Oscillation Regimes of Dynamic Parameters Changing in Couette Flow of Anomalous Thermoviscous Liquids

Available online at www.sciencedirect.com

Procedia IUTAM 8 (2013) 153 – 160

IUTAM Symposium on Waves in Fluids: Effects of Nonlinearity, Rotation, Stratification and Dissipation

Oscillation Regimes of Dynamic Parameters Changing in Couette Flow of Anomalous Thermoviscous Liquids S.F. Khizbullina*, S.F. Urmancheev Mavlutov Institute of Mechanics, 71, Prospekt Oktyabrya, Ufa, 450074, Russia

Abstract In this work results of researches of the hydrodynamic systems behavior describing a Taylor-Couette flow are presented. The subject being examined consisted in determining the influence of interdependence between viscosity and temperature on the structure of chaos in the above-noted systems. © S.F.Authors. Khizbullina and by S.F. Urmancheev. Published by Elsevier B.V. © 2013 2013 The Published Elsevier B.V. Selection and/orpeer-review peer-review under responsibility ofChashechkin Yuli Chashechkin and David Dritschel Selection and/or under responsibility of Yuli and David Dritschel Keywords: Couette flow; anomalous thermoviscous liquid; flow regimes.

1. Introduction Determinate chaos in dynamic systems has become one of the most important research trends both in modern physics and various fields of engineering. As a matter of fact, this line of research caused the setting of problems and the methods of their solution move to a new level. In this respect, the interpretation of the results obtained in the course of the problem solution is given priority. A distinctive feature of determinate chaos is the possibility of short-term forecasting of the dynamic system behavior. Due to multiple applications, analysis of determinate chaos in hydrodynamic systems is of enormous importance. Generally speaking, it was as a result of analyzing a problem on identification of convective fluxes in a liquid layer heated from below that Lorenz discovered the existence of chaos. This trend of hydrodynamics was later on marked by a dramatic increase in the number of publications. The problem of the viscous liquid flow between two rotating round cylinders attracts attention due to extensive technological applications, such as the problem of elimination of the heat generated in the rotator of an electric motor, the investigation of processes occurring in the clearance gap between a rotary shaft and a pillow block, the problem of fractionation of multi-component mixtures. Another trend is connected with the questions

* Corresponding author. Tel.: +7-347-235-5255; fax: +7-347-235-5255. E-mail address: [email protected]

2210-9838 © 2013 The Authors. Published by Elsevier B.V. Selection and/or peer-review under responsibility of Yuli Chashechkin and David Dritschel doi:10.1016/j.piutam.2013.04.019

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of the hydrodynamic stability theory initiated by Taylor's research. Along with Rayleigh-Benard convection, Couette flow presents a field where new ideas and methods of stability, nonlinear behavior and the transition to turbulence can be examined. Besides, the presence of a variable parameter in the hydrodynamic system may have an essential effect on its stability, which is used to control the liquid flow in various technology processes. Naturally, various aspects of a cylindrical Couette flow for a liquid with constant viscosity were studied both experimentally and theoretically. However, lately the liquids showing non-monotone viscosity dependences on temperature have attracted much attention. Hereinafter liquid whose viscosity is non-monotone function of temperature is called an anomalous thermoviscous liquid. The flow regularities of such medium are practically not studied and demand adequate problem statement for their theoretical and experimental research. The accounting of effects of a liquids flow caused by viscosity dependence on temperature represents the intricate problem interconnected with the application of modern computing systems and methods of mathematical modeling. By now the numerical modeling results of a flow of anomalous thermoviscous liquids in the plane channel and a cylindrical pipe with a not uniform temperature field have been received [5 – 8]. The essentially important fact of formation in a stream of the high-viscosity area caused by non-monotone viscosity dependence on temperature has been established. Thus, features of a liquid flow directly depend on distribution nature of high-viscosity area or else a viscous barrier. The main regularity of distributions of a velocity field, flow regimes and a rate flow of liquid depending on heat exchange conditions on walls of the channel which define space structure of a viscous barrier have been determined. In this work the unsteady flow of anomalous thermoviscous liquid between the rotating inner cylinder and the rest outer cylinder which have limited length is considered. 2. Formulation of the problem Consider the unsteady flow of anomalous thermoviscous liquid in the clearance between the coaxial cylinders having finite length L (see Fig. 1). The inner cylinder, with radius Ri, rotates at a constant angular velocity while the outer cylinder, with radius Ro, is at rest. The following assumptions are introduced in this model: the liquid is incompressible and the flow is axisymmetric; mass forces are missing; coefficients of heat conductivity and the thermal capacity of the liquid are constant. The system of cylindrical coordinates (r, , z) is connected with the rotation axis of the inner cylinder and axes z is directed along the symmetry axes of cylinders. Because of the symmetry, the considered computational domain represents the rectangle in the width d = Ro – Ri and the length L. At the initial time the liquid filling space between cylinders is at rest and has the temperature T0. At the top and bottom end faces of cylinders, and also on the outer cylinder we set no-slip conditions for a velocity vector u and constant temperature T0 while the inner cylinder has temperature Tw > T0. The unsteady flow is described by the system of differential equations of the motion, continuity and convective heat conductivity:

u t u cp Here u

u

u

0, D T t

u

p 0.5 T

(1)

u T

uT

(2)

D:

(3)

u, v, w is the velocity vector; p is the pressure of liquid; is the density of liquid; T is the tempera-

ture of liquid; cp and are the coefficients of heat conductivity and the thermal capacity of the liquid; stress deviator tensor; D is the strain tensor. The components of stress deviator tensor and strain tensor are interrelated in the following way:

is the

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S.F. Khizbullina and S.F. Urmancheev / Procedia IUTAM 8 (2013) 153 – 160

ij

2

T eij ,

T

min

1

BT T

Ae

2

,

A

max

1

(4)

min

zz

max

AA min min

RRii

L

RRoo

(1+Ae-1-1)) (1+Ae

1/2

1/2 2/B 2/B

min min

r

min min

T Tmin min Fig. 1. Geometry of problem

TT* *

TTmax max

Fig. 2. Non-monotone viscosity dependence on temperature

Here A and B > 0 are the anomaly parameters characterizing non-monotone viscosity dependence on temperature; T* = 0.5(Tw+T0) is the temperature at which liquid has the maximum value of viscosity. In Fig. 2 the diagram of the viscosity dependence on the temperature is shown and the geometrical meaning of parameters included in it is explained. It is visible that the anomaly parameter A characterizes the relation of the minimum and maximum values of viscosity. The anomaly parameter B is connected with the fullness degree of the given temperature range by viscosity anomaly: the increasing B shows narrowing of a temperature range on which there is a nonmonotone change of viscosity. In simulation, the coordinates and length variables are normalized by the cylinder gap width d = Ro – Ri; the velocity components are normalized by the rotation velocity of the inner cylinder u0 = Ri, and the pressure by

u02 . The dimensionless temperature and viscosity are defined by T T0 min , * Tw T0 max min So the dimensionless numbers are defined by

Re

u0d

, Pe

c p d Ri

, Ec

min

d Ri . c p Tw T0

Taking into account the accepted assumptions the equations system (1) – (4) in a dimensionless form become:

u t

1 ru 2 r r

2rA1 u Re r

v t

1 ruv r r

rA1 Re

w t

1 ruw r r

rA1 Re

v r u z

z v r

z w r

A1 Re

wu

w r

A1 v Re z

wv

z

u z

w2

2 A1 w Re z

A1 r Re

p r

v2 r

2 A1

v r

v r

uv r

p z

u r2

(5) (6) (7)

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S.F. Khizbullina and S.F. Urmancheev / Procedia IUTAM 8 (2013) 153 – 160

t Where

I2

r Pe r

1 ru r r

A1

1

4

u r

Ae

B Tw T0

w z

2

2

z 0.5

Ec I 2 A1 Re

1 Pe z

w

(8)

2

and

u w 4 r z

v r

v r

2

v z

2

u z

w r

2

Initial and boundary condition:

t z

0: 0, z

r

Ri d :

L d ,r

R0 d :

u u

0, T 0, T

v

v0 , u

0 0

(9)

w

0, T

1

The system of the differential equations (5) – (8) supplemented with initial and boundary conditions (9) completely describes an axisymmetric flow of incompressible anomalous thermoviscous liquid in a clearance between two coaxial cylinders having limited length. This mathematical model takes into account the fact the viscosity of a liquid is some function of temperature. And this model does not depend on a specific type of viscosity dependence on temperature and uses only minimal min and maximum max viscosity values. Therefore this model can be used for liquids with monotone viscosity dependence on temperature. 3. Results and discussion Mostly the Couette flow is characterized by three parameters: the radius ratio = Ri/Ro; the aspect ratio = L/d and the Reynolds number Re of the inner cylinder. In our work the system has a fixed radius ratio = 0.5 and the aspect ratio = 4, where Ri = 0.005m, Ro = 0.01m and L = 0.02m. The problem parameters are selected from a condition of that at these values the Couette flow for incompressible liquid with constant viscosity is steady [9]. The liquid has the following parameters: = 900kg/m3, = 1.13·10-7m2/s, min = 0.0045Pa·s, T0 = 20ºC, Tw = 90ºC. Besides results are also given in the work for anomaly parameter B = 0.005. The conducted numerical experiments showed that the flow regime depends both on the angular velocity and on the anomaly parameters A and B. For non-monotone temperature dependence of viscosity the variation of anomaly parameters leads to initiation of various flow regimes. While for the liquid with constant viscosity 2(T) = K min and the liquid with monotonous decreasing viscosity dependence on temperature 3(T) = mine-C(T-Tmax) the flow is steady at the same angular velocity. Parameters of these dependences are selected from a condition of equality of the following integrals Tmax

Tmax 1

Tmin

T dT

Tmax 2

Tmin

T dT

3

T dT .

Tmin

Here 1(T) is the model non-monotone viscosity dependence on temperature (4). The steady flow regimes characterized by stable streamlines are observed only at small angular velocity namely at 17s-1. In Fig. 3 the received steady vortex patterns are shown for different values of anomaly parameters A at angular velocity = 10s-1. We can notice that the parameter A = 0 corresponds to liquid flow with the constant minimum viscosity min. In this case streamlines are symmetric concerning the middle of length of cylinders. Changing of anomaly parameters A and B leads not only to deformation vortex patterns but also, depending on the anomaly parameters, to the appearance of new vortexes.

157

S.F. Khizbullina and S.F. Urmancheev / Procedia IUTAM 8 (2013) 153 – 160 0.02 0.02

0.02 0.02

0.02 0.02

0.02 0.02

0.015 0.015

0.015 0.015

0.015 0.015

0.015 0.015

zz 0.01 0.01

0.01 z 0.01

zz0.010.01

z z0.010.01

0.005 0.005

0.005 0.005

0.005 0.005

0.005 0.005

00 0.005 0.005

0.006 0.006

0.007 0.007

0.008

r

0.009

00 0.005 0.005

0.01 0.01

0.006 0.006

A=0

0.007 0.007

0.008 0.009 0.009 0.008

rr

0.01 0.01

AA == 11

0.006 0.007 0.007 0.008 0.008 0.009 0.009 0.01 0.01 0.006

0.006 0.007 0.0070.008 0.0080.009 0.009 0.010.01 0.006

rr

r r

AA==55

AA= =1010

Fig. 3. Steady vortex patterns for various values anomaly parameter A at angular velocity

= 10s-1

Increasing the angular velocity of the inner cylinder results not only in deformation of streamlines with change of anomaly parameters A and B but also in occurrence of streamlines oscillations concerning some steady condition that is, there is an oscillation flow regime. For example, for angular velocity = 20s-1 in a values range of anomaly parameter A = [0; 20] it was revealed that along with a steady flow regime at values 0.51 A < 2.3 an oscillation flow regime appears. Both quasiperiodic and chaotic fluctuations were found alongside with periodic fluctuations, as well. Pictorial view of oscillation modes gives Fig. 4 where diagram of shear stress versus time for different values of anomaly parameter A are presented. Shear stress was calculated in the center of the computational domain. 0

0

0

-0.05 -0.05

A = 0.4 A = 0.4

-0.1 -0.1

-0.15 -0.15

-0.15 -0.15

A = 2.4 A = 2.4

-0.25 3400 -0.25 3400

3600 3600

A = 0.54 A = 0.54

-0.25 3400 -0.25 3400

-0.1 -0.1

A = 2.14 A = 2.14

-0.15 -0.15

A = 0.78 A = 0.78

-0.2 A = 1.3 -0.2 A = 1.3 3800 time 4000 3800 time 4000

0

-0.05 -0.05

-0.05 -0.05

-0.1 -0.1

-0.2 -0.2

0

0

3600 3600

3800 time 4000 3800 time 4000

Fig. 4. Shear stress versus time at angular velocity

-0.2 -0.2 -0.25 3400 -0.25 3400

A = 1.98 A = 1.98

3600 3600

3800 time 4000 3800 time 4000

= 20s-1

For the analyses of the found oscillation flow regime the following diagrams were plotted: the dependence of variable on time; the corresponding phase portrait in projections of ( , '); the power spectrum F( ) for variable and the three-dimensional phase space of velocities. The power spectrum of variable was defined as follows

F where t and

t ei t dt , are the dimensionless time and the radian frequency accordingly.

158

S.F. Khizbullina and S.F. Urmancheev / Procedia IUTAM 8 (2013) 153 – 160

0

0.002

0.006

F( )

-0.05

0.001

0

(a)

w

0.004

-0.1

' 0

-2E-05

-0.15 0.002

-4E-05

0.3

-0.001

-0.2

v 0.35

-0.25 3400 0

3600

3800

time

4000

-0.002 -0.13 0.002

-0.125

-0.12

-0.115

0.1

0.2

0.3

0.4

0.4

0

-0.005

-0.01

u -0.015

-0.02

-6E-05 -0.025

F( )

-0.05

0.001

0

w

0.004

-0.1

(b)

0 0 0.006

'

-2E-05

0

-0.15 0.002

-4E-05

0.3

-0.001

-0.2

v

0.35

-0.25 3400 0

3600

3800

time

4000

-0.05

-0.125

-0.12

-0.115

0 0 0.012

0.1

0.2

0.3

0.4

0.4

0

-0.005

-0.01

u -0.015

-0.02

0.009

0.006

0.004

-0.15

'

0.003

0

0.006 0

-0.2

-0.004

0.3

0.003

-0.25

v

-0.008

3600

3800

time

4000

-0.05

0.35

-0.21

-0.18

-0.15

-0.12

0 0 0.012

0.1

0.2

0.3

-0.006 0.4 0.008

0.4

0.004

0

u

-0.004

0.009

0.006

0.004

0.003

-0.15

'

0

0.006 0

-0.2

-0.004

0.3

0.003

-0.25

v

-0.008

3600

3800

time

4000

-0.012 -0.24

0.35

-0.21

-0.18

-0.15

-0.12

0 0

0.1

0.2

0.3

0.4

w

-0.003 -0.006

-0.3 3400

-0.008

F( )

0.008

-0.1

(d)

-0.012 -0.24 0.012

w

-0.003

-0.3 3400 0

-6E-05 -0.025

F( )

0.008

-0.1

(c)

-0.002 -0.13 0.012

0.4 0.008

Fig. 5. Dynamical system characteristics in Couette flow for different values of anomaly parameter A, (a) A = 0.55, (b) A = 0.6, (c) A = 1.96, (d) A = 2.1

0.004

0

u

-0.004

-0.008

= 20s-1:

For different values of parameter of temperature viscosity anomaly A on Fig. 5 the following diagrams are presented: in the first column – the shear stress versus time; in the second one – the corresponding twodimensional projections of phase portrait; in the third one – the shear stress power spectrum; in the fourth one – three-dimensional phase space of velocities for various flow regimes. Let us note that at A = 0.6 in a phase-plane portrait there are no skew trajectories and in the power spectrum one observes only the main frequency and its harmonics (see Fig.5, b). Therefore such a flow is periodic. The typical quasiperiodic regime which is characterized by the occurrence of skew phase trajectories and existence of final number of peaks on the frequency spectrum is shown at A = 0.55 (see Fig. 5, a). In addition, quasiperiodic fluctuations in a value range of anomaly parameter 1.9 A < 2.15 chaotic fluctuations which are characterized by a non-repeating character, a strange attractor and a continuous frequency spectrum were found (see Fig. 5, d). Besides the transient regime going into a chaotic one which is characterized by wandering phase trajectories and the main frequency was discovered (see Fig. 5, c). It is visible that the variation of anomaly parameter A leads to the classical scheme of chaos transition through quasiperiodic fluctuations. There is a change of the main tone of fluctuations that is visible on graphics of a power spectrum and there is a chaos in low-frequency area. For anomaly parameter B = 0.0075 the oscillation flow regime arises at values of anomaly parameter 0.53 < A < 1.85 and for B = 0.01 – at values of 0.58 < A < 1.97. As well as for the case B = 0.005 there are qua-

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S.F. Khizbullina and S.F. Urmancheev / Procedia IUTAM 8 (2013) 153 – 160

siperiodic and chaotic fluctuations. And for B = 0.0075 chaotic fluctuations are found in a value range of 0.58 A < 0.93, and for B = 0.01 in a value range of 0.6 A < 0.98. It is also noted that in the value range of anomaly parameter A at which there is an oscillation regime, there can be narrow areas of a steady regime. For example, for the value of anomaly parameter B = 0.005 the steady regime is observed at A = 0.57, and for B = 0.0075 – at A = 1.82.

(a) (a)

0.04 0.04

0.04 0.04

0.04 0.04

0.04 0.04

0.04 0.04

0.04 0.04

0.04 0.04

0.02 0.02

0.02 0.02

0.02 0.02

0.02 0.02

0.02 0.02

0.02 0.02

0.02 0.02

0 0

00

''

''

''

-0.02 -0.02

-0.02 -0.02

-0.02 -0.02

-0.02 -0.02

-0.04 -0.04 -0.4 -0.4

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.04 -0.04 -0.4 -0.4

0.002 z z==0.002

(b) (b)

00

00

' '

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.04 -0.04 -0.4 -0.4

0.0047 zz==0.0047

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.04 -0.04 -0.4 -0.4

0.0074 zz == 0.0074

'

0

00

-0.02

-0.3 -0.3

-0.2 -0.2

-0.1

-0.02 -0.02

-0.04 -0.4

z = 0.01

00

''

''

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.02 -0.02

-0.04 -0.04 -0.4 -0.4

z = 0.0127

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.04 -0.04 -0.4 -0.4

zz == 0.0154 0.0154

0.06 0.06

0.06 0.06

0.06 0.06

0.06 0.06

0.06 0.06

0.06 0.06

0.03 0.03

0.03 0.03

0.03 0.03

0.03 0.03

0.03 0.03

0.03 0.03

0.03 0.03

' 0' 0

' 0' 0

' '00

'' 00

'' 00

'' 00

' '0 0

-0.03 -0.03

-0.03 -0.03

-0.03 -0.03

-0.03 -0.03

-0.03 -0.03

-0.03 -0.03

-0.03 -0.03

-0.09 -0.09 -0.4 -0.4

-0.2 -0.2

-0.1 -0.1

-0.09 -0.09 -0.4 -0.4

-0.06 -0.06

-0.06 -0.06

-0.06 -0.06 -0.3 -0.3

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.09 -0.09 -0.4 -0.4

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.09 -0.09 -0.4 -0.4

-0.06 -0.06 -0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.09 -0.09 -0.4 -0.4

-0.2 -0.2

-0.1 -0.1

Fig. 6. Phase portrait of shear stress at different points of computational domain. r = 0.5(Ro-Ri),

-0.09 -0.09 -0.4 -0.4

-0.1 -0.1

-0.06 -0.06

-0.06 -0.06 -0.3 -0.3

-0.2 -0.2

zz==0.018 0.018

0.06 0.06

-0.06 -0.06

-0.3 -0.3

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

-0.09 -0.09 -0.4 -0.4

-0.3 -0.3

-0.2 -0.2

-0.1 -0.1

= 30s-1: (a) A = 1.9; (b) A = 1.2

For the found oscillation flow regime phase portraits of shear stress are presented at angular velocity = 30s-1 for A = 1.9 (see Fig. 6, a) and A = 1.2 (see Fig. 6, b) in various points on an axis z in section of r = 0.5 (Ro-Ri). It is visible that at z = 0.0074 and z = 0.0127 arise high-peak fluctuations while in other points the fluctuations are small. Therefore the best mixing arises in an average part of a clearance between cylinders, and near end faces of cylinders dead space are formed. 4. Conclusion By methods of computer experiments it is shown that non-monotone viscosity dependence on temperature leads to essential changing of dynamic flow parameters and to initiation of various flow regimes in an annular gap between the coaxial cylinders having a finite length. Along with steady regimes oscillation flow regimes including chaotic ones are also observed. It is established that with changing of anomaly parameter A flow character undergoes several distinct transitions beginning with a periodic flow and ending with a chaotic flow and back from a chaotic flow to a periodic flow. Clearly, non-isothermal Taylor-Couette problem is of current interest as an example of interaction between two different mechanisms - rotation and heating-up. Each of the given mechanisms can have both stabilizing and destabilizing influence on the liquid. In this case, as it was discovered in the course of research, the transitions to the complex flows are observed at small Reynolds numbers. Acknowledgements The present work was supported by the Program of basic research fund of the Russian Academy of Sciences (OE-13), Program of support of young scientists of Presidium of the Russian Academy of Sciences and Program of NSh-834.2012.1.

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References [1] Betchov R, Criminale WO. Stability of parallel flows. N.Y.: Academic Press, 1967; 330 pp. [2] Lin CC The theory of hydrodynamic stability. Cambridge: University Press, 1955. [3] Taylor GI. Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. Roy. Soc. London Ser. A 1923;223:289 – 343. [4] Gershuni GZ, Juhovickiy EM. Convective stability of incompressible liquids. Moscow: Nauka, 1972; 392pp. (in Russian) [5] Urmancheev SF, Kireev VN et al A numerical investigation of anomalously viscous liquid flowing along the heat exchanger channel. Proceedings of the Third International Conference on Multiphase Flow, ICMF’98, Lyon, France, 1998, paper No. 375. [6] Urmancheev SF, Kireev VN. Influence of heat exchange on structure of anomalous-viscous fluid flow. 5th Euromech Fluid Mechanics Conference EFMC’2003, Toulouse, France, August 24 – 28. Book of abstracts, paper No. 261. [7] Urmancheev SF, Kireev VN. Steady flow of a fluid with an anomalous temperature dependence of viscosity. Doklady Physics 2004;49(5):328 – 31. [8] Khizbullina SF, Urmancheev SF, Kireev VN. Numerical investigation of thermostructured liquids. Proceedings of International Conference Flux and structures in Fluids. St. Petersburg, 2007. P. 304 – 307 [9] Smieszek M, Egbers C. Flow structures and stability in Newtonian and non-Newtonian Taylor – Couette flow. J. Physics: Conference 2005. Series 14:72 – 77.