Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science

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358 K. Gürlebeck and W. Sprößig

Remark The Prandtl number is a dimensionless parameter of a convecting system

that gives the regime of convection. It has the formula

m =Pr =

viscous diffusion rate

thermal diffusion rate

where μ is again the

kinematic viscosity, and a is the thermal diffusivity of the ﬂuid.

Remark The

Reynolds number is named after Osborne Reynolds, who proposed it

in 1883. It is the ratio of inertial forces to viscous forces in a ﬂuid. The expression

for this dimensionless measure is

Re =

inertial forces

viscous forces

where L is the characteristic length, u is the average velocity of the ﬂow, and μ is

the kinematic viscosity of the ﬂuid. A ﬂuid ﬂow in a pipe is laminar for Reynolds

numbers less than 2000; for values greater than 4000, the ﬂow is turbulent. The

Reynolds number is, roughly speaking, the product of Grashof number and Prandtl

number.

We consider the following iteration procedure:

=−R



M(u

n−1

) −

n−1



−

, (3.26)

0 =ScDR



M(u

n−1

) −

n−1



−

, (3.27)

=−

R Sc(u

D)w

+Rg. (3.28)

The computation of w

will be done by the inner iteration

R Sc(u

D)w

j−1

+Rg. (3.29)

The initial values of the iterations are u

= 0 is possible) and w

= 0is

possible, w

n−1

is better).

Theorem 3.6 (Convergence result)

1. Let u

∈

. Further, let m =4κ and u

<κ/mKC. The sequence {w

{j}

}

j∈N

converges in W

(G).

2. Let F ∈L

(G,H), g ∈L

(G), f :W

(G,H) →L

(G,H) with f(u)−f(v)

≤Lu −v

2,1

, and f(0) =0. Under the additional conditions

(i)

F 

K|d|

−1

g

16K



d :=



4 −



, (3.30)

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 359

(ii) g



1 −

√



ηd



32K



, (3.31)

(iii)m<4κ, (3.32)

the sequence {u

}

{n∈N}

converges in W

× W

× L

to the unique so-

lution (u,w,p)∈

(G,H) ×

(G,H) ×L

(G) of the originally boundary-

value problem, where p is unique up to a real constant.

Remark We note that conditions (i) and (ii) can always be realized for ﬂuids with

big enough viscosity number.

3.6 Continuous and Discrete Teodorescu Transforms

Let be G

a lattice G

with meshwidth h. We study the operators T

. All basic rela-

tions (Borel–Pompeiu formula, Hilbert-space decomposition) have a corresponding

discrete version, too. This enables us to ﬁnd strong relations between both the con-

tinuous and discrete transforms under weak conditions.

The discrete Teodorescu transform is suitably chosen. One can estimate the “dis-

tance” between the continuous operator and the discrete operator.

Let f be a Riemann-integrable function that belongs to L

∞

(G). Then



f −Tf



(G)

→0ash →0.

If f ∈C

0,β

(G), 0 <β≤1, then we have



f −Tf



(G)

≤ C



G, f 

q,h

, f 



−2+

|lnh| (3.33)

p,G

|h|

f 

(G)

(3.34)

for p<3/2, 1/p +1/q =1, where ·

denotes the norm in L

, and ·

q,h

is the

corresponding discrete norm. Restricting the range for p, we get for f ∈R∩L

∞

(G)

and p ∈(



f −Tf



p,G

→0ash →0.

Under the assumptions that f ∈C

0,β

(G), p ∈(

), 0 <β≤1, we obtain



f −Tf



p,G

≤ Cf 

∞,G

−2+



f 



,h,G

, f 





−2+



|ln h| (3.35)



f 

0,β

(G)h

. (3.36)

Here R describes the class of Riemann-integrable functions.

The proof of these properties requires some work, but there are just technical

difﬁculties. The proof can be found in [7].

360 K. Gürlebeck and W. Sprößig

3.7 Discrete Version of Navier–Stokes Equations

A corresponding discrete version of Navier–Stokes problem on a lattice is given by

−Δ

u +

∇

p +



u, ∇

−



u =

f in int G

, (3.37)

div

−

u = 0inintG

, (3.38)

u = 0on∂G

, (3.39)

where ∇

:=(D

1,h

2,h

3,h

),div

−

v :=



i=1

−

i,h

, and

k,h

...i

:=±

...i

±1,...i

−u

...i

). (3.40)

It can be proved that



∗,−

(u)



p,h





u, ∇

−





p,h

≤ 9C

u

2,1,h



q<6,p=2q/(2 +q)



. (3.41)

Applying the described discrete operator calculus as described in [7], we get the

equivalent system

u =−T

−

(u) −

−

p in G

, (3.42)

ScQ

−

(u) =

ScQ

p in G

. (3.43)

The operator Q

denotes the orthoprojection onto the (discrete) Hilbert subspace

−

(

2,h

)), and M

−

(u) :=M

∗,−

(u) −

f .

It can be proved that this discrete problem has a unique solution u

in G

under

the same conditions as in the continuous case. If f ∈C

0,β

(G) with 0 <β<1, then

we have the estimate

u −u



2,1;h

≤C



+h|lnh|



for h<h

. (3.44)

3.8 Stationary Magneto-Hydromechanics

Magnetic ﬂuids are very important from the technical point of view. These equations

read as follows:

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 361

−Δu +

(u ·∇)u +

∇p −

μη

B ×rot B =

f in G, (3.45)

ΔB +μσ rot(B ×u) = g in G, (3.46)

div B = 0inG, (3.47)

div u = 0inG, (3.48)

B =u =0onΓ, (3.49)

where μ is the

permeability, and σ is a measure of the electric charge. B describes

the

ﬁeld induction.

In quaternionic notation, we obtain

DDu +

Dp+M(u) =

μη

B ×(D ×B) in G, (3.50)

DDB +μσ [u, B]=g in G, (3.51)

ScDu = 0inG, (3.52)

ScDB = 0inG, (3.53)

B =u =0onΓ. (3.54)

We add the following trivial problems:

Δu

= 0,ΔB

=0inG, (3.55)

= 0,B

=0onΓ. (3.56)

This allows us to formulate the equivalent problem for quaternion-valued functions

u =u

+u and B =B

+B instead of vector-valued functions u and B. The latter

class is not adapted to the structure of the algebra H. We introduce the abbreviations

M(u) :=



∗

(u) −f



∗

(u) :=(u ·D)u, (3.57)

[u, B]:=D ×(B ×u) =(u ·D)B −(B ·D)u. (3.58)

Notice that [u, B] is just a Lie bracket, f =f, and g =g.

We get the following operator integral representation:

u =−

M(u) −

Qp +

μη



B ×(D ×B)



, (3.59)

B =T

g −μσ T

[u, B], (3.60)

0 = ScρQT

M(u) +Sc Qp −

ScQT



B ×(D ×B)



, (3.61)

0 = ScQT

g −μσ Sc QT

[u, B]. (3.62)

362 K. Gürlebeck and W. Sprößig

We know that any element Qu of imQ has the representation Qu = Dv with a

suitable v ∈

(G,H). Then it follows that

Qu =tr

Dv =tr

v −tr

trv =0. (3.63)

Let g =0. The solution {u, p, 0} solves the corresponding Navier–Stokes equations.

We look for solutions with B ≡ 0 and consider the following iteration method. For

n =0, 1, 2,..., we calculate

=−T



M(u

n−1

) −

μη

n−1

×(D ×B

n−1

)



−

, (3.64)

0 =Sc[ρQT

M(u

n−1

) +Qp

−



n−1

×(D ×B

n−1

)



, (3.65)

(j)

=−μσ T



(j−1)



g(j=1, 2,...), (3.66)

0 =−μσ Sc QT



(j−1)



+Sc QT

g. (3.67)

Here B

will be computed by using again an “inner” iteration.

We underline that at each step of the iteration one has to solve only a linear Stokes

problem.

3.8.1 Estimations

Let u, B ∈

(G,H) and 1 <p<3/2. Then we obtain



[u, B]



≤ 2



j,i=1

u



2,1

, (3.68)

B



2,1

≤ 18C

u

2,1

B

2,1

. (3.69)

The embedding constant C can be calculated by the estimate

u

=T

Du

≤T



]

Du

≤T



]

u

2,1

, (3.70)

which leads to T



]

= C for q<6,p =2q/(2 +q). Set now C

= 9

1/p

We get



[u, B]



≤2C

u

1,2

B

2,1

. (3.71)

Next, we consider the inner iteration

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 363

(j)

= μσ T



(j−1)

·D



−B

(j−1)

·D)



g, (3.72)

(0)

= 0. (3.73)

We abbreviate:

K :=T



∩im Q,W

]

T



]

. (3.74)

It follows that



(j)



2,1

≤μσ K2C

u



2,1

B

(j−1)



2,1

+Kg

. (3.75)

Because of

u



2,1

μσ K

, (3.76)

we have



(j)



2,1

≤2Kg

. (3.77)

3.8.2 Inner Iteration

Hence, (B

(j)

) is bounded in W

(G,H), and there exists a weakly convergent sub-

sequence. On the other hand, we have



(j)

−B

(j−1)



2,1

= 2C

μσ





(j−1)

−B

(j−2)





2,1

(3.78)

≤ 2C

Kμσu



2,1



(j−1)

−B

(j−2)



2,1



(j−1)

−B

(j−2)



2,1

. (3.79)

From these relations we get the strong convergence of (B

(j)

) in W

(G,H) to the

limit function B

which satisﬁes the estimate

B



2,1

≤2Kg

. (3.80)

3.8.3 A Priori Estimates

It is easy to see that

Du

≥

T



[im Q∩L

]

u

2,1

. (3.81)

364 K. Gürlebeck and W. Sprößig

Furthermore, one can obtain

T



[imQ,W

]

u

2,1

Qp

≤ 2

1/2

T



]





∗

(u)



(3.82)

f 

μη

B

2,1



, (3.83)

and so

T



[im Q,W

]

u

2,1

≤ 2T



]



u

2,1

f 

μη

B

2,1



, (3.84)

u

2,1

≤ 2K

u

2,1

+2K

f 

μη

B

2,1

(3.85)

We have the following quadratic inequality for u

2,1

u

2,1

−u

2,1

+2K

f 

μη

B

2,1

≥ 0, (3.86)

u

2,1

−

2KρC

u

2,1

f 

ρμC

B

2,1

= 0, (3.87)

so that

4KρC

−



16K

−

f 

−

ρμ

B

2,1

(3.88)

≤u

2,1

≤

4KρC



16K

−

f 

−

ρμ

B

2,1

. (3.89)

There arises the necessary condition

f 

ρμ

B

2,1

16K

. (3.90)

On the other hand, we have to realize for any u

the condition

u



2,1

≤

μσ K

. (3.91)

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 365

3.8.4 Convergence Results

Nowweareabletoestimate

u

−u

n−1



2,1

≤





M(u

n−1

) −M(u

n−2

)





2,1

(3.92)



μη



n−1

×(D ×B

n−1

) −B

n−2

×(D ×B

n−2

)





2,1

(3.93)



Q(p

−p

n−1

)



2,1

. (3.94)

From the a priori estimate we get

√

2T



[im Q,W

]



N(u,B)



≥u

2,1

T



[im Q,W

]

Qp (3.95)

with

N(u,B)=



ρf +

∗

(B) −ρM

∗

(u) −

2μ

D|B|



. (3.96)

For vector ﬁelds B

and B

,wehave

×(D ×B

) =(B

·D)B

−D(B

·B

DB

·B



≤C

B



1,2

B



1,2

(3.97)

Further, we obtain

u

−u

n−1



2,1

≤ 3T







∗

n−1

) −M

∗

n−2

)





2,1

(3.98)



μη



n−1



D ×(B

n−2

n−1

)





2,1

(3.99)



n−1

−B

n−2

) ×(D ×B

n−2

)



2,1

(3.100)

≤

3KC



ρu

n−1

−u

n−2



2,1



u

n−1



2,1

+u

n−2



2,1



(3.101)

B

n−1

−B

n−2



2,1



B

n−1



2,1

+B

n−2



2,1





(3.102)

On the other hand,

B

−B

n−1



2,1

≤8μσ KC

g

u

−u

n−1



2,1

. (3.103)

366 K. Gürlebeck and W. Sprößig

Now we have

μσ

(3.104)

and

u

≤

12μσ KC

, (3.105)

u

−u

n−1



2,1

≤



384K

g2



u

n−1

−u

n−2



2,1

. (3.106)

A sufﬁcient condition should be

f 

ρμ

g

. (3.107)

By straightforward calculation one will ﬁnd that the sequence {u



2,1

} is decreas-

ing and separated from zero. We can now formulate the following theorem:

Theorem 3.7 Let f, g ∈L

(G,H). We assume that

(i)

μσ η

< 1,

(ii) f 

ρμ

g

(iii) g

768K

with

K :=T



∩im Q,W

]

T



]

:= 9

T 

]



q<6,p=

2 +q



Then the operator integral equation has the unique solution {u, B, p}∈

(G,H)×

(G,H) ×L

(G,R), where u, B are uniquely deﬁned, and p is unique up to an

additive real constant. Our iteration method converges in

(G,H) ×

(G,R) to this solution if B

∈

(G,H) ∩ker div and are sufﬁciently small.

4 Time-Dependent Fluid Flow Problems

4.1 Characterization of Fluids

The so-called deviatoric stress tensor (τ

k

) determines the character of the ﬂuid

ﬂow. One has to distinguish the following ﬂuids:

Fluid Flow Problems with Quaternionic Analysis—An Alternative Conception 367

Ideal ﬂuids:Letτ

k

= 0, i.e., the stress only depends on the hydrostatical pres-

sure. Then we have

∂

u +(u ·∇)u +

∇p =0

(Euler equations), (4.1)

∇·u = 0

(incompressibility). (4.2)

Newtonian ﬂuids:Letτ

k

= 2μ ˙γ

k

with ˙γ

k

(∂



+ ∂



); μ denotes the

dynamic viscosity, and ˙γ

k

is the so-called shear rate tensor. The incompressibility

condition reads now

div u =



=1

˙γ

k

=0. (4.3)

Fluid ﬂow is now governed by the Navier–Stokes equations

∂

u +(u ·∇)u +

∇p −νΔu =

. (4.4)

The expression ν∇u with

kinematic viscosity ν := μ/ρ is called viscous term,

which is responsible for the dissipation effect (loss of energy). The expression

(u ·∇)u is called

convection term.

Newtonian compressible ﬂuids: The deviatoric stress tensor is now more compli-

cated and given by

k

=2μ ˙γ

k

+λ ˙γ

k

. (4.5)

λ,μ are called

Lamé coefﬁcients, which are not independent. Saint Venant found in

1843 that λ =−

μ. The Navier–Stokes equations transmute to

ρ∂

u −μΔu −(μ +λ)∇div u +∇ρ = ρf −ρ(u ·∇)u, (4.6)

∇·(ρu) = 0

(Stokes–Poisson equations), (4.7)

Φ(p,ρ) = 0. (4.8)

Now we assume the following nonlinear relation:

k

=τ

k

(u) =:Mu. (4.9)

Further we assume that ∇·u =0. Then we have the so-called

generalized Navier–

Stokes equations

[10, 15]

ρ∂

u −div M(u)∇u +∇p +ρ(u ·∇)u =ρf. (4.10)

Let Mu := ν(α +|˙γ

k

(u)|

p−2

) ˙γ

k

(ν, α are constants) The case p = 2,να =: μ

describes a Newtonian ﬂuid. For arbitrary p, these ﬂuids are called

non-Newtonian

ﬂuids with

p-structure.