Page 241 - Intro to Tensor Calculus

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entering the region if ~v · ˆn is negative and the change of mass leaving the region if ~v · ˆn is positive, as ˆn is
always an exterior unit normal vector. Equating the relations from equations (2.3.69) and (2.3.70) we obtain
a mathematical statement for mass conservation

ZZZ ZZ
∂m ∂%
= dτ = − %~v · ~ndσ. (2.3.71)
∂t R ∂t S
The equation (2.3.71) implies that the rate at which the mass contained in R increases must equal the rate
at which the mass flows into R through the surface S. The negative sign changes the direction of the exterior
normal so that we consider flow of material into the region. Employing the Gauss divergence theorem, the
surface integral in equation (2.3.71) can be replaced by a volume integral and the law of conservation of
mass is then expressible in the form

ZZZ
∂%
+ div (%~v) dτ =0. (2.3.72)
∂t
R
Since the region R is an arbitrary volume we conclude that the term inside the brackets must equal zero.
This gives us the continuity equation
∂%
+ div (%~v)= 0 (2.3.73)
∂t
which represents the mass conservation law in terms of velocity components. This is the Eulerian represen-
tation of continuity of mass flow.
Equivalent forms of the continuity equation are:
∂%
+ ~v · grad % + % div ~v =0
∂t
∂% ∂% ∂v i
+ v i i + % i =0
∂t ∂x ∂x
D% ∂v i
+ % =0
Dt ∂x i
D% ∂% ∂% dx i ∂% ∂%
where = + = + v i is called the material derivative of the density %. Note that the
i
Dt ∂t ∂x dt ∂t ∂x i
∂%
material derivative contains the expression ∂x i v i which is known as the convective or advection term. If the
density % = %(x, y, z, t) is a constant we have
D% ∂% ∂% dx ∂% dy ∂% dz ∂% ∂% dx i
= + + + = + =0 (2.3.74)
i
Dt ∂t ∂x dt ∂y dt ∂z dt ∂t ∂x dt
and hence the continuity equation reduces to div (~v)= 0. Thus, if div (~v) is zero, then the material is
incompressible.
EXAMPLE 2.3-2. (Continuity Equation) Find the Lagrangian representation of mass conservation.
Solution: Let (X, Y, Z) denote the initial position of a fluid particle and denote the density of the fluid by
%(X, Y, Z, t)sothat %(X, Y, Z, 0) denotes the density at the time t =0. Consider a simple closed region in
our continuum and denote this region by R(0) at time t =0 and by R(t)atsome later time t. That is, all
the points in R(0) move in a one-to-one fashion to points in R(t). Initially the mass of material in R(0) is
ZZZ
m(0) = %(X, Y, Z, 0) dτ(0) where dτ(0) = dXdY dZ is an element of volume in R(0). We have after a
R(0)