unsteady portion of the flow:
If we are talking about velocity, then the above term is local acceleration.
In steady flow, the above term goes to zero.
If we are talking about temperature, and V = 0, we still have heat transfer
because of the following term:
0 0 0
=
32. Acceleration Field: Unsteady Effects
Consider flow in a constant diameter pipe, where the flow is assumed to be
spatially uniform:
0
0 00 0
33. Acceleration Field: Convective Effects
The portion of the material derivative represented by the spatial derivatives is
termed the convective term or convective accleration:
It represents the fact the flow property associated with a fluid particle may
vary due to the motion of the particle from one point in space to another.
Convective effects may exist whether the flow is steady or unsteady.
Example 1:
Example 2:
Acceleration = Deceleration
34. Control Volume and System Representations
Systems of Fluid: a specific identifiable quantity of matter that may consist of
a relatively large amount of mass (the earth’s atmosphere) or a single fluid
particle. They are always the same fluid particles which may interact with their
surroundings.
Control Volume: is a volume or space through which the fluid may flow,
usually associated with the geometry.
Example: following a system the fluid passing through a compressor
We can apply the equations of motion to the fluid mass to describe their
behavior, but in practice it is very difficult to follow a specific quantity of matter.
When we are most interested in determining the forces put on a fan, airplane,
or automobile by the air flow past the object rather than following the fluid as it
flows along past the object.
Identify the specific volume in space and analyze the fluid flow within,
through, or around that volume.
35. Control Volume and System Representations
Fixed Control Volume:
Fixed or Moving
Control Volume:
Deforming Control
Volume:
Surface of the Pipe
Surface of the Fluid
Volume Around The
Engine
Inflow
Outflow
Outflow Deforming Volume
36. Reynolds Transport Theorem: Preliminary Concepts
All the laws of governing the motion of a fluid are stated in their basic form in
terms of a system approach, and not in terms of a control volume.
The Reynolds Transport Theorem allows us to shift from the system
approach to the control volume approach, and back.
General Concepts:
B represents any of the fluid properties, m represent the mass, and b
represents the amount of the parameter per unit volume.
Examples:
Mass b = 1
Kinetic Energy b = V2/2
Momentum b = V (vector)
B is termed an extensive property, and b is an intensive property. B is
directly proportional to mass, and b is independent of mass.
37. Reynolds Transport Theorem: Preliminary Concepts
For a System: The amount of an extensive property can be calculated by
adding up the amount associated with each fluid particle.
Now, the time rate of change of that system:
Now, for control volume:
For the control volume, we only integrate over the control volume, this is
different integrating over the system, though there are instance when
they could be the same.
38. Reynolds Transport Theorem: Derivation
Consider a 1D flow through a fixed control volume between (1) and (2):
CV, and system at t1
System at t2
System at t2
Writing equation in terms of the extensive parameter:
Originally,
At time 2:
Divide by δt:
39. Reynolds Transport Theorem: Derivation
Noting,
Let,
Time rate of change of mass within the control volume:
The rate at which the extensive property flows out of the control surface:
(1) (2) (3) (4)
(1)
(2)
(4)
40. Reynolds Transport Theorem: Derivation
The rate at which the extensive property flows into the control surface:
(3)
Now, collecting the terms:
or
Restrictions for the above Equation:
1) Fixed control volume
2) One inlet and one outlet
3) Uniform properties
4) Normal velocity to section (1) and (2)
41. Reynolds Transport Theorem: Derivation
The Reynolds Transport Theorem can be derived for more general conditions.
Result:
This form is for a fixed non-deforming control volume.
42. Reynolds Transport Theorem: Physical Interpretation
(1) (2) (3)
(1) The time rate of change of the extensive parameter of a system, mass,
momentum, energy.
(2) The time rate of change of the extensive parameter within the control
volume.
(3) The net flow rate of the extensive parameter across the entire control
surface. “outflow across the surface”
“inflow across the surface”
“no flow across the surface”
Mass flow rate:
43. Reynolds Transport Theorem: Analogous to Material Derivative
Time dependant Portion Convective Portion
Steady Effects:
Unsteady Effects (inflow = outflow):
44. Kinematic Description
In fluid mechanics, an element
may undergo four fundamental
types of motion.
a) Translation
b) Rotation
c) Linear strain
d) Shear strain
Fluids motion and deformation is
best described in terms of rates
a) velocity: rate of translation
b) angular velocity: rate of rotation
c) linear strain rate: rate of linear
strain
d) shear strain rate: rate of shear
strain
45. Rate of Translation and Rotation
To be useful, these rates must be expressed in terms of velocity
and derivatives of velocity
The rate of translation vector is described as the velocity vector.
In Cartesian coordinates:
Rate of rotation at a point is defined as the average rotation
rate of two initially perpendicular lines that intersect at that
point. The rate of rotation vector in Cartesian coordinates:
V ui vj wk= + +
1 1 1
2 2 2
w v u w v u
i j k
y z z x x y
ω
∂ ∂ ∂ ∂ ∂ ∂
= − + − + − ∂ ∂ ∂ ∂ ∂ ∂
46. Linear Strain Rate
Linear Strain Rate is defined as the rate of increase in length per unit length.
In Cartesian coordinates
Volumetric strain rate in Cartesian coordinates
Since the volume of a fluid element is constant for an incompressible flow, the
volumetric strain rate must be zero.
, ,xx yy zz
u v w
x y z
ε ε ε
∂ ∂ ∂
= = =
∂ ∂ ∂
1
xx yy zz
DV u v w
V Dt x y z
ε ε ε
∂ ∂ ∂
= + + = + +
∂ ∂ ∂
47. Shear Strain Rate
Shear Strain Rate at a point is defined as half of the rate of
decrease of the angle between two initially perpendicular lines
that intersect at a point.
Shear strain rate can be expressed in Cartesian coordinates
as:
1 1 1
, ,
2 2 2
xy zx yz
u v w u v w
y x x z z y
ε ε ε
∂ ∂ ∂ ∂ ∂ ∂
= + = + = + ∂ ∂ ∂ ∂ ∂ ∂
48. Vorticity and Rotationality
The vorticity vector is defined as the curl of the velocity vector
Vorticity is equal to twice the angular velocity of a fluid particle.
Cartesian coordinates
In regions where ζ = 0, the flow is called irrotational.
Elsewhere, the flow is called rotational.
Vζ = ∇×
2ζ ω=
w v u w v u
i j k
y z z x x y
ζ
∂ ∂ ∂ ∂ ∂ ∂
= − + − + − ∂ ∂ ∂ ∂ ∂ ∂