Conservation of Mass Continuity Equation Spatial Form

12 If we consider an arbitrary volume V, fixed in space, and bounded by a surface S. If a continuous medium of density p fills the volume at time t, then the total mass in V is

where p(x, t) is a continuous function called the mass density. We note that this spatial form in terms of x is most common in fluid mechanics.

13 The rate of increase of the total mass in the volume is dM r dp

14 The Law of conservation of mass requires that the mass of a specific portion of the continuum remains constant. Hence, if no mass is created or destroyed inside V, then the preceding equation must eqaul the inflow of mass (of flux) through the surface. The outflow is equal to v-n, thus the inflow will be equal to -v-n.

i (-pvn)dS = - [ pv^ndS = - [ V-(pv)dV J s J s J v must be equal to QM - Thus v dp + v-(pv)

since the integral must hold for any arbitrary choice of dV, then we obtain dP + -(-)« | + ^ = 0

15 The chain rule will in turn give d(pvi) = pdvi + v dp

16 It can be shown that the rate of change of the density in the neighborhood of a particle instantaneously at x by

dp dp ^ ^ dp ^ dp dt dt dt dxi where the first term gives the local rate of change of the density in the neighborhood of the place of x, while the second term gives the convective rate of change of the density in the neighborhood of a s s particle as it moves to a place having a different density. The first term vanishes in a steady flow, while the second term vanishes in a uniform flow.

17 Upon substitution in the last three equations, we obtain the continuity equation dp dvi dp _

dt dxi dt

The vector form is independent of any choice of coordinates. This equation shows that the divergence of the velocity vector field equals (-1/p)(dp/dt) and measures the rate of flow of material away from the particle and is equal to the unit rate of decrease of density p in the neighborhood of the particle.

is If the material is incompressible, so that the density in the neighborhood of each material particle remains constant as it moves, then the continuity equation takes the simpler form dvi dxi

this is the condition of incompressibility

6.2.2 Material Form

19 If material coordinates X are used, the conservation of mass, and using Eq. 4.41 (dV = \J\dV0), implies

and for an arbitrary volume dV0, the integrand must vanish. If we also suppose that the initial density p0 is everywhere positive in Vq (no empty spaces), and at time t = t0, J = 1, then we can write pJ — po or

which is the continuity equation due to Euler, or the Lagrangian differential form of the continuity equation.

20 We note that this is the same equation as Eq. 6.16 which was expressed in spatial form. Those two equations can be derived one from the other.

21 The more commonly used form if the continuity equation is Eq. 6.16.

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