Gauss Theorem

1
Gauss Theorem

• Using the notation of the figure below, Gauss
  theorem is stated as
• For an arbitrary continuous fixed volume V and
  associated closed surface area dA with a unit
  vector pointing outward normal to the
  surface, the following equation holds

2
Mass conservation

• Conservation of mass is the concept that matter
  can neither be created nor destroyed in a system.
  In application to an arbitrary fixed volume this
  means that any changes in mass over time must be
  due to the mass flow or flux into or out of the
  volume. For a fluid of density, r, with mass,
  this concept is quantitatively
  expressed as
• For density field, r(x,y,z,t), the fact that the
  volume integral is over the entire spatial domain
  leads to the equality

3
Mass conservation

• Apply Gauss theorem on the flux term
• This leads to the following integral form of
  conservation of mass
• For an arbitrary volume in space, the only way
  for the above integral to equal 0 is for the
  integrand to equal 0 so we obtain the result

4
Mass conservation

• Use of the material derivative of the density
  field,
• conservation of mass can also be expressed in
  terms of the fractional rate of change of density
  as
• This form shows us that fractional rate of change
  of the density (or volume) of a fluid element
  fluid is related to the divergence of the flow
  field

5
Incompressibility

• If the density field is constant or uniform over
  the fluid domain then the conservation of mass
  expression reduces to the simple form
• The question arises when is it appropriate to use
  this simple form of mass conservation for steady
  flows?
• If flows were unsteady and density was assumed
  to be strictly constant in the fluid medium then
  propagating waves would have infinite energy (See
  Waves and Tides). One additional requirement is
  that we dont consider vertical lengths scales so
  large that hydrostatic pressure causes large
  density variations.

6
Example - 1

• Find a value of a to make the following flow
  field incompressible?

7
Incompressibility Conditions of applicability

• One easy way to determine the conditions under
  which the incompressible equation holds is to
  perform dimensional analysis. For a flow field
  with characteristic density, r, speed, U, and
  length scale, L, we wish for the fractional rate
  of change of density to be much less than the
  characteristic dimensions of the divergence
  field.

8
Incompressibility Conditions of applicability

• For simplicity of the analysis, assume a
  Barotropic fluid so
• Also we will use the following well defined
  thermodynamic relation between the variation of
  density with respect to pressure and the sound
  speed
• Then by use of the chain rule on the fractional
  rate of density we obtain

9
Incompressibility Conditions of applicability

• Dimensionally, pressure is the force per unit
  area which we can relate to the characteristic
  density and velocity
• And we can then obtain the dimensional parameters
  of the material derivative of the pressure field
  as

10
Incompressibility Conditions of applicability

• Returning to the original inequality of interest
• So we can approximate a fluid as incompressible
  provided the square of the Mach number, M, is
  small.