fluid motion&continuity.ppt
TRANSCRIPT
1
Fluid in Motion
2
Kinematics of Fluid Mechanics
Fluid Motion
Flow Field
Stream lines
(Displacement, velocity, acceleration, ..etc) .
Flow
Non-viscous Flow Real Flow
(Ideal) (Real)
3
Steady and Unsteady Flow
Steady
Unsteady
A steady flow is one in which the conditions (velocity, pressure and cross-section) may differfrom point to point but DO NOT change with time
If at any point in the fluid, the conditions change with time, the flow is described as unsteady
Fluid Valve
4
Uniform and Non-uniform flow
Uniform flow: If the flow velocity is the same magnitude and direction at every point in the flow it is said to be uniform. That is, the flow conditions DO NOT change with position.
Non-uniform: If at a given instant, the velocity is not the same at every point the flow is non-uniform.
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Stream Line, Stream Tube and Pathline
vv
v
A stream line is a line that is everywhere tangent to the velocity vector at a given instant of time. A streamline is hence an instantaneous pattern.
6
A tubular surface formed by streamlines along which the fluid flows is known as a stream tube, which is a tube whose walls are streamlines. Since the velocity is tangent to a streamline, no fluid can cross the walls of a stream tube.
A pathline is the actual path traversed by a given (marked) fluid particle.
Stream tube
Pathline
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One, Two and Three Dimensional Flows
Stream line, being mathematically line
Stream lines of flow picture are essentially straight and parallel.
One dimensional Flow:
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Two and Three Dimensional Flows:
Stream lines of flow pictures describe a flow field.
Two dimensional in a single plane ( Flow over weir and around wings ).
Flow over weir Flow over and around wings
Two Dimensional Flow
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Two axisymmetric Three dimensional
( Stream line surfaces and stream tubes )
Three dimensional in space.
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Velocity and acceleration:
Streamline
S
0
dS
v
a s
a r
r
c
Fixed point
For one dimensional flow:
dt
dSv
S : is the displacement in time dt.
as : is the tangent component of acceln.
Where:
2
2
s dt
Sda
dt
dS
dt
d
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Sd
dvva s
dt
dS
dS
dv
dt
dva s
and
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For generalization: y
x
u
v V
)x,y(
- For Cartesian Coordinates:
dt
dxu
u = u(x,y) and v = v(x,y)
dt
dua x and
dt
dyv
dt
dva y and
and for steady state condition:
dyy
udx
x
udu
and then
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ydy
vdx
xd
v
v
y
vv
x
vua y
Dividing the above equation by dt, we obtain:
y
uv
x
uua x
Dividing the above equation by dt, we obtain:
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Equation of Continuity
Control volume
A control volume is a finite region, chosen carefully by the analyst for a particular problem, with open boundaries through which mass, momentum, and energy are allowed to cross
conservation of mass
For steady flowMass entering per unit time = Mass leaving per unit time
Mass entering per unit time = Mass leaving per unit time + Change of mass in the control volume per unit time
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Equation of Continuity:
1- For One dimension:A2A1
Flow
ds2ds1
dt
dsAρ
dt
dsAρ 2
221
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Conservation of mass
Dividing by dt, we obtain:
222111 VAρVAρ
V1
V2
ρ1A1ds1= ρ2A2ds2
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.ConstVAρVAρm 222111
Where: is the mass flow rate.m
= ρAV
0ρAVd
mFor constant ρ: A1V1 = A2V2= Q
0V
dV
A
dA
ρ
dρ
Where: Q is the volume flow rate.
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Equation of Continuity in General:
dx dy
dz
x
z
y
ρu dydzρu dydz + (ρu dydz )dx
xIn flow
Out flow
For x-direction
Net flow ρu dydz - [ρu dydz + (ρu) dxdydz]x
= - [(ρu) dxdydz ]x
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For y-direction
Net flow = - [(ρv) dxdydz ]x
For z-direction
Net flow = - [ (ρw) dxdydz ]x
The sum of net flow
- [ (ρu) + (ρv) + (ρw ) ] dxdydzx
x
x
)1(
The rate of change
dzdydxtt
Volume.
t
m
)2(
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The positive rate of change = The sum of the net flow ie; Equation (1) = equation (2)
0z
ρw
y
ρv
x
ρu
t
ρ
0t
ρ
For three dimensional incompressible steady state
0z
w
y
v
x
u
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For two dimensional steady state
0y
v
x
u
For one dimensional steady state
0x
u
This represent the uniform flow, V = Constant and does not change
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Flow of an Incompressible Ideal Fluid
1- ρ = Constant.
2- Viscous forces = 0 ( In-viscous Flow ).
3- Frictionless.
4- Can apply Newton’s law ( F = mass x acceleration)
5- External forces = Effective forces.
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Example-1
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Example-2
If pipe 1 diameter = 50mm, mean velocity 2m/s, pipe 2 diameter 40mm takes 30% of total discharge and pipe 3 diameter 60mm. What are the values of discharge and mean velocity in each pipe?