Fluid
Kinematics
Aiza A. Patadlas
Instructor
What is Fluid Kinematics?
• Branch of fluid mechanics which deals with response of fluids in motion without
consideringforcesand energiesinthem.
• Thestudy of kinematicsisoftenreferredtoasthegeometryof motion.
• It is generallyacontinuousfunctionin space and time.
Methods of Describing Fluid Motion
LangrangianMethod:
It describes a defined mass (position, velocity,acceleration, temperature ,
as functionsof time.
pressure, etc)
EulerianMethod:
It describes theflow field(velocity,acceleration,pressure, temperature,etc.)as functions of
position andtime.
Types ofFlow
Depending upon fluidproperties:
Ideal and Realflow.
Compressible and Incompressibleflow.
Depending upon properties offlow:
Laminar and turbulentflow.
Steady and unsteadyflow.
Uniform and Non-uniformflow.
Rotational and Irrotational flow.
One, two and three dimensionalflow.
Ideal and RealFlow
• Real fluid flows implies friction effects. Ideal fluid flowis
hypothetical.
• It assumes nofriction.
Velocity distribution of pipeflow.
Compressible andIncompressible Flows
Incompressible fluid flows assumes the fluid have constant density while in compressible fluid
flow,density is variableand becomesfunctionof temperatureandpressure.
Laminar and TurbulentFlow
The flow in which adjacent layer do not cross to each other and move along the well define
path iscalled as laminar flow.
LaminarFlow
TurbulentFlow
Reynold’s numberis usedto differentiatebetween laminarand turbulentflows.
Steady and UnsteadyFlows
Steadyflow:
It is the flow in which conditions of flow remains constant
withrespecttotime at a particular section but the condition
may be differentat differentsections.
e.g.,A constant discharge through apipe.
Unsteady flow:
It is the flow in which conditions of flow changes with
respecttotime at a
particularsection.
e.g.A variable discharge through apipe.
Uniform and Non-UniformFlow
Uniform flow:
Theflow in whichvelocity at a given time does not
change with respect to space (length
of
directionof flow)is calledas uniformflow.
e.g.Constant discharge though aconstant
diameterpipe.
Non – Uniform flow :
The flow in which velocity at a given time
changes with respect to space (length of
direction of flow) is called as non – uniform
flow.
e.g., Constant discharge through variable
diameterpipe.
One, Two and ThreeDimensional Flows
Ingeneralall fluids flow three-dimensionally, with pressures and velocities
and
other
flow propertiesvarying inall directions,in many cases the
greatest changes only
occurin two directions or evenonly in one.In these cases changes in the other
direction canbe effectively ignored making analysis muchmoresimple.
Flowis one dimensional:iftheflowparameters(such asvelocity,
pressure, depth etc.) at
agiveninstantintimeonlyvaryinthe
direction of flowand not across thecross-section.
Flowis two-dimensional:ifitcan beassumed thattheflowparameters varyinthedirectionofflow
andin one directionat rightangles to this direction.
Flowis three-dimensional:ifthe flowparametersvaryinallthree directionofflow.
Visualization of FlowPattern
Theflowvelocityisthebasicdescription ofhowa fluidmoves in time and
space,
but in orderto visualizetheflow pattern it is usefulto
definesome other properties of
theflow.These definitions
correspondto various experimental
methods of visualizing
fluid flow.
Theyare:
a. Streamlines- is a curvethat is everywheretangent to the velocityvector.
b. Pathlines- the actual path
travelled by an individual fluid particle over
some time period.
c. Streaklines - thelocus of fluid
particles that havepassed
sequentially through a
prescribed
point in theflow.
Velocity And Acceleration In Steady Flow
If the velocity of fluid particle has components 𝑢, 𝑣 and 𝑤 parallel to 𝑥, 𝑦 and 𝑧
axes, then for steady flow
𝑢𝑠𝑡 = 𝑢(𝑥, 𝑦, 𝑧)
𝑣𝑠𝑡 = 𝑣(𝑥, 𝑦, 𝑧)
𝑤𝑠𝑡 = 𝑤(𝑥, 𝑦, 𝑧)
Applying chain rule of partial differentiation, the acceleration of the fluid particle
for steady flow can be expressed as
𝑑
𝜕𝑉 𝑑𝑥 𝜕𝑉 𝑑𝑦 𝜕𝑉 𝑑𝑧
𝑎𝑠𝑡 = 𝑉 𝑥, 𝑦, 𝑧 =
+
+
𝑑𝑡
𝜕𝑥 𝑑𝑡 𝜕𝑦 𝑑𝑡 𝜕𝑧 𝑑𝑡
where
𝑉 = (𝑢2 + 𝑣 2 + 𝑤 2 )1/2
𝑑𝑥
𝑑𝑦
𝑑𝑧
Noting that = 𝑢, = 𝑣 𝑎𝑛𝑑 = 𝑤
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑎𝑠𝑡 =
𝑢+
𝑣+
𝑤
𝜕𝑥
𝜕𝑦
𝜕𝑧
This equations can be written as 3 scalar equations
𝜕𝑢
𝜕𝑢
𝜕𝑢
(𝑎𝑥 )𝑠𝑡 = 𝑢
+𝑣
+𝑤
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑣
𝜕𝑣
𝜕𝑣
(𝑎𝑦 )𝑠𝑡 = 𝑢
+𝑣
+𝑤
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑤
𝜕𝑤
𝜕𝑤
(𝑎𝑧 )𝑠𝑡 = 𝑢
+𝑣
+𝑤
𝜕𝑥
𝜕𝑦
𝜕𝑧
The fluid may possess an acceleration by virtue of a change in velocity with change in
position. (convective acceleration)
If the flow is unsteady
𝜕𝑢
𝜕𝑢
𝜕𝑢
𝜕𝑢
+𝑣
+𝑤
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑡
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑣
(𝑎𝑦 )𝑠𝑡 = 𝑢
+𝑣
+𝑤
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑡
𝜕𝑤
𝜕𝑤
𝜕𝑤
𝜕𝑤
(𝑎𝑧 )𝑠𝑡 = 𝑢
+𝑣
+𝑤
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑡
(𝑎𝑥 )𝑠𝑡 = 𝑢
(local acceleration)
For uniform flow (streamlines parallel to one another)
𝑑𝑉
𝑎=
𝑑𝑡
Generally, 𝑉 = 𝑉 𝑠, 𝑡
𝜕𝑉 𝜕𝑉
𝑎=𝑉
+
𝜕𝑠 𝜕𝑡
Problem 1:
A flow field is defined by 𝑢 = 2𝑥, 𝑣 = 𝑦. Derive expressions for the 𝑥 and 𝑦 components of
acceleration. Find the magnitude of the velocity and acceleration at the point 3,2 . Specify
units in terms of 𝐿 and 𝑇.
Problem 2:
A flow field is defined by 𝑢 = 2𝑦, 𝑣 = 𝑥. Derive expressions for the 𝑥 and 𝑦 components of
acceleration. Find the magnitude of the velocity and acceleration at the point 3,1 . Specify
units in terms of 𝐿 and 𝑇.
Problem 3:
An ideal liquid flows out the bottom of a large tank through a 100 − 𝑚𝑚 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 hole at
3
a steady rate of 0.80 𝑚 𝑠 . Assume the liquid approaches the center of the hole radially.
Find the velocities and convective accelerations at points 0.75 and 1.5 𝑚 from the center
of the hole.
THANK YOU!!