Page 28 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 13

| | x (oyu) + 5 (ou) dxdy =0

(a)

for any (0 ) of Oxy. Following the fundamental lemma (given by the

end of the next section) we could write

0 Mm 0 MH _
3, (PY u) + 57 (ey v) = 0,

a relation which is equivalent with the above continuity equation for the
plane or axially symmetric motions.
As the last relation expresses that py™ (udy — vdz) is an exact total
differential, there is afunction pow (x,y) ( po being a positive constant),
defined within an arbitrary additive constant, such that

py” (udy — vdx) = pody,

i1.e€., we can write

_ fo OY _ po OY

py™ Oy’ py™ Ox

and hence

k
va x grad wy.
py™

The function (2, y) is, by definition, the stream function of the con-
sidered steady (plane or axially symmetric) motion.
The above formulas show that the unknown functions u and v could be
replaced by the unique unknown function yw. The curves y% = const are
the streamlines in Oxy. Generally, (C) being an arc joining the points
A and B from the same plane, (27) po [W (B) — y (A)] represents the

mass flow rate through (Sc), the sense of n along (C) being determined
by the —% rotation of the (C) tangent (oriented from A to B).

1.3 Euler-Lagrange Criterion.
Euler’s and Reynolds’ (Transport) Theorems

Let us consider a material volume (closed system) D(t) whose surface
S(t) is formed of the same particles which move with the local continuum
velocity being thus a material surface. We intend to obtain a necessary
and sufficient condition, for an arbitrary boundary surface S(t) of equa-

tion f(r,t) =0, to be a material surface, ie., to be, during the motion,
a collection of the same continuum particles of fixed identity.