Fluid Mechanics Terminologies

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18 Ιουλ 2012 (πριν από 5 χρόνια και 2 μήνες)

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Fluid Mechanics Terminologies
1.
Reynolds Number(Re)
U=velocity; L=tube width; ρ=density;µ=viscosity
2.
Laminar Vs Turbulent Flow

Laminar: viscous force dominate; Re<2100 in tube

Turbulent: inertia force dominate but competes
with viscous force; Re>2100 in tube

Inviscid: viscous force negligible; Re >> 2100
µ
ρ
UL
=
Re
Viscous force
Inertia force
Fluid Mechanics Terminologies
3.
Steady Flow

Velocity of fluid in space is:

= (x,y,z,t) at position (x,y,z) at time t

Steady flow:

The velocity field of fluid, , at each point
in space is constant in time.

Not the same as mean velocity!
u
u
u
0
=


t
u
Fluid Mechanics Terminologies
4.
Streamline

A streamline is a curve that describes
directions tangent to the velocity vector in
space in a steady flow at any time t.

In other words, it traces the path of fluid
elements in a steady velocity field

Velocity perpendicular to streamline is zero
u
Fluid Mechanics Terminologies
5.
Rotational Vs
Irrotational
Flow

Irrotational: Fluid elements experience ZERO
net angular momentum at any point.

Rotational: Fluid elements experience finite
angular momentum
6.
Compressible and Incompressible Flow

Incompressible:

Compressible: e.g. supersonic flow
0
=
×

u
0
=


u
Fluid Mechanics Terminologies

Shear stress

Stress is an int
ernal distribution of forces within a body that
balances and reacts to the loads applied to it.

Shear stress is the stress that a fluid element experiences when
it is
deformed by a force at constant volume.

The deformation of fluid element can be expressed as rate of
deformation or rate of shear:
A
F
Moving at
velocity u
z
stationary
dz
du
dz
du
A
F
µ
τ
τ
τ
=


=
dz
du
Fluid Mechanics Terminologies
7.
Coefficient of Viscosity

Shear (shear stress)
For a deformable solid:
F/S = G θ; where G = shear modulus
In a fluid:
In other words, viscosity is the resistance of a fluid from
shear (flow) and is a measure of adhesive/cohesive or
frictional properties
Deformation
F
SS
dy
du
µ
τ
=
Fluid Mechanics Terminology
8.
Kinematic Viscosity
9.
Newtonian Fluid
A fluid for which the coefficient of viscosity is constant when
the shear stress and the rate of shear vary.
e.g. Water -
Newtonian fluid
Polymer -
Non-Newtonian fluid (viscoelastic
fluid is one type
of non-Newtonian fluid, i.e. viscosity changes with the rate of
shear.)
ρ
µ
υ
=
Dynamic viscosity
(“stickiness”)
Inertia of fluid element
dy
du
µ
τ
=
Fluid Mechanics Terminology
10.
No Slip Condition

Net velocity is ZERO at solid wall boundary
U=0 at bounding walls.

Nature of solid wall does not matter

Boundary Layer
11.
Continuity

Conservation of fluid elements in a control volume

i.e. what comes in must also goes out!
0

dz
du
IN
OUT
control volume
Fluid Mechanics: Poiseuille Flow

Pressure-driven flow in tube/pipe

Force balance:

Pressure force:

Shear force=(shear stress
) (surface area of cylinder)
Equate pressure and shear forces: Fp=Fv
P1
P2
R
Flow
velocity
u=u(r)
2
2
1
)
(
r
P
P
PA
F
tube
p
π

=


=
)
2
(
)
2
(
rL
dr
du
rL
F
v
π
µ
π
τ
=
=
L
rdr
P
du
rL
dr
du
r
P
µ
π
µ
π
2
)
2
(
2


=

=



Fluid Mechanics:
Poiseuille
Flow
Integrate using no-slip boundary conditions,
u=0 at r=R:
dr
L
r
P
du
r
R
u




=
µ
2
0
(
)
2
2
4
r
R
L
P
u



=

µ
[]
2
2
2
4
2
2
r
R
L
P
r
L
P
u
r
R



=








=
µ
µ
Fluid Mechanics:
Poiseuille
Flow
Volumetric flow rate, Q
udA
udA
Adu
dQ
Au
V
dt
d
Q
=
+
=
=
=
0
rdr
dA
π
2
=
()
()







=



=

=
=
=
=
R
R
R
rdr
r
R
L
P
rdr
r
R
L
P
Q
rdr
u
dQ
Q
rdr
u
udA
dQ
0
2
2
0
2
2
0
2
4
2
4
2
2
π
µ
π
µ
π
π
R
r
R
r
L
P
Q
0
4
2
2
4
2
4









=

µ
L
P
R
Q
µ
π
8
4

=
Q is proportional to R4