Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Zeitschrift für Naturforschung A

A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board: Fetecau, Corina / Kiefer, Claus


IMPACT FACTOR 2017: 1.414

CiteScore 2018: 1.15

SCImago Journal Rank (SJR) 2018: 0.370
Source Normalized Impact per Paper (SNIP) 2018: 0.431

Online
ISSN
1865-7109
See all formats and pricing
More options …
Volume 70, Issue 10

Issues

Self-aggregation Phenomenon and Stable Flow Conditions in a Two-Phase Flow Through a Minichannel

Grzegorz Górski
  • Faculty of Mechanical Engineering, Bialystok University of Technology, Wiejska 45 C, 15-351 Bialystok, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Grzegorz Litak
  • Corresponding author
  • Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, PL-20-618 Lublin, Poland
  • Laboratoire de Génie Electrique et Ferroélectricité, Institut National des Sciences Appliquées Lyon, 8 rue de la Physique, 69621 Villeurbanne cedex, France
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Romuald Mosdorf
  • Faculty of Mechanical Engineering, Bialystok University of Technology, Wiejska 45 C, 15-351 Bialystok, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Andrzej Rysak
  • Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, PL-20-618 Lublin, Poland
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-08-22 | DOI: https://doi.org/10.1515/zna-2015-0264

Abstract

By increasing a water flow rate of the two-phase (air–water) flow through a minichannel, both the partitioning of air slugs into air bubbles of different sizes and small air bubbles aggregation into larger air bubbles were identified. These phenomena were studied in detail by using the corresponding sequences of light transmission time series recorded with a laser-phototransistor sensor. To distinguish any instabilities in air slugs along with their break-ups and aggregations, the recurrence plots and recurrence quantification analysis were applied.

Keywords: Bubble Dynamics; Recurrence Plots; Recurrence Quantification Analysis; Two-Phase Flow

1 Introduction

Two-phase flow is typical of technical systems where, in addition to a liquid phase, a gas-phase appears. This phenomenon occurs in large boilers producing steam for turbines or in pumps, which operate close to a vapour pressure of a fluid being pumped. Many studies are carried out to investigate the nature of two-phase flow in pipes and minichannels [1–4]. Gas–liquid two-phase flow patterns were visualised and several distinctive flow patterns were identified by Triplett et al. [5] and Serizawa et al. [6].

Wang et al. [7] showed the flow patterns of an oil–gas–water mixture, identifying the flow patterns of air in water by using non-linear analysis methods, including the Hurst and Lyapunov exponents, and correlation dimension. On the other hand, Jin et al. [8] demonstrated that the correlation dimension and Kolmogorov entropy are sufficient to identify the flow patterns. Further, the results of non-linear analysis with respect to temperature and pressure fluctuations in microchannels are discussed by Mosdorf et al. [9].

Methods denote the dynamics of oil–water flow, that is, recurrence plots (RP) and recurrence quantification analysis (RQA), are used for identifying the flow patterns and to assess their complexity. Recently, Faszczewski et al. [10] used the RP to analyse the flow patterns in a vertical minichannel. Timung and Mandal [11] performed the modelling and identification of the flow patterns of gas–liquid flow through a circular microchannel using probabilistic neural networks.

This study focuses on identification and description of flow patterns that occur during the flow of a liquid–gas (water–air) mixture in a horizontal minichannel. More specifically, the RP- and RQA-based approaches are adopted [4, 10] and applied in order to analyse the laser light transmission time series, recorded by a laser-phototransistor sensor during the experiment. In this study, the bifurcation conditions are described, namely a transition from slugs or larger elongated bubble flows to the flow of smaller bubbles and vice versa. From the studies of Triplett et al. [5] and Serizawa et al. [6], where the stationary conditions (for fixed volume flow rate) were used, here, quasistatic conditions were used (by slowly changing the water volume flow rate). The main objective of this study is to identify the flow patterns by RQA parameters.

2 Experimental

During the experiment, it was analysed that the data recorded for different flow patterns (water–air at 21 °C) in a 3-mm-diameter circular channel. Figure 1 shows the schematic design of the experimental stand. Owing to the size of the minichannel, it was necessary to use a special generator of mini bubbles (Fig. 1a) to produce bubbly flow inside the channel. The proportional pressure regulator (metal work regtronic with an accuracy of 1 kPa) was used to maintain the constant overpressure in the supply tank (Fig. 1a) at 50 kPa. Flow patterns were recorded using the Phantom v1610 digital camera at 5000 fps (1280×64 pixels). The amount of air flowing through the minichannel was measured by a laser-phototransistor sensor (Fig. 1a). Data from the sensors were acquired by a acquisition system (Data translation 9804, an accuracy of 1 mV for voltages in the range of −10 V to 10 V), (Fig. 1a) at a sampling rate of 1 kHz. A schematic design of the laser-phototransistor sensor is shown in Figure 1b.

Schematic plot of the experimental stand. (a) Experimental set-up: 1. minichannel with a diameter of 3 mm, 2. pressure sensors (MPX12DP by Freescale Semiconductor Inc., Austin, TX, USA), 3. laser-phototransistor sensor (the laser-IMM Photonics by GmbH, Unterschleißheim, Germany, while the phototransistor-bpyp22 by Cemi S.A., Warsaw, Poland), 4. Phantom v. 1610 camera (by Vision Research, Inc. Wayne, NJ, USA), 5. lighting, 6. pumps (air or water), 7. flow meters (the water flow meter by Digmesa AG, Ipsach, Switzerland while the air flow meter by Medson s.c. Serv’Instrumentation ZI Broteau Nord Irigny, France), 8. mini bubbles generator (manufactured by Bialystok University of Technology, Bialystok, Poland), 9. air tank, 10. automatic valve (by Metal Work S.p.a. Concesio, Italy) to maintain a constant pressure in the tank 9, 11. data acquisition station (DT9800 by Data Translation, Inc., Marlborough, MA, USA), 12. computer, 13. water tank, 14. air tank. The total length of the minichannel was 1 m, while the laser was located 10 cm from the inlet end. (b) Laser-phototransistor sensor.
Figure 1:

Schematic plot of the experimental stand. (a) Experimental set-up: 1. minichannel with a diameter of 3 mm, 2. pressure sensors (MPX12DP by Freescale Semiconductor Inc., Austin, TX, USA), 3. laser-phototransistor sensor (the laser-IMM Photonics by GmbH, Unterschleißheim, Germany, while the phototransistor-bpyp22 by Cemi S.A., Warsaw, Poland), 4. Phantom v. 1610 camera (by Vision Research, Inc. Wayne, NJ, USA), 5. lighting, 6. pumps (air or water), 7. flow meters (the water flow meter by Digmesa AG, Ipsach, Switzerland while the air flow meter by Medson s.c. Serv’Instrumentation ZI Broteau Nord Irigny, France), 8. mini bubbles generator (manufactured by Bialystok University of Technology, Bialystok, Poland), 9. air tank, 10. automatic valve (by Metal Work S.p.a. Concesio, Italy) to maintain a constant pressure in the tank 9, 11. data acquisition station (DT9800 by Data Translation, Inc., Marlborough, MA, USA), 12. computer, 13. water tank, 14. air tank. The total length of the minichannel was 1 m, while the laser was located 10 cm from the inlet end. (b) Laser-phototransistor sensor.

Figure 2 illustrates the evolution of the flow with change in the water volume rate flow q, ranging from 0.021 l/min to 0.211 l/min. One can observe different patterns of slugs and bubbles aggregations during the flow. From the top (a), long slugs (extremely elongated large air bubbles) are observed, and then, with increase in q, the smaller slugs are noted, and finally, in cases (e–f), the slugs become very small, but they still have a substantially large length compared to the diameter of the minichannel. At the same time, small bubbles can also be observed in case (f).

At this stage, the pressure inside the pipe is high enough to destroy the slugs. Naturally, the new mechanism, combined with the different speeds of mixed size bubbles, leads to the new patterns. The longer bubbles, such as slugs, became more stable again [cases (j)–(o)]. They are separated by a larger volume of water and mediated by fairly small air bubbles. For larger q, the defragmentation of elongated bubbles to smaller ones and the pattern (u) that resembles the previous one (f) due to the presence of short slugs are observed. The differences in the new slugs formations are in a separation volume, which is covered by water with a large number of small bubbles.

Photographs of the selected flow cases at a decreasing water volume flow rate q[l/min]: (a) 0.021; (b) 0.042; (c) 0.042; (d) 0.063; (e) 0.084; (f) 0.1048; (g) 0.1053; (h) 0.116; (i) 0.126; (j) 0.126; (k) 0.147; (l) 0.137; (m) 0.158; (n) 0.158; (o) 0.158; (p) 0.169; (q) 0.179; (r) 0.179; (s) 0.190; (t) 0.190; (u) 0.211, respectively. The air flow rate was constant qa=0.1 l/min.
Figure 2:

Photographs of the selected flow cases at a decreasing water volume flow rate q[l/min]: (a) 0.021; (b) 0.042; (c) 0.042; (d) 0.063; (e) 0.084; (f) 0.1048; (g) 0.1053; (h) 0.116; (i) 0.126; (j) 0.126; (k) 0.147; (l) 0.137; (m) 0.158; (n) 0.158; (o) 0.158; (p) 0.169; (q) 0.179; (r) 0.179; (s) 0.190; (t) 0.190; (u) 0.211, respectively. The air flow rate was constant qa=0.1 l/min.

To classify these changes, more systematic studies were carried out using the laser-phototransitor sensor. More specifically, the time series of laser light transmission was recorded during the particular flows. Six representative cases are given in Figure 3a–f (upper panels). It can be noted that the laser light transmission increases with increase inwater volume rate. The recurrence statistical analysis was carried out after measurements, and the results were presented in the subsequent section. In these calculations, the Cross Recurrence Plot Toolbox for Matlab was used [12].

Normalised time series of the laser light transmission x′ versus i in sampling units (x′=(x − xav)/σ(x)), where x is the original signal measured in volts, xav denotes its average value and σ(x) its standard deviation, respectively. The corresponding recurrence plots for the cases 3a–f (with q=0.021, 0.042, 0.105, 0.137, 0.179, 0.211 l/min) were taken from the following regions in Fig. 2a,b,g,l,q,u, respectively. The embedding parameter dimension was applied as m=7, delay in the sampling units δi=7, while RQA threshold ϵ=3σ, where σ denotes the standard deviation of the corresponding x series. The sampling period was Δt=0.1 ms.
Figure 3:

Normalised time series of the laser light transmission x′ versus i in sampling units (x′=(xxav)/σ(x)), where x is the original signal measured in volts, xav denotes its average value and σ(x) its standard deviation, respectively. The corresponding recurrence plots for the cases 3a–f (with q=0.021, 0.042, 0.105, 0.137, 0.179, 0.211 l/min) were taken from the following regions in Fig. 2a,b,g,l,q,u, respectively. The embedding parameter dimension was applied as m=7, delay in the sampling units δi=7, while RQA threshold ϵ=3σ, where σ denotes the standard deviation of the corresponding x series. The sampling period was Δt=0.1 ms.

3 Recurrence Plots and Recurrence Quantification Analysis

The dynamics of the underlying phenomena can be investigated by means of recurrences that are calculated for each visited state of the reconstructed trajectory. This method was developed by Eckmann [13] and extended by Webber [14], Casdagli [15] later by Marwan et al. [16, 17] and others. This approach was used for both short deterministic and noise-affected experimental data [18, 19].

Two points on a trajectory are marked as neighbours if they are close enough to each other. This can be expressed by the distance matrix R with its element Rijϵ given by [13]:

Rijϵ=Θ(ϵxixj), (1)(1)

where ϵ is the threshold value and Θ(x) denotes the Heaviside function. The number of recurrence points depends on both the underlying dynamics and the cosing the threshold value. A standard technique for approximations is that the threshold value should not be higher than a few percentage of the total number of points [17].

According to Takens [20] and the laser light transmission x time series, the following vectors x in the embedding space are defined:

x(t)=(x(t),x(tδiΔt),x(t2δiΔt),,x(t(m1)δiΔt), (2)(2)

where Δt=0.1 ms is a sampling period. The phase space reconstruction is performed by using the standard methods of the first minimum of average mutual information and the nodal fraction of false neighbours [21–23]. The same embedding methods are used for comparison with the assumption of the smallest time delay δi and then the highest dimension m. In this way, the system could be described the dimension m=7 and the delay δi=7.

Figure 3a–f (in the lower panels) compare the recurrence plots (RP) of the threshold ϵ = 3σ, where σ denotes the standard deviation of a particular time series depicted in the upper panels. It is noted that one criterion is used to all the cases in order to obtain 7 % recurrences in the minimal case. To estimate the distances, the Euclidean distance in the reconstructed phase space of the laser light transmission time series for six representative cases, obtained from the flows shown in Figure 2, was used. The horizontal and vertical axes represent the time instants i and j to which the distance formula (1) is applied.

Here, some interesting observations can be made. The clear checkerboard structure, which can be seen in most of figures, implies two states of intermittency [24, 25] in the light transmission time series; however, its type can be assessed with changing q. Here, these intermittences are mostly related to switching between better and worse transparency depending on the presence of slugs and/or bubbles. It is worth noting that in the presence of very short slugs mediated by the small bubbles (Fig. 3f) the laser light is strongly scattered by moving surface of the slugs and bubbles resulting in fast oscillations of x.

Figure 3c shows a large number of short slugs rarely mediated by air bubbles. Figure 3c,e, and f illustrate the structure smearing that could correspond to the larger complexity in bubbles and slugs sizes, and, consequently, to their speeds. On the other hand, more regular points distributions are visible in Figure 3a,b, and d. These cases correspond to the observations of longer air slugs and elongated bubbles.

Webber and Zbilut [14] and later Marwan et al. [16, 17, 26] developed the recurrence quantification analysis (RQA) for recurrence plots. The RQA analysis includes the recurrence rate variable, RR, which expresses the system’s ability to return to the neighbourhood of previous state and has the following definition:

RR=1N2i,jiNRijϵ. (3)(3)

Furthermore, the RQA was used to identify diagonal lines through their maximal lengths, Lmax. The RQA provides the probability p(l) or p(v) of line distribution according to their lengths l or v (for diagonal and vertical lines). They are calculated as follows:

p(z)=Pϵ(z)z=zminNzPϵ(z), (4)(4)

where z=l or v depend on diagonal or vertical structures in a specific recurrence plot. Pϵ(z) denotes the histogram of z lengths and a fixed value of ϵ, Nz is the number of diagonal or vertical lines (depending on the definition of z). Measures, such as determinism DET, laminarity LAM, trapping time TT, and average length L, are based on probabilities Pϵ(z)

DET=l=lminNllPϵ(l)l=1NllPϵ(l), (5)(5)

LAM=v=vminNvvPϵ(v)v=1NvvPϵ(v), (6)(6)

L=l=lminNllPϵ(l)l=lminNlPϵ(l), (7)(7)

TT=v=vminNvvPϵ(v)v=vminNvPϵ(v), (8)(8)

On the other hand, the transitivity, TRAN, is a network topological parameter based on the distance matrix elements Rϵ:

TRAN=i=1Nj,k=1NRijϵRjkϵRkiϵi,j,k=1NRijϵRkjϵ. (9)(9)

These parameters were studied systematically with change in water volume rate flow q. The results are shown in Figure 4. It is noted that these figures show a non-monotonic behaviour of the studied flow phenomenon, namely they confirm the effect of resonating self-organisation (Figs. 2l and 3d for q=0.137 l/min). This is related to the values of q for which the elongated large bubbles are stable. They occur together with a low number of residual small bubbles. The recurrence rate, RR, (Fig. 1a) shows that the system returns to the same state most frequently at q ≈ 0.06 and 0.014, whereas between these value, the system behaves differently. It is concluded that for q=0.06 and 0.014, the fluctuations of laser transmitions appear between a small number of states, while for q ≈ 0.10, the number of states available significantly increases. This conclusion is supported by determinism, DET, and laminarity, LAM. At q ≈ 0.06, we observe the minimum, which indicates that the system becomes more random and unpredictable. The maxima 1/Lmax correspond to the dynamical divergence that increases for three values of q. The maxima 1/Lmax coincide with the minima in RR conforming the instabilities by increasing available states. The occurrence of intermittency is more probable for reaching such maxima in the dynamical divergence (1/Lmax). Particular attention should be payed on the trapping time parameter, TT. This measure signals the probability that the system will last in a single state. Once the clear increases in TT are observed, the corresponding cases analysed by the RP (Fig. 3a and d) are represented by larger and clear checkerboard. Interestingly, this is the measure that directly indicates about the shape stability of air slugs or elongated bubbles. This stability strongly depends on the change of water flow rate q (Fig. 4e). Then, the bubbles merge into one large elongated slug upon increase in the water flow rate. Finally, transitivity is a topological measure of RP network itself. This can be linked to spatial and temporal properties of the system. Surprisingly, the TRAN measure changes in a similar way to the trapping time, TT. Nonetheless, it should be noticed that a clear checkerboard formation leads to isolated square islands signalled by minimum in transitivity. Simultaneously, it is also possible to measure the cluster size of a square island. The smaller the size of square, the lower the value of transitivity. Consequently, for unstable slug and elongated bubble flows, the squares in the RP are irregular and of smaller sizes (Fig. 3c,e, and f), which is signalled by minimal TRAN (Fig. 4f). The multiple peaks in patterns (Fig. 4) are due to aggregation of small bubbles into larger bubbles and slugs.

RQA measures versus water volume flow rate q: (a) recurrence rate – RR; (b) determinism – DET; (c) laminarity – LAM; (d) divergence – DIV=1/Lmax; (e) trapping time – TT; (f) transitivity (causality) – TRAN, respectively.
Figure 4:

RQA measures versus water volume flow rate q: (a) recurrence rate – RR; (b) determinism – DET; (c) laminarity – LAM; (d) divergence – DIV=1/Lmax; (e) trapping time – TT; (f) transitivity (causality) – TRAN, respectively.

4 Conclusions

We observed multiple air slugs flow to bubbles flow bifurcations in the air–water flow patterns. The changes in pattern were caused by the instability of longer air slugs, which was increased for various values of water flow rate q. Using digital camera, the partitioning of air slugs to bubbles of different sizes with increasing q was identified. These patterns were also recorded by a simultaneous laser light transmission. The RQA analysis was applied to the transmission light time series. It was confirmed that the bifurcations significantly increases in all examined measures (RR, DET, TRAN, LAM, TT, and 1/Lmax). This information was used to automate identification of air slug or elongated bubble occurrence, their average length and stability. Taking into account the fact that the air flow rate is stable, the main problem analysed in the study is the observation of merging of multiple small air bubbles into air slugs or elongated bubbles. This merging can be associated with a phenomenon of self-organisation that can occur at certain water flow rates.

Acknowledgments

This research was funded by the Polish National Science Centre under the decision number DEC-2013/09/B/ST8/02850.

Nomenclature

q

Water volume flow rate

qa

Air volume flow rate

Recurrence plot

RP

Recurrence quantification analysis

RQA

x

Light transmission measured in [V]

x

Vector representation of x in the reconstructed (embedding) phase space

t

Time

Δt=0.1 ms

Sampling time

δi

Time delay in the sampling units

m

Embedding dimension

ϵ

Threshold value

σ

Standard deviation

R, Rϵ

Recurrence matrix

Θ

Heaviside function

RR

Recurrence rate

l or v

Lengths for diagonal and vertical lines

p(l) or p(v)

Probabilities of diagonal and vertical line distribution according to their lengths

DET

Determinism

LAM

Laminarity

DIV=1/Lmax

Divergence, inverse of the maximal diagonal line length

TT

Trapping time

TRAN

Transitivity

References

  • [1]

    L. Zhao and K. S. Rezkallah, Int. J. Multiphase Flow 19, 751 (1993).CrossrefGoogle Scholar

  • [2]

    S. Wongwises and M. Pipathattakul, Exp. Therm. Fluid. Sci. 30, 345 (2006).CrossrefGoogle Scholar

  • [3]

    L. Chen, Y. S. Tian, and T. G. Karayiannis, Int. J. Heat. Mass Tran. 49, 4220 (2006).Google Scholar

  • [4]

    Y. B. Zong, N. D. Jin, Z. Y. Wang, Z. K. Gao, and C. Wang, Int. J. Multiphase. Flow 36, 166 (2010).CrossrefGoogle Scholar

  • [5]

    K. A. Triplett, S. M. Ghiaasiaan, S. I. Abdel-Khalik, and D. L. Sadowski, Int. J. Multiphase. Flow 25, 377 (1999).CrossrefGoogle Scholar

  • [6]

    A. Serizawa, Z. Feng, and Z. Kawara, Exp. Therm. Fluid. Sci. 26, 703 (2002).CrossrefGoogle Scholar

  • [7]

    S. F. Wang, R. Mosdorf, and M. Shoji, Int. J. Heat. Mass. 46, 1519 (2003).Google Scholar

  • [8]

    N. D. Jin, X. B. Nie, Y. Y. Ren, and X. B. Liu, Flow Meas. Instrum. 14, 169 (2003).CrossrefGoogle Scholar

  • [9]

    R. Mosdorf, P. Cheng, H. Y. Wu, and M. Shoji, Int. J. Heat. Mass Tran. 48, 4667 (2005).CrossrefGoogle Scholar

  • [10]

    M. Faszczewski, G. Gorski, and R. Mosdorf, Acta Mach. Autom. 6, 31 (2012).Google Scholar

  • [11]

    S. Timung, T. K. Mandal, Appl. Soft. Comput. 13, 1674 (2013).Google Scholar

  • [12]

    N. Marwan, Cross Recurrence Plot Toolbox for Matlab, V 5.15, Release 28.10, http://tocsy.pik-potsdam.de (2014).

  • [13]

    J. P. Eckmann, S. O. Kamphorst, and D. Ruelle, Europhys. Lett. 5, 973 (1987).Google Scholar

  • [14]

    C. L. Webber, Jr. and J. P. Zbilut, J. Appl. Physiol. 76, 965 (1994).CrossrefGoogle Scholar

  • [15]

    M. Casdagli, J. R. Stat. Soc. 54, 303 (1992).Google Scholar

  • [16]

    N. Marwan, Encounters with Neighbours: Current Development of Concepts Based on Recurrence Plots and their Applications, PhD Thesis, Universitaet Potsdam, Potsdam 2003.Google Scholar

  • [17]

    N. Marwan, M. C. Romano, M. Thiel, and J. Kurths, Phys. Rep. 438, 237 (2007).Google Scholar

  • [18]

    G. Litak, M. Wiercigroch, B. W. Horton, and X. Xu, ZAMM. 90, 33 (2010).Google Scholar

  • [19]

    L. P. Yang, S. L. Ding, G. Litak, E. Z. Song, and X. Z. Ma, Chaos. 25, 013105 (2015).CrossrefGoogle Scholar

  • [20]

    F. Takens, in: Dynamical Systems and Turbulence, Lecture Notes in Mathematics (Eds. D. A. Rand, L.-S. Young), Springer, Berlin 1981, Vol. 898, pp. 366–381.Google Scholar

  • [21]

    A. M. Fraser and H. L. Swinney, Phys. Rev. A 33, 1134 (1986).Google Scholar

  • [22]

    M. B. Kennel, R. Brown, and H. D. I. Abarbanel, Phys. Rev. A 45, 3403 (1992).CrossrefGoogle Scholar

  • [23]

    H. Kantz and T. Schreiber, Non-linear Time Series Analysis, Cambridge University Press, Cambridge 1997.Google Scholar

  • [24]

    K. Klimazewska and J. J. Zebrowski, Phys. Rev. E 80, 026214 (2009).Google Scholar

  • [25]

    G. Litak, A. Syta, B. F. Yao, and G. X. Li, J. Theor. Appl. Mech. 47, 55 (2009).Google Scholar

  • [26]

    N. Marwan, J. F. Donges, Y. Zou, R. V. Donner, and J. Kurths, Phys. Lett. A 373, 4246 (2009).Google Scholar

About the article

Corresponding author: Grzegorz Litak, Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, PL-20-618 Lublin, Poland; and Laboratoire de Génie Electrique et Ferroélectricité, Institut National des Sciences Appliquées Lyon, 8 rue de la Physique, 69621 Villeurbanne cedex, France, E-mail: g.litak@pollub.pl


Received: 2015-03-31

Accepted: 2015-07-15

Published Online: 2015-08-22

Published in Print: 2015-10-01


Citation Information: Zeitschrift für Naturforschung A, Volume 70, Issue 10, Pages 843–849, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784, DOI: https://doi.org/10.1515/zna-2015-0264.

Export Citation

©2015 by De Gruyter.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Grzegorz Gorski, Grzegorz Litak, Romuald Mosdorf, and Andrzej Rysak
Zeitschrift für Naturforschung A, 2017, Volume 72, Number 6, Page 535
[2]
Romuald Mosdorf, Paweł Dzienis, and Grzegorz Litak
Meccanica, 2017, Volume 52, Number 11-12, Page 2641
[3]
Grzegorz Górski, Grzegorz Litak, Romuald Mosdorf, and Andrzej Rysak
The European Physical Journal Plus, 2016, Volume 131, Number 4

Comments (0)

Please log in or register to comment.
Log in