Gauss Theorem

1
Gauss Theorem

• Using the notation of the figure below, recall
  that Gauss theorem is stated as follows
• 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

4
Mass conservation

• There are two ways that the above integral is
  equal to 0.
• I) The first is that the boundary of the surface
• contains a unique symmetry such that the above
• integral is zero. As an example of this,
  consider
• the following 1-D integral
• This possibility is too restrictive as we wish
  our results to be
• applicable to any arbitrary shape.

5
Mass conservation

• II) The other possibility is that the integrand
  itself is equal
• zero throughout the integral domain. This leads
  to the result
• Although this possibility might seem trivial this
  is the exact
• result we are looking for.

6
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

7
Incompressibility

• If variations in density are small compared to a
  background density 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). Also, one additional
  requirement is that we dont consider vertical
  lengths scales so large that hydrostatic pressure
  causes large density variations.

8
Example - 1

• Find a value of a(x,y) to make the following flow
  field incompressible?

9
Example -2

• A well mixed ocean layer extends from the surface
  to 80m deep with a measured average horizontal
  divergence at the surface of
• The time averaged vertical flow at the surface is
  zero
• Find the vertical flow at a depth of 60m (z-60m)
• You can assume the flow is incompressible.
• Draw a simple diagram depicting your result.

10
Incompressibility Conditions of applicability

• One easy way to determine the conditions under
  which the incompressible equation holds is to
  perform dimensional scaling 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.
• Notice we are examining characteristic scales
  and not actual derivatives in the above
  expression

11
Incompressibility Conditions of applicability

• It is shown without proof that the restriction
• Leads to the result that
• So we can approximate a fluid as incompressible
  provided the square of the Mach number, M, is
  small.
• Details of the above limit will be provided in SO
  414