Modelling of Polymeric Fluid Flow Taking into Account the Electromagnetic Impacts and the Heat Dissipation*

*

Modelling of Polymeric Fluid Flow Taking into Account the Electromagnetic Impacts and the Heat Dissipation

^*

ALEXANDER BLOKHIN

¹

, EKATERINA KRUGLOVA

^2,3

, BORIS SEMISALOV

^2,3

1

Sobolev Institute of Mathematics SB RAS, 630090 Novosibirsk, Russia

2

Novosibirsk State University, 630090 Novosibirsk, Russia

3

Institute of Computational Technologies SB RAS, 630090 Novosibirsk, Russia blokhin@math.nsc.ru, sl2857@mail.ru, ViBiS@ngs.ru

Abstract: A new mathematical model describing non-isothermal flow of incompressible viscoelastic polymeric liquid between two coaxial cylinders has been developed on the basis of the mesoscopic approach to polymer dynamics. This model is a system of non-linear PDEs taking into account the electromagnetic impacts and the dissipation of heat. Integral expression for determining the velocity of flow is derived and boundary value problem for temperature is posed. For calculating the velocity and temperature profiles Chebyshev approximations were used and the pseudospectral numerical algorithm was constructed. The stationary numerical solutions are obtained for wide range of values of physical parameters and for record-low values of the radius 𝑟𝑟₀ of the inner cylinder.

Keywords: polymer dynamics, mesoscopic approach, magnetohydrodynamics, heat dissipation, coaxial cylinders, singularly perturbed problem, pseudospectral method, Chebyshev polynomials.

1 Introduction

In the present times the additive manufacturing (3D printing) using polymer materials is one of the most demanded and fast-growing industries. This work is devoted to the reliable mathematical modelling and accurate numerical simulation of flows of polymeric solutions and melts through the channels of printing machines on the basis of mesoscopic theory of polymer dynamics [1,2]. It is the natural extension of the series of works [1–7], where the development and validation of modified rheological Pokrovskii–

Vinogradov model that takes into account thermomechanical and electromagnetic properties of polymeric liquids were done; the derivation of resolving systems of quasi-linear PDEs describing the flows through channels of different forms that are qualitatively similar to the classical Poiseuille and Couette ones was given; and the methods and algorithms for solving these PDEs were designed.

In [7] the problem of non-isothermal flows of polymeric liquid between two coaxial cylinders has been posed and solved numerically for various values of physical and geometrical parameters. In this work we consider the similar problem, but here a new way of deriving the resolving equations is used, and that is the main novelty, we supplement a model with the equations for the components of magnetic field intensity and with the heat-

conducting equation having the specific heat dissipation term. This leads to significant changes in the velocity profiles and other characteristics of the flow and allows for increasing its physical reliability by taking into account a number of effects inspired by the dissipation of heat and the magnetic impacts.

For solving the stationary boundary-value problem describing the non-isothermal flow of polymeric liquid between to coaxial cylinders the pseudospectral algorithm developed in [8] and based on Chebyshev approximations has been used. It is worth noting that the asymptotic of error of Chebyshev approximations and of the applied algorithm strictly corresponds to that of the best polynomial approximations [9,10] (in Russian literature this property is known as "the absence of saturation of the algorithm", see [11]). This enabled us to derive the reliable and accurate error estimates for approximate solutions of the simplified problem in [12]. In the more complex model considered in this work the application of such algorithm also gave us the opportunity to obtain the solution for singularly perturbed problem with extremely small values of the radius of inner cylinder.

2 Preliminary information

2.1 Governing equations of

magnetohydrodynamics

Following the results of the books [13–17] and of the papers [4,18] let us write governing system of equations of a new mathematical model describing magnetohydrodynamical non-isothermal flows of an incompressible polymeric liquid. These equations written in the dimensionless form in the cylindrical coordinate system are as follows:

div = u 1 v w u = 0,

u r r ϕ z r

∂ + ∂ + ∂ +

∂ ∂ ∂



(1)

1 ( ) 1

div = rL M N = 0,

H r r r ϕ z

∂ + ∂ + ∂

∂ ∂ ∂



(2)

2

{

2

( )

( )

1 1

Re

( )

,

rr r

rz rr

m

Ya Ya

u P

t r r r r

a a

Ya v L

Y L

z r r r

M L L M

r N z r

ϕ

ϕϕ

σ

ϕ

∂ ∂

∂∂ +∂∂ =  ∂ + ∂ +

− 

∂ ∂ + + + ∂∂ +

∂ ∂ 

+ ∂ + ∂ − 

(3)

( ) ( )

1 1 1

= Re

( ) 2

,

r

z

r

m

Ya Ya

dv P

dt r r r

Ya uv

z r Ya r

M M M M LM

L N

r r z r

ϕ ϕϕ

ϕ

ϕ

ϕ ϕ

σ ϕ

∂ ∂

∂ 

+ ∂   ∂ + ∂ +

∂ 

+ ∂ +   − +



∂ ∂ ∂

+    ∂ + ∂ + ∂ +  

(4)

( )

( )

1 1

= Re

( ) 1

Ga( 1),

rz z

zz

rz m

Ya Ya dw P

dt z r r

Ya N

Ya L

z r r

M N N

N Y

r z

ϕ

ϕ σ

ϕ

∂ ∂

+ ∂ ∂   ∂ + ∂ +

∂   ∂

+ ∂ +   +   ∂ +



∂ ∂

+ ∂ + ∂   + −

(5)

, , 2 2

2 = 0,

m r z

dL u M u u

L N

dt r r z

M L

b L

r r

ϕ

ϕ

ϕ

 ∂ ∂ ∂ 

−   ∂ + ∂ + ∂   −

 ∂ 

−   ∆ − ∂ −  

(6)

, , 2 2

2 0,

m r z

dM vL v M v v Mu

L N

dt r r r z r

L M

b M

r r

ϕ

ϕ

ϕ

 ∂ ∂ ∂ 

+ −   ∂ + ∂ + ∂ +   −

 ∂ 

−   ∆ + ∂ −   =

(7)

, ,

= 0,

m r z

dN w M w w

L N

dt r r z

b

_ϕ

N ϕ

 ∂ ∂ ∂ 

−   ∂ + ∂ + ∂   −

− ∆

(8)

2

^r

= 0,

rr

r rz rr

da u a u u

A a

dt r r z

ϕ

ϕ

 ∂ ∂ ∂ 

−   ∂ + ∂ + ∂   + L

(9)

2 2 1 ( )

= 0,

r

z

da v v v

a u A

dt r r r

a v z

ϕϕ

ϕ ϕ

ϕ ϕϕ

ϕ



∂ ∂

 

+   − ∂   −   + ∂ + + ∂   +

∂  L

(10)

2

= 0,

zz z

rz z

zz

da w a w u

a A

dt r r z

ϕ

ϕ

 ∂ ∂ ∂ 

−   ∂ + ∂ + ∂   + + L

(11)

= 0,

r

r r rz

z r

da v v w v

A a a

dt r r z z

A u u

r a z

ϕ

ϕ

ϕ

ϕ ϕ

ϕ

∂ ∂ ∂

  

+   − ∂   +   ∂ − ∂ −



∂ ∂

− ∂ − ∂   + L

(12)

= 0,

rz r

rz r

z

z rz

da u w w a w

a A

dt r z r r

a u u

r A z

ϕ

ϕ

ϕ

ϕ

∂ ∂  ∂ ∂

 

− ∂ + ∂  − ∂ + ∂ +

∂ ∂ 

+ ∂ + ∂ +L

(13)

= 0,

z

rz z z

r z

da v v u v

a a A

dt r r r z

w A w

a r r

ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

∂ ∂ ∂

  

+   − ∂   −   ∂ + ∂ +



∂ ∂

+ ∂ + ∂   + L

(14)

1 _{, ,}

= .

Pr Pr Pr

m

r z m

A

dY A

Y Y

dt ∆ ^ϕ + Φ + Φ (15)

Here

t

is the time variable,

u

,

v

,

w

are the components of the velocity vector

u  ,

L, M ,

N

are the components of the vector of magnetic field intensity H

in the cylindrical coordinate system;

2

= ^m 2

P p+

σ

H

  _, p is the pressure,

2= ( , )

H H H

  

  ^;

a

rr,...,a_ϕz are the components of the symmetric tensor of anisotropy

Π

having the second rank;

2

= (

r

) /

0

( )

rr

K a

I rr

+ β a  τ Y

L  

,

= ( , , )

r rr r rz

a a a

_ϕ

a



,

2

= ( K a

_I

a

ϕ

) /

0

( ) Y

ϕϕ ϕϕ

+ β  τ

L  

^,

= (

_r

, ,

_z

) a

ϕ

a

ϕ

a

ϕϕ

a

ϕ



,

2 0

= (

z

) / ( )

zz

K a

I zz

+ β a  τ Y

L  

^,

= ( , , )

z rz z zz

a a a

_ϕ

a



,

= ( (

r

, )) /

0

( )

rϕ

K a

I rϕ

+ β a a  

ϕ

τ Y

L

,

= ( (

r

,

z

)) /

0

( )

rz

K a

I rz

+ β a a   τ Y

L

,

= ( ( ,

z

)) /

0

( )

z

K a

I z

a a

ϕ

Y

ϕ ϕ

+ β   τ

L

,

=

1 I

3

K W

⁻

+ k I

, k =k−

β

, I =a_rr+a_ϕϕ +a_zz is the first invariant of the tensor of anisotropy Π;

k

and β ( 0 <β < 1) are the phenomenological parameters of the rheological Pokrovsky–

Vinogradov model (see [4–7,13]);

= 1

r rr

A a +W⁻ ,

A

_ϕ

= a

_ϕϕ

+ W

⁻¹, A_z =a_zz+W⁻¹;

= /

0

Y T T

, T is the temperature, T₀ = 300^K is the temperature of surrounding environment;

0

( ) = J Y( )

Y Y

τ

, 1

( ) = exp A

J Y E Y

Y

− − 

 

 ^;

Φ and

Φ

_m are the dissipative functions (see Remark 1);

2 2 2

, , 2 2 2 2

1 1

r z

=

r r z r r

ϕ

ϕ

∂ ∂ ∂ ∂

∆ + + +

∂ ∂ ∂ ∂

^,

d = v

u w

dt t r r ϕ z

∂ + ∂ + ∂ + ∂

∂ ∂ ∂ ∂

^;

Ga = Ra

Pr is the Grashof number,

0

= ^A

A

E E

T ^{; the} constants Ra (Rayleigh number), Pr (Prandtl number), Re (Reynolds number),

E

_A(activation energy) are described in [4, 7] in detail;

W

is the Weissenberg number;

A,

A

_m are the dissipative coefficients;

2

0 0

=

2 m

H

H u σ µ

ρ

is the magnetic pressure coefficient,

0

= 1 , Re =

m Re m m H

m

b

σ µ

u l is the magnetic

Reynolds number,

µ

₀ is the magnetic permeability of vacuum,

σ

_m is the electrical conductivity of the medium.

The variables

t

, r, z,

u

,

v

,

w

, p, L, M , ,

N

a

_rr,...,a_ϕz in the system (1)–(15) are related to

/

_H

l u

,

l

,

u

_H,

ρ

u_H² ,

H

₀,

W / 3

, where

l

is the characteristic length,

u

_H is the characteristic velocity,

ρ

=(const) is the density of medium,

H

₀ is the characteristic value of magnetic field intensity.

Magnetohydrodynamic equations (1)–(15) have been derived using the Maxwell's equations (see [14,16]), where the vector of magnetic induction B

was taken as follows:

=

0

. B µ H

 

In fact a more general relation is valid (see [20,21]):

B  = µµ

₀

H  = (1 + χ µ )

₀

H  ,

(16) where

µ

is the magnetic permeability,

χ

is the magnetic susceptibility, and besides

χ χ =

₀

/ , Y

(17) where

χ

₀ is the magnetic susceptibility for

= 0(= 300 ).

T T ^K

Remark 1. The equation (15) differs from the temperature equation in [4,7,18] by the presence of summands with dissipative functions

Φ Φ ,

_m. Following [19] we specify the functions Φ,

Φ

_m in the form:

3

1 11

, =1 1

3

1 2 1

12 13

2 1 3 1

3 3

2 2

22 23 33

2 3 2 3

= : = =

,

j ij

i j i

u u

u a a

x x

u

u u u

a a

x x x x

u u

u u

a a a

x x x x

∂  ∂

Φ Π ∇ ∂  ∂ +

 

∂ ∂  ∂ ∂

+∂ + ∂  +∂ +∂  +

 ∂  ∂ 

∂ ∂ 

+ ∂ +∂ +∂  + ∂ 



∑

(18)

3

, =1

2 1 2 2 2 3

1 2 3

1 2 3

3

1 2 1

1 2 1 3

2 1 3 1

3 2

2 3

3 2

= : = =

.

j

m m m m i j

i j i

m

u L L u x u

u u

L L L

x x x

u

u u u

L L L L

x x x x

u

u L L

x x

σ σ

σ

Φ Π ∇ ∂

∂

 ∂ + ∂ + ∂ +

 ∂ ∂ ∂



 

∂ ∂  ∂ ∂

+ + +  +

∂ ∂  ∂ ∂

   

∂ ∂  

+∂ +∂  



∑

(19)

Here

a

₁₁,...,

a

₃₃ are the components of the tensor of anisotropy written in the Cartesian coordinate system

x

₁

(= ) x

,

x

₂

(= ) y

,

x

₃

(= ) z

;

u

1,

u

₂,

u

₃

(= w )

and

L

₁,

L

₂,

L

₃

= ( ) N

are the components of the velocity vector

u 

and of the vector of magnetic field intensity H

written in the Cartesian coordinate system x y z, , ;

= ( ), , = 1, 2, 3.

m L Li j i j

Π

In Figure 1 the vector g=−g(0, 0,1)^T

, where g is the gravitational acceleration. This quantity enters the definition of the Rayleigh and Grashof numbers that specify the convective flows of liquid in gravity.

Below we shall seek the stationary solutions of the system (1)–(15) describing the flows of incompressible polymeric liquid between two coaxial cylinders. The outer cylinder has a wall of finite thickness Rh , and the inner one is a heating element (see Figure 1). The characteristic length

l

has been taken equal to R, therefore the dimensionless thickness of the outer cylinder is

h

. The dimensionless radius of the heating element was denoted by r₀: 0 <r₀< 1. The relations (1)–(15) should be supplemented with the boundary conditions:

r = r u

₀

:  = 0, Y = Y

_H

, r = 1: u  = 0, Y = 1.

(20) The quantity

Y

_H and other boundary conditions are discussed below. Here we only note that

Y

_H

> 1

is the heating from inside (in this case we take

Ga Re = 1 ⋅

) and

Y

_H

< 1

is the heating from outside of the channel (in this case we take

Ga Re = 1 ⋅ −

).

In Figure 1 the magnetic field is also shown. The characteristic value of the magnetic field intensity was set equal to the value of certain outer magnetic field intensity

H

₀

= N ˆ

₀, therefore the dimensionless

N ˆ = 1

₀ . Notations '0' and '1' refer to the domains: and respectively, 'H' refers to the heating element. We assume, that in the domains '0' and '1' the temperature T is constant and equals to

T

0 (T₀ = 300^K).

Fig.1. Channel formed by two coaxial cylinders: the channel in Cartesian coordinates x y z, , and the cross-section of channel in polar coordinates.

2.2 Description of the heating element

We assume the heating element 'H' to be a long straight-line cylinder of the radius

r

₀

: 0 < r

₀

< 1

manufactured of paramagnetic material (for example, aluminum or tungsten) with magnetic susceptibility

χ

_OH

> 0

(the value of

χ

_OH is taken from reference tables, [20]). The current-conducting wire is wound on the heating element and there are

n

turns per each meter (see Figure 2). An electric current is passed through the winding, the current strength is

I

H. Following [20–22], let us express the component of magnetic field intensity

ˆ

N

H in the cylinder:

ˆ = A ,

H m NH nI  

  

 (21) therefore the dimensionless magnetic field intensity

ˆ

N

H is the following :

0

ˆ = .

ˆ

H H

N nI N



(22)

Fig.2. Sketch of the heating element In (21) the value of

ˆ

N

H is given in the SI (see [20]) system and the corresponding dimension is given in brackets. We assume the material of the heating element to be the ideal paramagnetic, therefore the Curie's law is satisfied, [20–23]:

^ m

H

= χ

H

N ˆ

H

,

(23) here H

 m

is the magnetization. Taking into account (17), (21), (23) we obtain

Y

_H on the boundary of heating element

ˆ

=

^{OH H}

=

^H

= ,

H OH OH

H H

N nI a

Y b

χ χ  χ

 

m m

⁽²⁴⁾

where

0

= ˆ nI

H

a N



,

0

= ˆ b

H

N

 m

and the value of

χ

_OH can be taken from the reference tables [20].

It is well-known (see for instance [21, 22]), that a surface current appears on the surface of the paramagnetic whose linear density is equal to H

 m

. Thus, in (1)–(15)

L = N = 0

and on the boundary

=

0

r r

this system should be supplemented with one more condition (see (22), (24)):

₀

0

ˆ ˆ

= ( ) / = ,

ˆ

H

H H H

N N N nI

N µ

+  

m

(25)

where

µ

_H

= (1 + χ

_OH

/ Y

_H

)

, see (17).

3 Stationary system of resolving equations

Here we derive a system of non-linear equations describing the flows between two co-axial cylinders that are qualitatively similar to classical Poiseuille ones. To this end let

( , , , ) = ˆ( ), ˆ ˆ

( , , , ) = ( ) , U t r z r

P t r z P r Az

ϕ

ϕ

−

 U

(26)

where:

= ( , , , , , ,

_rr

,...,

_z

, )

^T

U  u v w L M N a a

_ϕ

Y

,

ˆ = (0, 0, ( ), 0, 0, w r ˆ N r a ˆ ( ), ˆ

_rr

( ),..., r a ˆ

_ϕz

( ), ( )) , r Y r ˆ

^T U

ˆ ( )2

ˆ( ) = ˆ( )

m 2 N r P r p r +

σ

,

ˆ > 0

A

is the dimensionless constant pressure drop along

Oz

axis.

For calculating the functions ˆ ( )w r , N rˆ ( ),

ˆ ( )

_rr

a r

,...,aˆ ( )_ϕz r , Y rˆ( ), P rˆ ( ) the relations (3)–(5), (9)–(15), (8) can be correspondingly rewritten as follows:

ˆˆ 0

ˆ = (ˆ ˆ ) , ( ) =ˆ ,

Re Re

rr

rr

Ya d

P a a Y

ϕϕ σ d

σ

ξ

σ

 

− − ⋅

 

  ⁽²⁷⁾

Yaˆˆ =_rϕ e^{2ξ σ0} _rϕ, (28)

0 2 0

* 20 0

0 0

ˆˆ =

+Ga ˆ = ,

Ya

rz

e De

e Yd e Z

ξ σ ξ σ

σ ξ λ ξ σ

ξ λ

−

−

 − +



 

∫ 







(29)

ˆ

_rr

ˆ

_r ²

= 0,

K a

I_

+ β   a 

(30)

ˆ

2

ˆ = 0,

K a

I_{ ϕϕ}

+ β   a 

_ϕ (31)

ˆ

2

ˆ ˆ ˆ

ˆ

_{zz z}

2 ˆ

_rz

ˆ ( ) / = 0, K a

I_

+ β   a  + a ν w J Y

_σ

Y

⁽³²⁾

ˆ

_r

( , ˆ ˆ

_r

) = 0,

K a

I_{ ϕ}

+ β a a  

_ϕ (33)

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ

_rz

( ,

_{r z}

)

_r

ˆ ( ) / = 0,

K a

I_

+ β a a   + A w J Y ν

_σ

Y

₍₃₄₎

ˆ ˆ ˆ ˆ ˆ

ˆ

_z

( ,

_z

) ˆ

_r

ˆ ( ) / = 0, K a

I_{ ϕ}

+ β a a  

_ϕ

+ a

_ϕ

ν w J Y

_σ

Y

(35)

ˆ ˆˆ ˆ = 0, ˆ ^rz

Y_σσ AYa w_σ

−

ν

(36)

1 2

ˆ = 0, ˆ = .

N

_σσ

N  σ + 

(37) Here

σ ξ ξ = /

₀, r =e^−ξ ,

ξ

₀

= − ln r

₀,

0 ≤ ≤ σ 1

,

0

ˆ = e

^{ξ σ}

/

0

ν ξ

, ˆ 1 ˆ

ˆ = =

ˆ

dw dw

w^σ d

σ

⁻

ν

dr ^,

20 * 2 0 0

0

= Ga ˆ

Z De e Yd

ξ σ

ξ

σ ξ λ

λ

− −

− +

∫

   ^,

ˆ

=0

= a

_rz

|

_σ

+  D



^{, = ˆ Ga*}

2

D D− _, Ga = GaRe , ^*

ˆ = Re ˆ

D A

,_rϕ^,



1^,



2 are the certain constants,

ˆ

_r

= ˆ

_rr 1

A a + W

⁻ ,

=

¹

,

I

3

K

_

W

⁻

+ k I 

ˆ ˆ ˆ

= _{rr zz}

I a +a_ϕϕ +a

 _.

It is worth noting that (36) has been obtained from (15), where the summand 1

r r

∂

∂ from the expression of ∆_{r, ,ϕ z} was annihilated with a summand in the expression of

2

2

d Yˆ

dr written for new variable

σ

. The relations (27)–(37) should be supplemented with the conditions (see (20), (25)):

=0,1 =0 =1

=1 1 2

0

ˆ ˆ

ˆ | = 0, | = 1, | = ,

ˆ | = = .

ˆ

H

H H

w Y Y Y

N nI

N

σ σ σ

σ

µ

 

 +

 

  

(38)

The dissipative functions

Φ Φ ,

_m (see (18), (19)) computed on the vector-functions Uˆ ( )r (see (26)) are as follows:

ˆ = ˆ_rz dwˆ , ˆ_m = 0.

a dr

Φ Φ

This has been used while deriving (36).

Note also that in (29) we can set ˆ = 1D by specific choosing the characteristic velocity

u

_H (see [4–7]).

We will assume for later use that the outer cylinder (it is denoted by the number ‘1’ in the Figure 1) is made of the material with the electrical conductivity

σ

_m =

σ

_m¹ =0, therefore in it

, , = 0

r_ϕzN

∆ , i.e.

(1) (1)

1 1 2

= ˆ ( ) = ln , 1 < < 1 , N N r −  r +  r + h

(39) (39)

here in accordance with [21, 22] and (37)

(1)



1 2 2 1

0

ˆ (1) = = = nI

^H _H

. N

N  µ −

  

⁽⁴⁰⁾

On the boundary

r = 1 + h

one gets

(1)



1 1

0

1 = ln(1 ) nI

^H _H

. h

N µ

− + +  −

 

(41)

Let us assume that the inner wall of the outer cylinder has the absolute conductivity, i.e. (see for instance [15,16]):

₁

=0

ˆ

= = 0.

N

σ

σ

∂

∂  (42) From (41), (42) we obtain

₁⁽¹⁾

0

= 1 / ln(1 )

ˆ

H H

nI h

N

µ

 

− + +

 

 

 

 

and

1

0 0

ˆ ( ) = ln 1 ,

ˆ ln(1 ) ˆ

1 < < 1 .

H H

H H

nI r nI

N r N h N

r h

µ

− + ^

µ

− ^ +

 

If the inner wall of the outer cylinder is insulator, see [15,16], we obtain

N ˆ (0) = 

₂

= 0

and using (39)–(41) derive:

_{1 1}⁽¹⁾

0

= , = 1

ˆ ln(1 )

H H

nI

N µ − h

+

  

and ₁

0

ˆ ( ) = ln , ˆ ( ) = . ln(1 ) ˆ

H H

r nI

N r N

h σ N µ σ

+



A careful analysis of the relations (28)–(35) performed in [18] on the basis of [6] yields

ˆ = ˆ _z = ˆ_r = 0, a_ϕϕ a_ϕ a_ϕ

i.e _rϕ = 0 and for calculation of functions

a ˆ

_rr,

a ˆ

_zz ,

w ˆ

we get the following relations (here the function

a ˆ

_rz can be obtained from (29)):

(

²

)

²

ˆ_rr ˆ_rr ˆ = 0, ˆ = ˆ_rz,

K aI_ +

β

a +g g a (43)

0

ˆ = ˆˆ = ,

ˆ ˆ

ˆ ˆ ˆ

( ) ( )

= .

I rz I

r r

I I

K Ya K

w Z

J Y A J Y A

K K I

σ

ξ

ν β

− −

+

 

 

 



 

(44)

Substituting this into (32) one obtains

ˆ ˆ

²

2 ˆ = 0.

ˆ

I

zz z

I

r

K a a g K

β A

+ −

^



 

 

(45)

Following [4,18] it is easy to obtain from (43), (45) the relations

ˆ ˆ

ˆ = ˆ 2 , = 2 ˆ .

ˆ ˆ

zz rr rr

r r

g g

a a I a

A A

   

+ +

   

   

   



Hereinafter we shall assume that

k = 0

(k =β ).

Thus, from (43) it follows that

ˆ

2

1 1 ˆ

ˆ = , = 4 ˆ 1,

2

= .

rr

a G G W g

W

W W β

− ± −

 ≤





(46)

Later for definiteness we shall consider only the branch of the function

a ˆ

_rr with the ‘+’ sign.

Accounting for (38) from (44) one gets

w ˆ = ξ

₀

  − F

₀

( , ) σ   + F

₁

( , ) , σ   

(47) where

₀

0

( , ) = ( , )s ds,

σ

σ

∫

F  F 

20 * 2 0

1 0

0 0

( , ) = ( , ) s De – Ga e Yd ˆ ds ,

σ σ

ξ σ ξ λ

σ ^ _

⁻

ξ

⁻

λ ^ _

 

∫ ^ ∫

F  F 

( )

( , ) = ( ) =

ˆ ( ( ))ˆ ( )

ˆ ˆ ˆ

1 2 ( ) ( ) / ( ) ˆ( ) 1

exp .

ˆ( ) 1 ˆ ( )

I

r

rr r

A

rr

s K s

J Y s A s

W a s g s A s E Y s

Wa s Y s

+ +

 − 

  +

 





F 

Since

w ˆ |

_σ=1

= 0

(see (38)), then from (47) the relation for calculation of the constant  follows:

¹

0

(1, )

= .

(1, ) F F

 



(48) Making the following substitution into (36)

ˆ

ˆ ( ) = ( ) ˆ ( ), ˆ ( ) = 1 (

_H

1) , Y σ Q σ + V σ V σ + Y − σ

one can get the following boundary value problem

2 2

ˆ

0

[ ( )] ˆ ( , , ) = 0, ˆ ˆ (0) = ˆ (1) = 0.

Q A Z Q Q

Q Q

σσ

+ ξ  F σ 

(49) Here we have underlined the nonlinearity of differential equation for Qˆ by denoting Z =Z Q( )ˆ and F( , ) =

σ

 F( , , )

σ

 Qˆ ^.

A simple inspection reveals that the boundary value problem (49) has the strong singularity in the case of small values of radius of the inner cylinder

r

0, see Figure 1. Indeed, since

ξ

₀

= − ln r

₀ and

ξ → ∞

0 as

r

₀

→ 0

, then the equation (49) can be regarded as one with the small parameter 1 /

ξ

₀² at the highest order derivative. The mentioned singularity cardinally effects not only the distribution of temperature, but also the distribution

of the velocity of flow. For solving this problem we use special pseudospectral algorithm based on Chebyshev approximations. Its detailed description is given in [7,8]. To use it, it is convenient to drop

"^" above "Q" and rewrite (49) as follows

2 2

=

0

[ ( )] ( , , ) = ( , ), (0) = (1) = 0.

Q A Z Q Q f Q

Q Q

σσ

− ξ  F σ  σ

(50)

4 Description of the pseudospectral numerical algorithm

In this section the fast pseudospectral algorithm is developed on the basis of Chebyshev approximations. The remarkable properties of such approximations and spectral methods based on them are described in vast literature (see [9,10,24] for instance). Below we detail the most important of them.

For solving the equation (50) we apply the iterative stabilization (or relaxation) technique. It requires to introduce a fictive time variable

t

that will be used for iterations and the regularizing operator (the regularization)

B

_t. Assuming that

= ( , )

Q Q

σ

t we shall seek the solution of (50) as the limit of solutions of the evolution equation

= ( , )

B Q

t

Q

_σσ

− f σ Q

as

t → ∞

. In this work the Sobolev's regularization

2

1 2 2

t

=

B k k

σ t

 ∂  ∂

 − ∂  ∂

 

is used, where

k k

₁

,

₂

> 0

are constants. By introducing the grid with respect to t with the step

τ

and the nodes

t

_n

= n τ

,

= 1, 2,...

n and by denoting

[ ] [ ]

= ( ) = ( , )

n n

Q Q

σ

Q

σ

tn we approximate the derivative

Q

_t by the difference quotient

[ ] [ 1]

( Q

ⁿ

− Q

ⁿ⁻

) / τ

. Using the Sobolev's regularization of (50) we obtain:

2

[ ] [ ] [ 1]

1 2 1 2 2

[ 1] [ 1]

( ) =

( , ) = ( , ).

n n n

n n

k Q k Q k k Q

f Q f Q

τ

σσ

σ

τ σ σ

−

− −

 ∂ 

− +  − ∂  −

− 

(51)

The idea of the method consists in finding the solution

Q

^{[ ]n} to (51) on the current time step using its values

Q

^[n−1] from the previous one. This

process should continue till the solution stabilized.

More precisely, the stopping criteria is

 B Q

_t

 ≤ ε

_S

,

(52) where

ε

_S is the error (or residual) of the stabilization process,  ⋅ is the uniform norm of function (all the functions are assumed to be continuous).

In order to approximate in regularized equation (51) the values of

Q

^{[ ]n} and the values of its derivatives with respect to

σ

, we use the modified interpolation polynomial with Chebyshev nodes

written in the Lagrange form (see [11])

=1

( ) ( , ) = ( , ) ( )

( ),

( ) '( )

2 1

= cos 2

N N

j N

j

j j N j

j

Q p Q

s T

T Q j

N

σ σ

σ σ σ

σ σ σ σ

σ π

≈

= −

−

∑

_ ^{  } (53)

where

2

2

( , ) =1

j 1

j

s

σ σ σ

σ

−

−

  ^,

( ) = cos( arccos )

T

N

σ  N σ 

^,

σ

 ^{= 2}

σ

− ∈ −^{1 [ 1,1],}

= ( 1) / 2

j j

σ



σ

+ and the dash means the

differentiation with respect to

σ

.

We called this polynomial "modified" due to the factor s( ,

σ σ

_{ j}) that was introduced to satisfy automatically the homogenous boundary conditions of (50).

For solving the equation (51) we use the (53) and the collocation method. The expansion (53) was differentiated with respect to

σ

. Further

[ ]

= ⁿ = ( ( 1),..., ( _N))^T Q^{ }Q Q

σ

_ Q

σ

_ ,

[ ]

= ⁿ = ( ( 1),..., ( _N))^T Q_σ Q_σ Q′

σ

Q′

σ

  ^,

[ ]

= ⁿ = ( ( 1),..., ( _N))^T Q_σσ Q_σσ Q′′

σ

Q′′

σ

  ^.

Here and in what follows the upper index "[ ]n "

of the components of vectors Q

,

Q 

_σ ,

Q 

_σσ

and the arrows above vector notations are omitted.

 

1

=1 2

2

2

( , ) = ( 1)

2 1

( ) ( )

1 ( ) ,

( )

N j

N j

j

j

N

j j

N

j j

p Q H Q

N T

T

σ

σσ σ σ σ γ σ

γ σ

γ σ σ

′ − − ⋅

 − −

 +

 −



+ − − 

∑

 

  (54)



 

 

 

2 1

=1

2 2 2

3

2

2 2

( , ) = ( 1)

2 ( )

( ) ( )

( 3 2)

1 ( )

( )

N j

j j

j j

N

j j

j

N

j j

p Q H Q

N T

N

T

σ

γ σ σ

σ σ γ σ

σ σσ

γγ σ σ σ

′′ − − ⋅

 − −

 −

 −



− + 

− − 

− 

∑

(55)

where

γ

= 1−

σ γ

², _j = 1−

σ

²_j, H =2.The limit expressions of the first and second derivatives (54), (55) as

σ → σ 

_i were obtained using l'Hospital's rule.

2

=1,

( , ) = ( 1) 3 ,

( ) 2

N

i j i i

N i j i

j j i j i

j i

p Q σ H γ Q σ Q

γ σ σ γ

+

≠

 

 

′ − −

 − 

 ∑ 

2

2 1

=1, 2

2 2 2

4

2 3 ( )

( , ) = ( 1)

( )

( 5) 3

3

N i j i i i j

N i j

j j i i j

j i

i i

i i

p Q H Q

N Q

γ σ σ σ

σ γ γ σ σ

γ σ γ

+ −

≠

 + −

′′  − −

 −



+ + 

− 



∑

Let us denote

2 1

2

2 3 ( )

= ( 1) , = ( 1) ,

( ) ( )

, = 1,..., , ,

i j

i i i j

i i j

ij ij

j i j j i i j

a

i j N i j

γ σ σ σ γ

γ σ σ γ γ σ σ

+ + −

+ −

− −

− −

≠ a

2 2 2 2 4

= 3 / (2 ), = (( 5) 3 ) / (3 ), = 1,...,

i i i i N i i i

i N

σ γ ν γ σ γ

− − + +

n

and make up N×N matrices