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 
NonNewtonian fluid (viscoelastic
fluid is one type
of nonNewtonian 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
•
Pressuredriven 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 noslip 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
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