Chapter 4: description of fluids in motion

the streamline is a curve that is

parallel (or tangent) to the velocity

vector (in any point of the fluid

domain)

steady motion:

the flow velocity does not change in time

particular streamline

First: fluid motion is defined by the velocity vector v = v(x,y,z)

let us define V=V(s,t)

remember that V has a

magnitude and a direction

Uniform flow : 𝜕𝑉(𝑠,𝑡)

𝜕𝑠= 0

Steady flow : 𝜕𝑉(𝑠,𝑡)

𝜕𝑡= 0

straight parallel streamlines ensure uniformity

uniform = homogeneous

the velocity does not change along the fluid path

steady? Uniform?

This is uniform, but

what if the flow is accelerating

in the pipe? V magn increases V direction changes

Steady flow: at every point in space, the velocity is independent of time; e.g. qin & qout constant

Uniform flow: the velocity is the same from one point in space to the next.

In non-uniform flow, the velocity may be different at different positions even if the flow is steady.

In the non-uniform flow, the fluid acceleration is generally not equal to zero, even if the flow is steady

qin

qout

we define acceleration along a pathline

nt er

Ve

dt

dV

ds

dVVa

2

BB6AB

Total acceleration is often defined as the Lagrangian acceleration

(perceived by an observer moving with the fluid)

nt er

Ve

t

V

s

VVa

2

in this example we assumed

steady “flow” and straight path,

so

tes

VVa

example BLACKBOARD

6C

In a general x,y,z reference system:

acceleration in a (un-)steady (non-) uniform flow

Acceleration:

Steady flow:

Uniform flow: 0z/v,y/v,x/v

0t/v

dt/vda

t

v

z

vv

y

vv

x

vv

dt

dva xx

z

x

y

x

x

x

x

convective acceleration Local

acceleration

Total

acceleration

kajaiaazyx

Euler equation

LazpL

Applying F = ma along an arbitrary direction L

for an inviscid, incompressible flow

Euler equation is valid point-wise.

Let us now apply the Euler equation along the pathline (precisely, along a tangent line )

the acceleration will thus have a component tangent to the pathline as

t

V

s

VVa

azps

s

s

BB6DE

t

v

s

vvazp

s

ssss

Along the direction s (assuming steady flow!)

= 0

2

2

ssss

v

ss

vvazp

s

02

vzp

s

2

s

constant2

2

svzp

along the pathline s,

which is not changing in time

(steady assumption)

Bernoulli equation

the pathline s in steady flow is a streamline

the streamline is a curve that is

parallel (or tangent) to the

velocity vector (in any point of

the fluid domain)

Bernoulli illustrated

constant2

vzp

2

s

constantg2

vz

p2

s

constant2g

vh

2

s

pressure head +

elevation head +

+ kinetic (or velocity) head

= constant along the streamline

piezometric

head

Think please ....

constant2

2

sVzp

along the pathline s,

which is not changing in time

(steady assumption)

can a fluid particle travel without losing any energy ?

is it a perpetual motion ? Like a perfect pendulum (never stops)

Remember we neglected viscosity !!!

It is not a dissipative system.

It is an ideal fluid,

frictionless

Application of Bernoulli equation to flow measurements

Stagnation tube BLACKBOARD

Application of Bernoulli equation to flow measurements

Pitot tube BLACKBOARD

Bernoulli – a simple application

?? ??

?? ??

(a) (b) (c)

BLACKBOARD