Numerical Simulation of Three-Dimensional Viscous Flows with ...

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ISMAIL-ZADE et al.
obtain a zero row, because
πl qlmn
----a
l 1ijk
1
πm qlmn πn qlmn
+ ------- a 2ijk + ----- a 3ijk = 0, q = 123, , , ijklmn , , , , , ∈ N 0 .
l 2
As a result, we set to zero the last set of rows in the matrix of system (28). The last set of elements in the
column of absolute terms will remain zero as well, because
πl 1
----ρ
l flmn , ,
1
This means that the vectors ψ (1) , ψ (2) , and ψ (3) satisfy the system of linear equations
Moreover, the linear dependence of rows in the matrix of system (28) implies that systems (28) and (29)
are equivalent. These systems are solvable since the vector field y satisfies the variational equation.
The columns of the matrices of systems (28) and (29) are also linearly dependent, because
πi 1lmn
----a
l pijk
1
Therefore, the solvability of (29) entails the solvability of the system
l 3
( ) πm
( )
l 2
( 2) πn
l 3
+ ------- ρ flmn ( , , ) + ----- 0 = 0, lmn , , ∈ N 0 .
⎛ A 11 A 12 A ⎞⎛
13 ψ ( 1)
⎞
⎜
⎟⎜
⎟
⎜ A 21 A 22 A
⎜
23
⎟⎜
ψ ( 2)
⎟ =
⎟⎜
⎟
⎝ 0 0 0 ⎠⎝
⎠
ψ ( 3)
⎛
⎜
⎜
⎜
⎝
ρ ( 1)
ρ ( 2)
πj 2lmn πk 3lmn
+ ----a pijk + -----a pijk = 0, p = 123, , , ijklmn , , , , , ∈ N 0 .
l 2
l 3
0
⎞
⎟
⎟.
⎟
⎠
(29)
⎛ A 11 A 12
⎞
⎛ ( 1)
⎞
⎜ ⎟
⎜ ψ *
⎟
⎝ A 21 A 22 ⎠
⎜ ( 2)
⎟
⎝ ψ * ⎠
=
⎛
⎜
⎜
⎝
ρ ( 1)
ρ ( 2)
⎞
⎟,
⎟
⎠
(30)
( 1)
( 2)
and its solution is given by the vectors ψ and ψ whose components are
* *
( 1)
( )
ψ *
fi, jk ,
( 1) il
ψ 3
fi ( , jk , )
( 3)
( 2)
= – ------ ψ
kl fi ( , jk , ) , ψ = ψ
1
*
fi ( , jk , ) fi, jk ,
i, j,
k ∈ N 0 , k ≠ 0,
( 2) jl 3
( ) –
( 3)
------ ψ
kl fi ( , jk , ) ,
2
( 1)
( )
ψ *
fi, j0 ,
( 2)
= 0, ψ = *
fi ( , j0 , )
0, i,
j ∈ N 0 .
( 1)
( 2)
The reverse proposition is also valid: if vectors ψ and ψ constitute a solution to system (30), then the
* *
vectors ψ (1) ( 1)
= , ψ (2) ( 2)
ψ = ψ , and ψ (3) = 0 constitute a solution to both (28) and (29).
* *
The analysis above implies that one may set ψ (3) = 0 when considering systems (28) and (29). Therefore,
the velocity field u can be represented as in (18), where the potential y = (ψ 1 , ψ 2 , 0) satisfies the more
restrictive boundary conditions corresponding to impermeability with perfect slip.
Now, we consider the case of no-slip conditions. The corresponding representation (18) subject to the
more restrictive no-slip conditions is relatively simple to find when viscosity has the form µ = µ(t, x 3 ) or
µ = µ(t, x 1 , x 2 ). This assertion can be validated by an analysis that does not rely on the scheme developed
above for the impermeability conditions with perfect slip. However, the desired representation (18) has been
proved in the case when viscosity has the general form µ = µ(t, x) in [28] with the use of Chandrasekhar’s
functions [16].
Remark. Note that the problem formulated for a horizontally uniform viscosity µ = µ(t, x 3 ) can be substantially
simplified by replacing a three-component potential y = (ψ 1 , ψ 2 , ψ 3 ) or a two-component potential
y = (ψ 1 , ψ 2 , 0) with the single-component potential
y = curl( ϕe 3 ) = ( ∂ϕ/∂x 2 , – ∂ϕ/∂x 1 , 0),
(31)
where ϕ = ϕ(t, x) is a scalar function satisfying appropriate boundary conditions. This can be done because
the velocity field u can be represented as u = curl curl(ϕe 3 ) (note that this representation is not unique).
COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 41 No. 9 2001