Page 26 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 11

Obviously, of great interest is how the circulation along a material

closed simple contour changes while the contour moves with the contin-
B
uum. To analyze this aspect let us evaluate # J v -dr, i.e., the rate

of change (in time) of the circulation about a material contour joining

the points A and B as it moves with the medium. Considering then
r=r(s,t),for 0<s <J, we have

B I l
D D dr D dr
afve-gl. Ge-[E( Ze
A 0 0

? Dv i 1
=f pees [vd v= facies So),
A A

where v = |v|. If A and B coincide so as to form a simple closed curve
(C) in motion, obviously # ¢v-dr = fa-dr,i., the rate of change of
Cc C
circulation of velocity is equal to the circulation of acceleration along the

same closed contour (C). If the acceleration comes from a potential, 1.e.,
a= grad U, then the circulation of the velocity along the closed contour
does not change as the curve moves, the respective motion being called
circulation preserving.
For the fluids, under some additional hypotheses a very important
result connected with the circulation conservation will be given later on

(the Thompson Kelvin theorem).

1.2.5 Stream Function for Plane and Axially Symmetric
(Revolution) Motions

By extending the already given kinematic definition to the dynamics
case, a motion is supposed to be steady (permanent) if all the (kinematic,
kinetic, dynamic) parameters characterizing the medium state and ex-
pressed with Euler variables x1, Z2, £3, ¢, are not (explicitly) dependent
on f.
All the partial time derivatives of the mentioned parameters being zero
=
(2 = =0), we have (from the continuity equation) that div (pv) 0, i.e.,
the vector field pv is conservative (solenoidal).

The above equation allows us to decrease the number of the unknown
functions to be determined; we will show that in the particular, but ex-