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Unformatted text preview: must have:
2
2
(x y) = x ; y
22
In this context, is called a velocity potential. The Divergence
~
The divergence of a vector V is de ned as the dot product of the gradient
~:
and the V
@i @j @^
~
^
r V = @x^ + @y ^ + @z k V1^ + V2^ + V3k
i
j
! 6 ~
The divergence arises in conservation laws. If V is the velocity eld then the
above equation is an expression of conservation of mass for incompressible
ow. The Vorticity
The vorticity is an important quantity in uid mechanics and it is de ned
by taking the cross product of the gradient and the velocity:
~
!
~ =r V
(1)
We will discuss this quantity the nature of this quantity in uid mechanics
later. PROBLEMS
~
~~
V1. For the given velocity eld, nd V dA where dA is given:
~
~
(a) V = ax2^ ; 2axy^ with dA = dA^
i
j
i
~
~
(b) V = ax2^ ; 2axy^ with dA = dAx^ + dAy^
i
j
i
j
~
~
ij
V2. For V = x22 ^ ; y22 ^, nd r V :
V3. For each of the following velocity elds, nd the vorticity: ~
(a) V = ax2^ ; 2axy^
i
j
~
(b) V = (2y ; y2)^
i 7...
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This note was uploaded on 02/17/2013 for the course AME 331 taught by Professor Zohar during the Spring '08 term at University of Arizona Tucson.
 Spring '08
 ZOHAR

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