Fluids - Northern Illinois University
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Transcript Fluids - Northern Illinois University
Fluids
Eulerian View
r (r0 , t )
In a Lagrangian view each
body is described at each
point in space.
• Difficult for a fluid with many
particles.
(r , t )
v (r , t )
In an Eulerian view the
points in space are
described.
• Bulk properties of density
and velocity
Compressibility
A change in pressure on a
fluid can cause deformation.
Compressibility measures
the relationship between
volume change and
pressure.
V
p
V
p
V
B
Vp 1
V
• Usually expressed as a bulk
modulus B
Ideal liquids are
incompressible.
Fluid Change
A change in a property like
pressure depends on the
view.
In a Lagrangian view the
total time derivative depends
on position and time.
An Eulerian view is just the
partial derivative with time.
• Points are fixed
dp p p dx p dy p dz
dt t x dt y dt z dt
dp p
v p
dt t
d
F v2k l x l
dt t
2
dp
p
dt r const t
x
2
l x2
2
Volume Change
Consider a fixed amount of
fluid in a volume dV.
• Cubic, Cartesian geometry
• Dimensions dx, dy, dz.
The change in dV is related
to the divergence.
• Incompressible fluids must
have no velocity divergence
v
d
dx x dx
dt
x
v y
d
dy
dy
dt
y
d
v
dz z dz
dt
z
vx v y vz
d
dxdydz
dV
dt
x y z
d
dV v dV
dt
Continuity Equation
A mass element must remain
constant in time.
• Conservation of mass
Combine with divergence
relationship.
Write in terms of a point in
space.
( v ) 0
t
dm dV
d
d
dm d V 0
dt
dt
d
ddV
dV
dt
dt
d
dV v dV 0
dt
d
v 0
dt
v v 0
t
Stress
F
A
F
A
A stress measures the
surface force per unit area.
• A normal stress acts normal
to a surface.
• A shear stress acts parallel
to a surface.
A fluid at rest cannot support
a shear stress.
Force in Fluids
P(dS1 )
dS1
P(dS2 )
• Describe the stress P at any
point.
• Normal area vectors S form
a triangle.
dS 2
dS1 dS2
P(dS1 dS2 )
Consider a small prism of
fluid in a continuous fluid.
The stress function is linear.
P(cdS ) cP(dS )
P(dS ) P(dS )
P(dS1 ) P(dS2 ) P(dS1 dS2 )
Stress Tensor
Represent the stress
function by a tensor.
P(dS ) P dS
• Symmetric
• Specified by 6 components
If the only stress is pressure
the tensor is diagonal.
The total force is found by
integration.
P(dS ) P dS p1 dS
F P dS
S
Force Density
F P nˆ dS
S
F PdV
The force on a closed
volume can be found
through Gauss’ law.
• Use outward unit vectors
V
f S P
A force density due to stress
can be defined from the
tensor.
• Due to differences in stress
as a function of position
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