What is the Mass Balance Equation Density?

The mass balance equation, expressed in terms of density, is ˙ρ + ρ div v = 0, where ˙ρ represents the material time derivative of the density in the spatial frame, and is defined as ˙ρ = ∂ρ/∂t + grad ρ ⋅ v.

This equation is a statement of mass conservation in fluid dynamics and continuum mechanics. It essentially says that the rate of change of density following a fluid particle is related to the divergence of the velocity field. Let's break down the components:

• ρ (rho): Density, which represents the mass per unit volume. It can vary with both time and position (ρ(x, t)).
• v: Velocity vector field of the fluid or continuum.
• ˙ρ: Material derivative (also known as substantial derivative or total derivative) of the density. It represents the rate of change of density as observed by an observer moving with the fluid.
• ∂ρ/∂t: Local time derivative of density, representing the rate of change of density at a fixed point in space.
• grad ρ ⋅ v: Advective term, representing the change in density due to the movement of the fluid carrying different densities. grad ρ represents the gradient of the density, and the dot product with v gives the rate at which the density changes along the direction of the velocity.
• div v: Divergence of the velocity field. This represents the rate at which volume is expanding (or contracting) at a point.

The equation ˙ρ + ρ div v = 0 can be rewritten as:

∂ρ/∂t + grad ρ ⋅ v + ρ div v = 0

This equation states that the local rate of change of density (∂ρ/∂t) plus the advective term (grad ρ ⋅ v) equals the negative of the density times the divergence of the velocity field (-ρ div v). In other words, changes in density at a point are due to the flow of density into or out of that region (advection) and the expansion or compression of the fluid.

For an incompressible fluid, the density ρ is constant, so ˙ρ = 0 and ∂ρ/∂t = 0. In this case, the mass balance equation simplifies to:

div v = 0

This means that for an incompressible fluid, the divergence of the velocity field is zero, implying that the volume of the fluid is conserved.

In summary, the mass balance equation in terms of density is a fundamental equation expressing the principle of mass conservation for fluids and other continuous materials. It relates the rate of change of density to the velocity field and its divergence.