Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science
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368 K. Gürlebeck and W. Sprößig
Fluids can be for a fixed time structure viscous (=convex curve in the shear-rate–
shear-stress system (sss)),
Newtonian (= straight line in (sss)), or dilatant (concave
curve in (sss)). Structure viscous means shear-thinning. For instance, solid color
becomes liquid while stiring. Dilatant fluids mean shear-thickening. An example
would be walking in wet sands at the beach. The depth of sinking depends on the
velocity inverse proportionally.
In the case of a constant shear-rate we have a convex curve (increasing viscosity
in the time). Such fluids are called
rheopex. Usually, for a fluid which has a decreas-
ing viscosity in the time, one uses the notation
thixotrop. Thixotrop fluids correspond
with structure viscous fluids.
5 Rothe’s Method of Semi-Discretization
5.1 Time-Dependent Stokes Problem
The time-dependent Stokes’ problem reads as follows:
1
μ
∂
t
u −Δu +
1
μ
∇p =
1
μ
f in (0,T)×G, (5.1)
div u = 0in(0,T)×G, (5.2)
u(0, ·) = u
0
in {0}×G, (5.3)
u = g on (0,T)×Γ. (5.4)
Let now be
T =nτ, T > 0,τ meshwidth, and u
k
:=u(kτ, ·), p
k
:=p(kτ,·). (5.5)
We approximate the partial derivative with respect to the time t by forward differ-
ences:
∂
t
u ∼
u
k+1
−u
k
τ
. (5.6)
5.1.1 Semi-Discretization
We obtain from (5.1) the following discretized equation:
ρ
μτ
u
k+1
+DDu
k+1
+
1
μ
Dp
k+1
=
f
μ
+
ρ
μτ
u
k
(k =0, 1,...,n−1). (5.7)
A suitable factorization leads to
(D +a)(D −a)u
k+1
+1/μ D p
k+1
=f/μ +a
2
0
u
k
(k =0, 1,...,n−1).
(5.8)
Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 369
This equation permits the operator integral representation
u
k+1
=−
1
μ
T
−a
Q
a
T
a
Dp
k+1
+T
−a
Q
a
T
a
f
μ
+a
2
0
u
k
+H
k
. (5.9)
H
k
is explicitly described. We have
H
k
:= T
−a
F
Γ,a
(tr
Γ
V
a
)
−1
Q
Γ,−a
g +F
Γ,−a
g, (5.10)
u
k+1
=−
1
μ
T
−a
Q
a
˜p
k+1
+T
−a
Q
a
T
f
μ
+a
2
0
u
k
+H
k
, (5.11)
˜p
k+1
= p
k+1
−aT
a
p
k+1
. (5.12)
Moreover,
(D
a
˜p
k+1
=Dp
k+1
!), (5.13)
which yields the well-known form.
5.2 Oseen’s Equation
f
μ
=1/ν∂
t
u −Δu +1/ν(v ·∇u) +1/μ∇p
in (0,T]×G
(Oseen equation), (5.14)
div u =0
(Continuity condition), (5.15)
u =g
(Boundary condition), (5.16)
u(0, ·) =u
0
(Initial value condition), (5.17)
where v is a dominant flow vector (suitably chosen), ν is the
kinematic viscosity, μ
is the
dynamic viscosity, and n is the unit vector of the outer normal.
5.3 A Special Discretization Method
Let T>0,T= nτ with meshwidth τ . We abbreviate u
k
:= u(kτ, ·), p
k
:=
p(kτ,·), f
k
=1/τ
(k+1)τ
kτ
f(t,x)dt. Using the forward differences
(∂
t
u)(kτ, ·) ∼
u
k+1
−u
k
τ
(k =0, 1,...,n−1), (5.18)
370 K. Gürlebeck and W. Sprößig
we get
u
k+1
−u
k
=−τ(v ·D)u
k
−τDp
k+1
+τνΔu
k+1
+
τf
k
ρ
(∗). (5.19)
Using the decomposition idea of Kalthoff/Schwarzer/Herrmann (Stuttgart), we in-
troduce the function u
∗
by
u
k+1
−u
∗
−
u
k
−u
∗
=−τ(v ·D)u
k
−τDp
k+1
+
τ
ρ
f
k
+Δu
k+1
(ντ ). (5.20)
It follows that
u
k+1
−u
∗
=−τDp
k+1
+ντΔu
k+1
;
u
k
−u
∗
=−τ(v ·D)u
k
−
τ
ρ
f
k
.
(5.21)
We have now to consider the problem
div u
k+1
=divu
∗
−τΔp
k+1
+ντΔdivu
k+1
=0 (5.22)
with u
∗
=u
k
+τ(v ·D)u
k
+
τ
ρ
f
k
.
It remains to deal with the following equations:
Δ(−ν divu
k+1
+p
k+1
) =+
div u
∗
τ
, (5.23)
Δη =+
div u
∗
τ
. (5.24)
Borel–Pompeiu’s formula yields
Dp
k+1
=T
div u
∗
τ
+φ
k+1
,φ
k+1
∈ker D. (5.25)
Inserting this into (∗), we can conclude
u
k+1
−ντΔu
k+1
=−τ(v ·D)u
k
+T div u
∗
+τφ
k+1
+
τ
ρ
f
k
+u
k
, (5.26)
and with a
2
0
=
1
ντ
, we arrive at
a
2
0
−Δ
u
k+1
=−
1
ν
(v ·D)u
k
+a
2
0
u
k
+
f
k
μ
+a
2
0
T div u
∗
+
φ
k+1
ν
. (5.27)
Remark Each
˜
φ
k+1
:=
φ
k+1
ν
defines another approximation!
Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 371
With S :=a
2
0
(−
1
ν
(v ·D) +I) and R
k
:=
f
k
μ
+a
2
0
T div u
∗
, we obtain
D
ia
D
−ia
u
k+1
=Su
k
+R
k
+
˜
φ
k+1
. (5.28)
Borel–Pompeiu’s formula leads to
u
k+1
= T
−ia
T
ia
(Su
k
+R
k
) +T
−ia
T
i
a
˜
φ
k+1
+T
−ia
φ
+ia,k+1
+φ
−ia,k+1
, (5.29)
where φ
±ia,k+1
∈ker D
±a
.
It remains to determine φ
ia,k+1
and φ
−ia,k+1
.
•F
Γ,−ia
u
k+1
=φ
−ia,k+1
=F
Γ,−ia
g
k+1
; g
k+1
=g
(k +1)τ, ·
. (5.30)
5.4 (D +ia)-Holomorphic Functions
Let us now compute (D +ia)-holomorphic functions. As for tr
Γ
u
k+1
=g
k+1
,we
have
tr
Γ
+T
−ia
T
ia
(Su
k
+R
k
) +T
−ia
T
ia
˜
φ
k+1
+T
−ia
φ
ia,k+1
=Q
Γ,−ia
g
k+1
. (5.31)
We know that
tr
Γ
:(I −F
Γ,−ia
)g
k+1
→Q
Γ,−ia
g
k+1
. (5.32)
Therefore,
tr
Γ
φ
ia,k+1
= (tr
Γ
T
−ia
F
Γ,ia
)
−1
−T
−ia
T
ia
Su
k
+
R
k
−
˜
φ
k+1
(5.33)
+(tr
Γ
T
−ia
F
Γ,ia
)
−1
Q
Γ,−ia
g
k+1
. (5.34)
Consequently, we get
φ
ia,k+1
= F
Γ,ia
(tr
Γ
T
−ia
F
Γ,ia
)
−1
−T
−a
T
ia
Su
k
+
˜
R
k
(5.35)
+F
Γ,ia
(tr
Γ
T
−ia
F
Γ,ia
)
−1
Q
Γ,−ia
g
k+1
. (5.36)
5.4.1 Final Representation Formula
After substitution and straight forward calculation it follows
u
k+1
= T
−ia
T
ia
Su
k
+
˜
R
k
(5.37)
−F
Γ,ia
(tr
Γ
T
−ia
F
Γ,ia
)
−1
T
−ia
T
ia
Su
k
+
˜
R
k
(5.38)
372 K. Gürlebeck and W. Sprößig
+T
−ia
F
Γ,ia
(tr
Γ
T
ia
F
Γ,ia
)
−1
Q
Γ,−ia
g
k+1
+F
Γ,−ia
g
k+1
, (5.39)
u
k+1
= T
−ia
Q
ia
T
ia
Su
k
+
˜
R
k
+T
−ia
F
Γ,ia
(tr
Γ
T
−ia
F
Γ,ia
)
−1
Q
Γ,−ia
g
k+1
+F
Γ,−ia
g
k+1
,
with the
Bergman projection P
ia
=F
Γ,a
(tr
Γ
T
−ia
F
Γ,ia
)
−1
T
−ia
and Q
ia
=I −P
ia
.
5.4.2 Influence of the Boundary Condition H
k+1
We have
H
k+1
=F
Γ,−ia
g
k+1
+T
−ia
P
i
aD
−ia
H
k+1
(k =0,...,n−1), (5.40)
where H
k+1
is a smooth continuation into the domain G.If
˜
H
k+1
is another exten-
sion, then it follows
T
−ia
P
ia
D
−ia
H
k+1
−
˜
H
k+1
=0 (5.41)
and
D
−ia
˚
W
1
2
(G) =imQ
ia
. (5.42)
Notice that tr
Γ
H
k+1
=tr
Γ
˜
H
k+1
.
5.4.3 Boundary-Value Problem for H
k+1
H
k+1
solves the following boundary-value problem
(D +ia)(D −ia)v
k+1
= 0inG, (5.43)
v
k+1
= g
k+1
on Γ. (5.44)
Using Borel–Pompeiu’s formula and the formulae of Plemelj–Sokhotzkij type, we
obtain
tr
Γ
T
−ia
Q
ia
T
ia
Su
k
+
˜
R
k
=0. (5.45)
5.4.4 Relation to the Initial-Value Condition
Let us abbreviate: T
−ia
Q
ia
T
ia
=:X
a
. Then we obtain
u
k+1
= X
a
Su
k
+H
k+1
+X
a
˜
R
k
= X
a
S
X
a
Su
k−1
+H
k
+X
a
˜
R
k−1
+H
k+1
+X
a
˜
R
k
= (X
a
S)
2
u
k−1
+X
a
SH
k
+X
a
SX
a
˜
R
k−1
+H
k+1
+X
a
˜
R
k
Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 373
= (X
a
S)
2
u
k−1
+(X
a
S)
1
H
k
+X
a
˜
R
k−1
+(X
a
S)
0
H
k+1
+X
a
˜
R
k
= (X
a
S)
k+1
u
0
+
k
l=1
(X
a
S)
l
X
a
˜
R
k−l
+H
k−l+1
+X
a
˜
R
k
+H
k+1
.
The boundary condition remains fulfilled, i.e.,
tr
Γ
X
a
...=0. (5.46)
6 Approximation and Stability
6.1 Approximation Property
Lu :=
1
ν
∂
t
u −
1
ν
(v ·D)u +
1
ν
+DDu, t =kτ;
L
τ
u :=
1
ντ
u(t +τ)
−u(t, ·) +DDu(t +τ,·)
+
1
ν
Dp(t +τ,·) −
1
ν
(v ·D)u(t, ·);
|L
τ
u
k
−Lu
k
|≤
1
ντ
(u
k+1
−u
k
) +DD u
k+1
+
1
ν
Dp
k+1
−
1
ν
(v ·D)u
k
−
1
ν
∂
t
u
k
+
1
ν
(v ·D)u
k
−
1
ν
Dp
k
−DD u
k
=
1
ντ
u
k
+τ∂
t
u
k
+τ
2
∂
2
t
u(kτ +θτ,·)
−
1
ντ
u
k
−
1
ν
∂
t
u
k
(6.1)
−
1
ν
∂
t
u
k+1
−
1
ν
∂
t
u
k
−
1
μ
(f
k+1
−f
k
)
≤ max
G
τ
2
1
ν
∂
2
tt
u(kτ +θτ,·)
+max
G
1
ν
∂
tt
u(kτ +θ
1
τ,·)
+max
G
τ
μ
∂
t
u(kτ +θ
2
τ,·)
≤ 2
τ
ν
max
G
∂
2
tt
u
k +θ
τ,·
+max
G
τ
μ
∂
t
u
(k +θ
2
)τ, ·
≤ Cτ →0.
374 K. Gürlebeck and W. Sprößig
6.2 Stability
We follow now the results in [1], where the following estimates of the norm of the
operator T
ia
0
were obtained:
T
±ia
0
L(C)
≤
c
a
0
u
C
. (6.2)
It remains to understand that X
a
S is bounded in a suitable Sobolev space. First, we
have to state that X
a
is smoothing, and therefore X
a
(v · D) has to be studied. Let
a =ia
0
. Then we have
D(v u) =−vDu +(v ·D)u + and D =D
a
−a +(Dv)u,
T
a
(v ·D)u = T
a
D
a
u −aT
a
u +T
a
vD
a
u +T
a
vau−T
a
(Dv)u
= (I −F
a
−aT
a
+T
a
vD
a
+T
a
va)u −T
a
(Dv)u.
As for Sc v =0, it follows that
T
a
(v ·D)u =(1 +v)(I −F
a
) +a(v −1)T
a
−T
a
(Dv)u,
and this means that T
a
S =T
a
(I +τ(v·D)) is uniformly bounded with respect to τ .
6.3 Representation Formulae
From
u
k+1
−τνΔu
k+1
=−τ(v ·D)u
k
+
τ
ρ
f
k
+u
k
−
1
ν
Dp
k+1
,
F(u
k
) =
I −τ(v ·D)
u
k
a
2
0
+
f
k
μ
it follows that
u
k+1
=−
1
ν
T
−a
Q
a
T
a
Dp
k+1
+T
−a
Q
a
T
−a
F(u
k
) +H
k
and
T
a
Dp
k+1
= T
a
(D +a)p
k+1
−aT
a
p
k+1
=p
k+1
−F
a
p
k+1
−aT
a
p
k+1
=˜p
k+1
−F
a
p
k+1
Q
a
T
a
Dp
k+1
= Q
a
˜p
k+1
(F
a
p
k+1
∈im P
a
).
Because of
˜p
k+1
=p
k+1
−aT
a
p
k+1
,
D
a
˜p
k+1
=Dp
k+1
,
Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 375
we obtain
u
k+1
=−
1
ν
T
−a
Q
a
˜p
k+1
+T
−a
Q
a
T
a
F(u
k
).
7 More Relevant Problems in Fluid Dynamics
7.1 Magnetic Benard’s Problem
Governing equations:
∂
t
u +(u ·∇)u −
1
μρ
(B ·∇)B +
1
2ρμ
∇
|B|
2
+
1
ρ
∇p −ηΔu =f, (7.1)
∂
t
B +(u ·∇)B −(B ·∇)u =
1
μσ
ΔB.
(7.2)
Divergence free fields:
div u = 0, (7.3)
div B = 0. (7.4)
Boundary conditions:
u =0onΓ, (7.5)
B =0onΓ. (7.6)
7.1.1 Time-Discretization of the Magnetic Benard Problem
Let T>0,T =nτ , and τ the meshwidth. We abbreviate again u
k
:=u(kτ, ·), p
k
:=
p(kτ,·), B
k
= B(kτ, ·). Further we set f
k
= (1/τ)
(k+1)τ
kτ
f(t, x) dt and g
k
=
(1/τ )
(k+1)τ
kτ
g(t,x)dt and approximate
(∂
t
u)(kτ, ·) ∼
u
k+1
−u
k
τ
(k =0, 1,...,n−1). (7.7)
Then we obtain for (k =0, 1,...,n−1):
u
k+1
−u
k
= τ(u
k
·D)u
k
+
τ
μρ
(B
k
·D)B
k
−
τ
2ρμ
D|B
k
|
2
−
τ
ρ
Dp
k+1
(7.8)
+τηΔu
k+1
+τf
k
, (7.9)
376 K. Gürlebeck and W. Sprößig
B
k+1
−B
k
=−τ(u
k
·D)B
k
+B
k
·D)u
k
+
τ
μσ
ΔB
k+1
+τg
k
, (7.10)
u
k+1
−ητ Δu
k+1
+
τ
ρ
Dp
k+1
=−τ(u
k
·D)u
k
−
τ
μρ
(B
k
·D)B
k
(7.11)
−
τ
2ρμ
D|B
k
|
2
+τf
k
=: τ F(u
k
,B
k
) +τf
k
, (7.12)
B
k+1
−
τ
μσ
ΔB
k+1
= B
k
−τ(u
k
·D)B
k
+(B
k
·D)u
k
+τg
k
(7.13)
=: τ G(u
k
,B
k
) +τg
k
. (7.14)
We introduce virtual functions u
∗
and B
∗
. It then follows that
u
k+1
−u
∗
+
u
∗
−u
k
= τ F(u
k
,B
k
) +τf
k
−
τ
ρ
Dp
k+1
+τηΔu
k+1
, (7.15)
B
k+1
−B
∗
+
B
∗
−B
k
= τ G(u
k
,B
k
) +τg
k
τ
μσ
ΔB
k+1
. (7.16)
We work with the ansatz
B
k+1
−B
∗
=
τ
μσ
ΔB
k+1
,B
∗
−B
k
=τ G(u
k
,B
k
) +τg
k
, (7.17)
u
k+1
−u
∗
=−
τ
ρ
Dp
k+1
+τηΔu
k+1
,
(7.18)
u
∗
−u
k
= τ F(u
k
,B
k
) +τf
k
.
Setting divu
k+1
=:U
k+1
and divB
k+1
=:b
k+1
, it then follows:
U
k+1
−divu
∗
= Δ
−
τ
ρ
p
k+1
+τηU
k+1
, (7.19)
b
k+1
−divB
∗
=−
τ
μσ
Δb
k+1
. (7.20)
7.1.2 Divergence and Uniqueness
The Poisson equation
Δ
1
ρ
p
k+1
+ηU
k+1
=
div u
∗
τ
(7.21)
Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 377
is always solvable. If we take this result as the hydrostatical pressure multiplied by
(1/ρ), then the scalar function U
k+1
has to be 0. Uniqueness results are obtained
under the following growth and boundedness conditions:
∇
x
u(x, t)
,
∇
x
B(x,t)
≤const, (7.22)
p(x, t)
≤const
1 +|x|
−1/2
. (7.23)
Meanwhile several examples for nonuniqueness are known (cf. [11]) if the men-
tioned conditions are not satisfied.
7.1.3 Velocity Calculation
It is easy to see that
τ
ρ
Dp
k+1
=−T divu
∗
+φ
k+1
,φ
k+1
∈ker D. (7.24)
This leads to
u
k+1
−u
∗
=−ητ DDu
k+1
−T divu ∗+φ
k+1
. (7.25)
Thus we obtain with b
2
=1/(ητ )
DDu
k+1
+b
2
u
k+1
=F
u
∗
,φ
k+1
=:F
k+1
, (7.26)
and so
(D −ib)(D +ib)u
k+1
=F
k+1
. (7.27)
Factorization methods with an estimation of the truncation error and the approxima-
tion properties can be found under the assumption that the velocity is bounded at all
times (cf. [18] and [19]).
7.2 Boussinesq’s Formulation of Poisson–Stokes’ Problem
Let be the body-force f equal to the gravity acceleration g. Then we have
ρ
0
∂
t
u +ρ
0
(u ·∇)u =−∇p +μΔu +ρ
0
g
1 −α(w −w
0
)
, (7.28)
ρ
0
c∂
t
w +ρ
0
c(u ·∇)w = div(k∇w) +ρ
0
β +Φ, (7.29)
div u = 0, (7.30)
where the last term in the first equation is the
buoyancy force, β is the heat con-
sumption (loss)
through chemical reactions, c is the difference between the two heat
capacities, i.e., c = c
p
− c
v
, and k is the coefficient of thermal conductivity. Φ is
the so-called
dissipation function, which is, roughly speaking, connected with the
long-time average of entropy production, and α is the
volume expansion coefficient.