Page 60 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 45

be a smooth function vanishing in the closed outside of a domain S, the

curve & dividing the domain S into the disjoint subdomains S$; and S,
(S=S,US2 ). Then

0= [ sraae -Fdzdt = | sraas -Fdadt + [ grace - Fdzdt.

Ss Si So

Since u is a regular function in both S$; and So, if n is the unit nor-
mal vector oriented from 5S; to S2, then by applying Gauss’ divergence
formula and the validity of the relation divF = Oin 5S; andin S2 we are
led to

[1 -F:) nds = 0,

a
where F; and F2 are the F values for u taking the limiting values from

Si, respectively So.
As the above relation takes place for any ® , we will have [F-n] =O
on % where [F -n] = Fy -n — F2-n denotes the “jump” of F -n across
yu.
Suppose that 4 is given by the parametric equation x = x(t), so that
dz 1,—d
the displacement velocity of discontinuity is d = Further n =
dt Vite
and F being (f(u),u) , the above relation becomes

-dlul+[f(u]=0 pe ¥,

where again [] designates the jump of the quantity which is inside the
parentheses, when the point (z,t) is passing across © (from Sj; to S2).
A function wu satisfying the differential equation whenever it is possi-
ble (in our case in S; and S2) and the above jump relation across the
discontinuity surface & , will satisfy both the integral and the weak form

of the equation.
Obviously, all the above comments could be extended to the conserva-
tive laws systems. Let us consider, as a conservative system, the system
of equations for an isentropic gas in a one-dimensional flow, precisely

=
pt +m, 0,

m?
Mt + Ga = 0,
p t