Page 242 - Intro to Tensor Calculus

236

ZZZ
time t has elapsed the mass of material in the region R(t)given by m(t)= %(X, Y, Z, t) dτ(t)where
R(t)

dτ(t)= dxdydz is a deformed element of volume related to the dτ(0) by dτ(t)= J x,y,z dτ(0) where J is
X,Y,Z
the Jacobian of the Eulerian (x, y, z) variables with respect to the Lagrangian (X, Y, Z) representation. For
mass conservation we require that m(t)= m(0) for all t. This implies that

%(X, Y, Z, t)J = %(X, Y, Z, 0) (2.3.75)

for all time, since the initial region R(0) is arbitrary. The right hand side of equation (2.3.75) is independent
of time and so
d
(%(X, Y, Z, t)J)= 0. (2.3.76)
dt
This is the Lagrangian form of the continuity equation which expresses mass conservation. Using the result
dJ
~
that = Jdiv V, (see problem 28, Exercise 2.3), the equation (2.3.76) can be expanded and written in the
dt
form
D%
~
+ % div V =0 (2.3.77)
Dt
where D% is from equation (2.3.74). The form of the continuity equation (2.3.77) is one of the Eulerian forms
Dt
previously developed.

In the Eulerian coordinates the continuity equation is written ∂% + div (%~v) = 0, while in the Lagrangian
∂t
d(%J)
system the continuity equation is written =0. Note that the velocity carries the Lagrangian axes and
dt
the density change grad%. This is reflective of the advection term ~v · grad %. Thus, in order for mass to
be conserved it need not remain stationary. The mass can flow and the density can change. The material
derivative is a transport rule depicting the relation between the Eulerian and Lagrangian viewpoints.
In general, from a Lagrangian viewpoint, any quantity Q(x, y, z, t) which is a function of both position
and time is seen as being transported by the fluid velocity (v 1 ,v 2 ,v 3 )to Q(x+v 1 dt, y +v 2 dt, z +v 3 dt, t+dt).
∂Q
Then the time derivative of Q contains both and the advection term ~v ·∇Q. In terms of mass flow, the
∂t
Eulerian viewpoint sees flow into and out of a fixed volume in space, as depicted by the equation (2.3.71),
In contrast, the Lagrangian viewpoint sees the same volume moving with the fluid and consequently
ZZZ
D
ρdτ =0,
Dt R(t)
where R(t) represents the volume moving with the fluid. Both viewpoints produce the same continuity
equation reflecting the conservation of mass.

Summary of Basic Equations

Let us summarize the basic equations which are valid for all types of a continuum. We have derived:
• Conservation of mass (continuity equation)

∂% i
+(%v ) ,i =0
∂t