ABSTRACT
Let us assume that liquid is viscous, incompressible, and its flow is steady state. Using the equation of continuity
∂ ∂
∂ ∂
u
x
v
y + = 0 (3.1)
and the vorticity definition
ω
∂ ∂
∂ ∂
∂ ∂
∂ ∂
∂ ∂
= ∇× = = − ⎛
⎝
⎜
⎞
⎠
⎟
v
i j k
x y z u v
k v x
u
y 0
(3.2)
that conform to the boundary conditions, integral approximation [59] of the same can be obtained:
v x r x r
r x r dR x
v x
( )ξ pi
ω ξ ξ= −
× −( ) −
−
⎡
⎣
⎢
⎢
∫∫12 ( ) ( ) ( )( ) ( ) ( ) (2 ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )⋅ − ×( )×{ } −( )− ⎤∫
n x v x n x r x r
r x r dB x
ξ ξ 2
⎦
⎥
⎥
,
(3.3)
where R is an area occupied by liquid B is the flow area boundary x point is a variable of integration ξ is the point at which v = u v( ) ( ), ( )ξ ξ ξ{ } velocities are determined n x( ) is the outward normal ω( )x is x point vorticity vector r x r( ) ( ), ξ are radius vectors of points x and ξ
The vorticity can be determined from Navier-Stokes equation expressed in terms of vorticity:
∇× = × − ∇ω
ν ω
1 0( ),v h (3.4)
where
h p v
2 = +
ρ
is the total head
p is the relative static pressure ρmed is the medium densityv is the velocity modulus ν is the kinematic viscosity coefficient
An integral countertype of this equation is given by
ω ξ pi ν ω ξ
ξ( ) 1
2 ( ) ( ) ( ) ( )
( ) ( ) = −
×⎡ ⎣
⎤
⎦
× −( ) −
∫∫1 2v x x r x rr x r dR R
⎡
⎣
⎢
⎢
−
⋅ × + ⋅ − × ×{ } −h x n x x n x x n x r x r0( ) ( ) ( ) ( ) [ ( ) ( )] ( ) (/ν ω ω ξ ξ
) ( ) ( )
.
