Irrotational Flows

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FLUID

MECH

ANICS

Kinem at i cs

1. Intro du ction

Student today we are going to study the Kinematic aspect of

the fluid i.e., the of fluid or liquid in motion without

considering the forces causing motion. In this topic we will first

deal with different types of flow, which exists like laminar or

turbulent flow, path line or streak line .The study of these types

flow effect the different changes to occur suppose in case of

hydraulic turbine if flow is turbulent through the penstock then

the head available at the end of pipe will be low leading to less

energy available for the generation of power but if we know in

advance the kind of flow corrective can be taken to ensure that

laminar flow occurs leading to increase in the generation ofpower .

Like in the examples quoted above we as mechanical engineers

need to have to have good understanding of the different kind

of flow that exists. Hence the topics that will be under our

ambit of discussion need a through under standing .

Kinematics is defined as that branch of science, which deals

with the motion of particles without considering the forces

causing motion. The velocity at a point in a flow field at any

time is studied in this branch of fluid mechanics. Once the

velocity is known, then the pressure distribution and hence

forces acting on the fluid can be determined

2. Method of Describing Fluid MotionThe fluid motion is described by two methods. They are :

i. Lagrangian method

ii. Eulerian method.

In the Lagrangian method a single fluid particle is followed

during its motion and its velocity, acceleration, density etc are

described.

In case of Eulerian method, the velocity, acceleration, pressure,

density etc are described at a point in a flow field. The Eulerian

is commonly used in fluid mechanics.

3. Streamline

A streamline is an imaginary curve drawn through a flowingfluid in such away that the tangent to it at any point gives the

direction of velocity of the flow at that point. Since a fluid is

composed of fluid particles, the pattern of flow of fluid may

be represented by a series of stream lines, obtained by drawing a

series of curves through flowing fluid such that the velocity

vector at any point is tangential to the curves .The fig below

show some of the stream lines for a flow pattern in the xy

plane in which a stream line passing through appoint P(x,y) is

tangential to the velocity vector V at P.If u and v are the

components of V along x and y directions, then

v/ u = tan = dy/ dx

Where dy and dx are the y and x components of the differential

displacement ds along the streamline in the immediate vicinity

of P.Therefore the differential equation for the stream lines inthe xy plane may be written as

dx/ u = dy/ v; Or (udy vdx)

A general differential equation for three dimensional flow may

however be obtained in the manner as

dx/ u = dy/ v = dz/ w

LESSON 5:

NON-UNIFORM , LAMINAR, TURBULENT, ROTATIONAL,

IRROTATIONAL FLOWS, CONSERVATION OF MASS


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4. Stream Tube

A stream tube is a tube imagined to be formed by a group of

streamlines passing through a small closed curve, may or may

not be circular. Since the stream tube is bounded on all sides by

streamlines and since the velocity has no component normal to

a streamline, there can be no flow across the bounding surface

of stream tube. Therefore a fluid may enter or leave the stream

tube only at its ends. A stream tube with a cross-sectional areasmall enough for the variation of velocity over it to be

negligible is sometimes termed, as stream filament .The concept

of stream tube is quite useful in analyzing several fluid flow

problems, Since the entire flow field may be divided into large

number of stream tubes, thus yielding a clear picture of entire

pattern of flow. However, only in steady flow a stream tube is

fixed in space.

5. Type s of Fluid Flow

Ste ad y Flow

When the velocity at each location is constant, the velocity field

is invariant with time and the flow is said to be steady.

(dv/ dt)XoYoZo =0

Uniform Flow

Uniform flow occurs when the magnitude and direction of

velocity do not change from point to point in the fluid.

Flow of liquids through long pipelines of constant diameter is

uniform whether flow is steady or unsteady.

Non-uniform flow occurs when velocity, pressure etc., change

from point to point in the fluid.

Stea dy, Un iform Flow

Conditions do not change with position or time.

e.g., Flow of liquid through a pipe of uniform bore running

completely full at constant velocity.

Stea dy, Non-uniform Flow

Conditions change from point to point but do not with time.

e.g., Flow of a liquid at constant flow rate through a tapering

pipe running completely full.

Unsteady, uniform Flow: e.g. when a pump starts-up.

Unsteady, non-uniform Flow: e.g. Conditions of liquid during

pipetting out of liquid.

Laminar Flow

Laminar flow is defined as that type of flow in which the fluid

particles move along well defined path or stream line and all the

stream lines are straight and parallel. Thus the particle moves in

laminas or layers gliding smoothly over adjacent layer. This type

of flow is also called streamline or viscous flow.

Turbulent Flow

Turbulent flow is that type of flow in which the fluid particle

move in zig zag way .Due to the movement of fluid particle inthe zig zag way, the eddies formation take place which are

responsible for high energy loss .For a pipe flow, the type of

flow is determined by a non dimensional number VD/ called

the Reynolds number

Where

D = diameter of the pipe

V = Mean velocity of flow in the pipe

= Kinematic viscosity of the fluid

If the Reynolds number is less than 2000, the flow is called

Laminar flow .If the Reynolds number is more than 4000,it is

called turbulent flow.Rot at iona l Flow

Rotational flow is that type of flow in which in which fluid

particles while moving along a stream also rotate about their

own axis.

Irrot at iona l Flow

Irrotational flow is defined as that type of in which the fluid

particles while along stream line, do not rotate about their own

axis, that type of flow is called Irrotational flow.

6. Continuity Equation

Let us make the mass balance for a fluid element as shown

below: (an open-faced cube)


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Mass Balan ce

Accumulation rate of mass in the system = all mass flow rates

in - all mass flow rates out > 1

The mass in the system at any instant isx y z . The flow into

the system through face 1 is

And the flow out of the system through face 2 is

Similarly for the faces 3, 4, 5, and 6 are written as follows:

Substituting these quantities in equn.1, we get

Dividing the above equation by xyz :

Now we let xy and z each approach zero simultaneously,

so that the cube shrinks to a point. Taking the limit of the three

ratios on the right-hand side of this equation, we get the partial

derivatives.

This is the continuity equation for every point in a fluid flowwhether steady or unsteady, compressible or incompressible.

For steady, incompressible flow, the density r is constant and

the equation simplifies to

For two-dimensional incompressible flow this will simplify still

further to

Notes

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Irrotational Flows