7.7 Conservation of Mass: The Continuity Equation
Now lets derive the equation for the conservation of mass in a fluid. We begin by writing down the flow of mass into and out of a small box (Figure 7.5).
The mass flux into the volume must be (mass flow out) - (mass flow in)
The third term inside the parentheses becomes much smaller than the first two terms as δx approaches 0; and
In three dimensions:
The mass flux must be balanced by a change of mass inside the volume, which is:
and conservation of mass requires:
This is the continuity equation for compressible flow, first derived by Leonhard Euler (1707-1783).
The equation can be put in an alternate form by expanding the derivatives of products and rearranging terms to obtain:
The first four terms constitute the total derivative of density Dρ/Dt from (7.7), and we can write (7.17) as:
This is the alternate form for the continuity equation for a compressible fluid.
The Boussinesq Approximation
Boussinesq's assumption requires that:
The approximations are true for oceanic flows, and they ensure that oceanic flows are incompressible. See Kundu (1990: 79 and 112), Gill (1982: 85), Batchelor (1967: 167), or other texts on fluid dynamics for a more complete description of the approximation.
where V is volume, and p is pressure. For incompressible flows, β = 0, and:
because dp/dt does not equal 0. Remembering that density is mass m per unit volume V, and that mass is constant:
If the flow is incompressible, (7.18) becomes:
This is the Continuity Equation for Incompressible Flows.
|Department of Oceanography, Texas A&M University
Robert H. Stewart, email@example.com
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Updated on September 24, 2008