Stream Tube

Stream Tube
A region of the moving fluid bounded on the all sides by
streamlines is called a tube of flow or stream tube.
As streamline does not intersect each other, no fluid
enters or leaves across the sides.
V 1
V 2
δA 1
δA 2
When density do not depend explicitly on time then from
continuity equation, we have
∇⋅ ( vρ) = 0
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∫ ∇⋅
=
V
∫
s
( vρ) 0
( vρ ). dS=
0
Where V is the volume of the
streamtube between faces δA 1 and
δA 2
From divergence theorem
v
ρ and vρ are the mean values of ρvover δA and δA
1 2
vρ δA= vρ δA
1 1 2 2
1 2
For a homogeneous, incompressible, Non-viscous fluids
v
δA= v δA
1 1 2 2
Thus, whenever stream tube is constricted, i.e., wherever
streamlines gets crowded, the speed of flow of liquid is larger
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For the entire collection of the stream tubes occupying the
whole cross section of the passage through which the fluid
flows
∫ ρvdS
A
is constant along the passage
In the case of constant density
∫ vdS is constant
or
v
A= v A
1 1 2 2
A
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Bernoulli’s Principle and Physical Significance of
different terms
In the steady flow of homogeneous, incompressible non-viscous fluid,
P
1
v
gz
Constant where gravitional potential
2
ρ + 2
+ = Ω
=gz
Along a very narrow stream tube
In the case of the gases last term is negligible w.r.t. first two terms as the
Density is very low for the gases
Third term is gravitational potential energy of the unit mass
Second term is kinetic energy of the unit mass
First term also has units of energy per unit mass
What energy this term represents
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δl 1
δl 2
P 1
P 2
A B
1
1
A 2
B 2
Consider the volume of fluid element between A1 and A2 which is
moved after some time to volume between B1 and B2
δlδA = δl δA = δV
1 1 2 2
Work done during the motion by the pressure forces is given by
δlPδA− δlPδA = ( P−P)
δV
1 1 1 2 2 2 1 2
Work done during the motion by the pressure forces in moving the
volume element from A1B1 to A2B2 per unit mass is given by
P2
( P1−P2)
δV P1−P2
dp
= = ∫ −
ρδV
ρ ρ
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P1

P2 0
∫
P1
dp dp P
− = ∫− =
ρ ρ ρ
P
Therefore, P/ρ term represents the work that will be done on
the unit mass of the fluid by the pressure forces, as the
element flows from a point where pressure is P to a point
where pressure is zero.
At two different point in the stream tube
P 1 P 1
+ v + gz = + v + gz
ρ 2 ρ 2
1 2 2 2
1 1 2 2
P
− P
ρ
1 1
2 2
1 2
2 2
⇒ + gz (
1− z2)
= v2 − v1
In the steady flow of an incompressible, homogeneous, nonviscous
fluid, moving from position 1 to 2, the increase in
kinetic energy per unit mass is equal to the work done on unit
mass of the fluid by pressure and gravity forces.
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The three terms is therefore called as pressure energy,
kinetic energy and gravitational energy
2
P v
, , z are known as pressure head, velocity head
ρg 2g
and gravity head and each having the dimention of length
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Application 1: Velocity of efflux from a reservoir
P 1
P 2
Flow from a sharp
edged orifice in a tank
Vena Contracta
Following assumptions are made in order to simplify the problem
• steady state flow of Incompressible, homogeneous, nonviscous
fluid (frictional effects can be neglected)
•Cross sectional area of the reservoir is very large compared to
that of orifice. Hence the velocity of the fluid at the surface is
negligibly small. Practically hydrostatic equilibrium condition
exist for points at the surface.
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From the Bernoulli’s principle, we can write
P2
≈ P atm
v1 = 0
1 atm
1
P 1 P 1
+ v + z = + v + z
ρg 2g ρg 2g
1 2 2
2
1 1 2 2
z
2
= 0 when centre of the orifice is chosen
as the horizontal plane for the gravitaional head
P − P = ρg( h−z
) from the hydrostatic condition
Putting these all values
v
2
= 2
gh
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Identical relation for velocity of efflux is obtained for a
discharge from one reservoir to another reservoir where h
now stand for difference between the gravitational head
across the orifice
The rate of discharge Q, i.e., the volume of the fluid flowing
out of the reservoir per unit time is given by
Q Av A gh
actual
= αα
c v 2
= αd
2
Q= Av = A gh
2
2
Actual Velocity
where αv
= ; αc
= Cofficient of contraction
Ideal Velocity
and αα = α is called coefficient of discharge
c v d
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Application 2: Flow meters
From the Bernoulli’s
principle for the
streamline OA
1 2 1 2
P0 + ρv0 + ρgz0
= PA + ρvA + ρgzA
2 2
For steady state, and OA taken as horizontal and velocity at
A is zero from the definition of stagnation point
1 2 1 2
PA
= P0 + ρv0 or PA
− P0 = ρv0
2 2
Thus pressure at the stagnation point exceed the pressure
at any other point on the streamline OA exceeds by 1 2
2 ρv
Dynamic pressure
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Vortex Line
In the steady flow of homogeneous, incompressible,
non-viscous fluid the Eulers equation is given by
∂v
1 ⎛1
⎞
− v× ( ∇× v)
=− ∇p−∇ v
2
⎜ +Ω⎟
∂t
ρ ⎝2
⎠
Instead of taking a dot product with velocity vector as we
have taken earlier, we take dot product with vorticity vector
⎛ ⎞
χ χ φ
ρ
P 1
. v
2
. 0
2
∇ ⎜ + +Ω ⎟= ∇ =
⎝ ⎠
Therefore, the potential φ is constant along the vortex line
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P v
2
+
ρ 2
+ Ω =
C
May be satisfied by more than one stream line and more than one vortex line
Circular flow with straight streamline
Non-zero curl or circulation does not necessary mean
that the streamlines are circular or curved Ex.
v
∇ × v
=
v
( y)
ˆ
x
e x
⎡dvx
( y)
⎤
= −⎢
⎥eˆ
⎣ dy ⎦
z
≠ 0
Flow is rotational even though the stream lines are parallel straight lines
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