Towards the Continuity Equation
Fluid Dynamics
Dr Oliver Kerr
2010–11
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
The average density in a volume is
density = ρ =
mass
volume
For a fluid we can also define a local density
ρ(x, t)
Density can depend on position and time.
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
How to change the mass inside a volume
Consider a volume V fixed in space...
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
There are two ways you can change the mass of fluid inside this
volume:
I
Create/destroy mass
I
Move mass across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can be created
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Problem
How do you create or destroy mass?
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Problem
How do you create or destroy mass?
(may be a useful model)
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass can move across the boundary
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass of fluid, M, inside a boundary, V , is given by
Z Z Z
ρ dV
M=
V
where ρ(x, t) is the density (mass per unit volume).
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Flow across an area element
n
u
dA
volume flow across element = u · n dA
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
θ
θ
Width of stream crossing line is proportional to cos θ.
Flow across line is also proportional to the speed of flow, |u|, and
the width of the line, L.
So flow proportional to |u|L cos θ = u · n L where n is normal to the
line.
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
Density
How to change the mass inside a volume
Flow across an area element
Mass flow across an area element
n
u
dA
mass flow across element = ρ × (volume flow) = ρ u · n dA
Dr Oliver Kerr
Fluid Dynamics
Towards the Continuity Equation
dM
d
=
dt
dt
Density
How to change the mass inside a volume
Flow across an area element
Z Z Z
Z Z
ρ dV = −
V
ρ u · n dA
S
where S is the surface of the closed volume V . The minus sign is
there because the flow out of the volume decreases the mass inside.
Dr Oliver Kerr
Fluid Dynamics