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

Compiled and modifiedCompiled and modifiedbyby

Sharma, AdamSharma, Adam

Week – 5Class – 2

Kinematics of FluidsKinematics of Fluids

ReviewReview

• 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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Objectives

• Types of Motion - Translation, Deformation• Rotation• Vorticity• Existence of flow• Continuity equation• Irrotational flow

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OTHER KINEMATIC DESCRIPTIONS

Fundamental types of fluid element motion or

deformation: (a) translation, (b) rotation, (c) linear strain,

and (d) shear strain.

Types of Motion or Deformation of Fluid ElementsIn fluid mechanics, an element may undergo four fundamental types of motion or deformation:(a) translation, (b) rotation, (c) linear strain (also called extensional strain), and(d) shear strain. All four types of motion or deformation usually occur simultaneously.

It is preferable in fluid dynamics to describe the motion and deformation of fluid elements in terms of rates such as velocity (rate of translation), angular velocity (rate of rotation), linear strain rate (rate of linear strain), and shear strain rate (rate of shear strain). In order for these deformation rates to be useful in the calculation of fluid flows, we must express them in terms of velocity and derivatives of velocity.

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Volumetric strain rate or bulk strain rate: The rate of increase of volume of a fluid element per unit volume.This kinematic property is defined as positive when the volume increases. Another synonym of volumetric strain rate is also called rate of volumetric dilatation, (the iris of your eye dilates (enlarges) when exposed to dim light). The volumetric strain rate is the sum of the linear strain rates in three mutually orthogonal directions.

Air being compressed by a piston in a cylinder; the volume of a fluid element in the

cylinder decreases, corresponding to a negative rate of volumetric dilatation.

The volumetric strain rate is zero in an incompressible flow.

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A fluid element illustrating translation, rotation, linear strain, shear strain, and volumetric strain.

Figure shows a general (although two-dimensional) situation in a compressible fluid flow in which all possible motions and deformations are present simultaneously.

In particular, there is translation, rotation, linear strain, and shear strain.

Because of the compressible nature of the fluid flow, there is also volumetric strain (dilatation).

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• Vorticity is closed related to fluid rate of rotation.

• If the vorticity at a point in a flow field is nonzero, the fluid particle that happens to occupy that point in space is rotating; the flow in that region is called rotational.

• Likewise, if the vorticity in a region of the flow is zero (or negligibly small), fluid particles there are not rotating; the flow in that region is called irrotational.

• Physically, fluid particles in a rotational region of flow rotate end over end as they move along in the flow.

The difference between rotational and irrotational flow: fluid elements in arotational region of the flow rotate, but those in an irrotational region of the flow do not.

Existence of flow

• Velocity components in accordance with the mass conservation principle are said to constitute a possible fluid flow.

• Therefore the existence of a physically possible flow field is verified from the principle of conservation of mass.

Continuity EquationContinuity Equation• A rectangular parallelepiped is considered as the control

volume in a Cartesian frame. • Let the fluid enter across the left face with a velocity ,

and a density The velocity and density with which the fluid leave the right face will be and ,

neglecting the higher order terms in • The rate of mass entering the control volume is:

• The rate of mass leaving the control volume is:

u.

dxx

uu

dxx

.dx

dzdyu

dzdydxx

uudx

x

dzdydxux

u

neglecting higher order terms in dx

Continuity EquationContinuity Equation

• The net rate of mass leaving the control volume in the x direction

• Mass leaving the control volume – Mass entering the control volume

• In a similar manner, the net rate of mass leaving in the y direction

dzdydxudzdydxux

u

dzdydxux

dzdydxvdzdxdyy

vvdy

y

dzdydxvy

Continuity EquationContinuity Equation• In a similar manner, the net rate of mass leaving in the z

direction

• The rate of accumulation of mass within the control volume is

{the volume is invariant with time}.

From the law of conservation of mass for a control volume, it can be written that

For an steady, incompressible flow,

dzdydxwdydxdzz

wwdz

z

dzdydxwz

dzdydxt

dzdydx

0

dzdydxwz

vy

uxt

0

z

w

y

v

x

u

Average Velocity

- Actual and averaged velocity profile as shown For simplification, we need Vavg instead of Vn

- For the average velocity can be written as;

1

c

avg n cc A

V V dAA

- For the flow with uniform density value across the cross section;

avg cm V A

Mass and Volume flow rates

Volume of fluid flow through a cross section per unit time is called “volume flow rate”

-Volume flow rate:

- Mass flow rate and volume flow rate can be correlated as;

m V

Volume Flow Rate & Mass Conservation

conservation of mass in flow rate form;

CVin out

dmm m

dt

Conservation of Mass Principle;

Conservation of Mass Principle

Steady Flow Conservation

- For steady flow across CV, mass is CONSTANT

- This means that;

in out

m m

- For incompressible flow, it is possible to write;

n n n nin out

V A V A

Steady Mass Flow Conservation

Any questions?