FLUID MECHANICS – 1FLUID MECHANICS – 1Semester 1 2011 - 2012Semester 1 2011 - 2012

Compiled and modifiedCompiled and modified

byby

Sharma, AdamSharma, Adam

Week – 5Class – 1

Kinematics of FluidsKinematics of Fluids

ObjectivesObjectives

• Description of Fluid flow Description of Fluid flow

• StSteady and eady and unsteady unsteady flowflow

• Uniform and non uniform flowUniform and non uniform flow

• Dimensions of flowDimensions of flow

• Material derivative and accelerationMaterial derivative and acceleration

• Differentiate between streamlines, pathlines and Differentiate between streamlines, pathlines and streaklinesstreaklines

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Fluid DescriptionFluid Description• KKinematicsinematics:: TThe study of motion. he study of motion.

• FFluid kinematicsluid kinematics:: TThe study of how fluids flow and how to he study of how fluids flow and how to describe fluiddescribe fluid motion.motion.

• There are distinct ways to describe motion of fluid There are distinct ways to describe motion of fluid particles:particles:

a) Lagrangian a) Lagrangian

b) Eulerianb) Eulerian

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Lagrangian descriptionLagrangian description

• This method requires us to track the position and velocity This method requires us to track the position and velocity of each individual fluid of each individual fluid particleparticle..

• If the number of objects is small, suchIf the number of objects is small, such as billiard balls on as billiard balls on a pool table,a pool table, individual objects can be tracked.individual objects can be tracked.

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However, if a fluid lump However, if a fluid lump changes its shape, size changes its shape, size and state as its moves with and state as its moves with time, it is difficult to trace time, it is difficult to trace the lumpthe lump in in Lagrangian Lagrangian description.description.

Eulerian Description

• Instead of tracking individual fluid particles, we define field variables such as velocity, pressure as functions of space and time, within the control volume.

• The field variable at a particular location at a particular time is the value of the variable for whichever fluid particle happens to occupy that location at that time.

• Eulerian description is often more convenient for fluid Eulerian description is often more convenient for fluid mechanicsmechanics applications.applications. EExperimental measurements are xperimental measurements are generally moregenerally more suited with Eulerian approach.suited with Eulerian approach. 5

• To describe the fluid flow, a flow domain of a finite volume or control volume is defined, through which fluid flows in and out of control volume.

 Variation Of Flow ParametersVariation Of Flow Parameters

Steady and Unsteady flowSteady and Unsteady flow

• Steady flow is defined as the flow in which pressure and Steady flow is defined as the flow in which pressure and density do not change with time in a control volumedensity do not change with time in a control volume

• In the Lagrangian approach, time is inherent in In the Lagrangian approach, time is inherent in describing the trajectory of any particle. But in steady describing the trajectory of any particle. But in steady flow, the velocities of all particles passing through any flow, the velocities of all particles passing through any fixed point at different times will be same. fixed point at different times will be same.

Uniform and Non-uniform flow Uniform and Non-uniform flow

• When velocity and other hydrodynamic parameters at When velocity and other hydrodynamic parameters at any instant of time do not change from point to point in a any instant of time do not change from point to point in a flow field, the flow is said to be uniform. Hence for a flow field, the flow is said to be uniform. Hence for a uniform flow, the velocity is a function of time only. uniform flow, the velocity is a function of time only.

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Streamlines and Streamtubes

Streamline: A curve that is everywhere tangent to the instantaneous local velocity vector.

Stream line at any instant can be defined as an imaginary curve or line in the flow field, so that the tangent to the curve at any point, represents the direction of the instantaneous velocity at that point.

Streamlines are useful as indicators of the instantaneous direction of fluid motion throughout the flow field.

Properties of stream linesProperties of stream lines

• The component of velocity, normal to a streamline is zero, there can be no flow across a streamline.

• Since the instantaneous velocity at a point in a fluid flow must be unique in magnitude and direction, the same point cannot belong to more than one streamline.

• In other words, a streamline cannot intersect itself nor can any streamline intersect another streamline.

• In a steady flow, the orientation or the pattern of streamlines will be fixed.

• In an unsteady flow where the velocity vector changes with time, the pattern of streamlines also changes from instant to instant.

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Equation Of Stream Lines Equation Of Stream Lines

• Consider a streamline in a plane flow in the x-y plane. Consider a streamline in a plane flow in the x-y plane. By definition, the velocity vector U at a point P must be By definition, the velocity vector U at a point P must be tangential to the streamline at that point. It follows thattangential to the streamline at that point. It follows that

;;

where u and v are velocity components along x and y where u and v are velocity components along x and y directions respectively. The velocity vector is expressed directions respectively. The velocity vector is expressed asas

This shows that the velocity may vary along a streamline This shows that the velocity may vary along a streamline direction as well as with the passage of time.direction as well as with the passage of time.

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u

vtan

dx

dy 0 dxvdyu

t,sUU

EQUATION OF STREAM LINEEQUATION OF STREAM LINE

• Consider an elementary displacement element Consider an elementary displacement element along a along a general streamline where the velocity U such thatgeneral streamline where the velocity U such that

oror

i.e.,i.e.,

which is the which is the equation of a streamline.equation of a streamline. 10

wkvjuiU kzji yxs

0sx Ui j k

u v w 0

x y zd d d

0 kxvyujxwzui yw-zv

dx dy dz

u v w

STREAM TUBE• Stream tube: A bundle of neighboring stream lines may

be imagined to form a passage through which the fluid flows. This passage (non necessarily circular in cross-section) is known as a stream tube.

• Since a stream tube is bounded on all sides by streamlines, velocity does not exist across a streamline, no fluid may enter or leave a stream tube except through its ends.

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In an incompressible flow field, a stream tube In an incompressible flow field, a stream tube ((aa) decreases in diameter as the flow ) decreases in diameter as the flow

accelerates or converges and (accelerates or converges and (bb) increases in ) increases in diameter as the flow decelerates or diverges.diameter as the flow decelerates or diverges.

A A stream tube consists of a bundle of streamlines much like a communications cable consists of a bundle of fiber-optic cables.

Both streamlines and stream tubes are instantaneous quantities, defined at a particular instant in time according to the velocity field at that instant.

Dimensions Of Flow

• In general, fluid flow is three dimensional. This means In general, fluid flow is three dimensional. This means that the flow parameters like velocity, pressure vary in all that the flow parameters like velocity, pressure vary in all the three coordinate directions. the three coordinate directions.

• Sometimes simplification is made in the analysis of Sometimes simplification is made in the analysis of different fluid flow problems by selecting the coordinate different fluid flow problems by selecting the coordinate directions so that appreciable variation of the directions so that appreciable variation of the hydrodynamic parameters take place in only two hydrodynamic parameters take place in only two directions or even in only one.directions or even in only one.

• So in one dimensional flow, all the flow parameters are So in one dimensional flow, all the flow parameters are expressed as functions of time and one space expressed as functions of time and one space coordinate only. This single coordinate is usually the coordinate only. This single coordinate is usually the distance measure along the centre line of some conduit distance measure along the centre line of some conduit In which the fluid is flowing.In which the fluid is flowing.

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Material DerivativeMaterial Derivative

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The material derivative The material derivative D/Dt D/Dt isis defined by following a fluid defined by following a fluid particleparticle as it moves throughout the flow field.as it moves throughout the flow field. In this In this illustrations, the fluid particle isillustrations, the fluid particle is accelerating to the right accelerating to the right as it movesas it moves up and to the right.up and to the right.

Particle Acceleration

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Material DerivativeMaterial Derivative

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• The total derivative operator The total derivative operator dd//dt dt in this equation in this equation is given is given a special name, thea special name, the material derivativematerial derivative; it is assigned a ; it is assigned a special notation, special notation, DD//DtDt, in order to emphasize, in order to emphasize that it is that it is formed by formed by following a fluid particle as it moves throughfollowing a fluid particle as it moves through the flow fieldthe flow field. .

• Other names for the material derivative includeOther names for the material derivative include total, total, particle, Lagrangian, Eulerianparticle, Lagrangian, Eulerian, and , and substantial derivativesubstantial derivative..

Even under steady flow, Even under steady flow, a fluid particle can be a fluid particle can be accelerated as in the accelerated as in the flow throughflow through a nozzle a nozzle

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Any questions?Any questions?

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