# 2016 • no. 4

# TABLE OF CONTENTS

**Arustamyan D.A., Chebotarev S.N., Lunin L.S., Lunina M.L., Kazakova A.E., Pashchenko A.S.**

Modelling of the functional characteristics of photoelectric converters based on multi-component solid solution Al_{x}Ga_{1-x}As, received from a liquid phase epitaxy

**Babenko I.D., Galutskiy V.V., Ivashko S.S., Stroganova E.V.**

Temperature dependence of spectral-kinetic properties of crystals Er_{3+}, Yb_{3+}:LiNbO^{3}

**Bocharova O.V., Sedov A.V., Andjikovich I.E., Kalinchuk V.V.**

On efficient method of signal processing in problems of low-frequency defetoscopy

**Gorbacheva E.V., Kalaydin E.N.**

Electrohydrodynamics of conductive micro- and nanofilms under DC electric field

**Gusev A.A., Shvetsova N.A., Voloshin A.E., Yakovenko N.A.**

The study of the applicability of the tool for fuzzy inductive reasoning to the problem of solar activity prediction

**Demekhin E.A., Morshneva I.V., Kalaydin E.N.**

Mathematical modeling of electrodynamics of bipolar membranes with water dissociation and chemical reaction of ionizable groups

**Drobotenko M.I., Volchenko N.N., Samkov A.A., Svidlov A.A.**

Mathematical modeling of processes in the microbial fuel cell membrane

**Kamalyan S.R., Kamalyan R.Z.**

To the dynamics of saturated soils

**Kovalenko A.V.**

Mathematical classification of electroconvection in electro-membrane systems

**Kochergin V.S., Kochergin S.V.**

The parameters of identification algorithm of instant point source pollution in the Azov Sea on the basis of the method of adjoint equations

**Rubtsov S.E., Pavlova A.V.**

To the study of the mixed dynamic problems for a limited volume of fluid on an elastic foundation

**Syromyatnikov P.V.**

The simulation of surface disturbances elastic semi-infinite medium caused by moving oscillating source

**Fomenko S.I., Alexandrov A.A.**

Wavefields and band-gaps in layered piezoelectric phononic crystals

## ABSTRACTS

*Subjects*: physics

**Arustamyan D.A.**^{1}, **Chebotarev S.N.**^{1,2}, **Lunin L.S.**^{1,2}, **Lunina M.L.**^{2}, **Kazakova A.E.**^{1}, **Pashchenko A.S.**^{2}

**Modelling of the functional characteristics of photoelectric converters based on multi-component solid solution Al _{x}Ga_{1-x}As, received from a liquid phase epitaxy**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 5-12.

The most important technological problems of solar energetics are reduction of production costs of photoelectric converters and increase their efficiency. This paper presented a theoretical and experimental study of solar cells based on solid solution Al_{x}Ga_{1-x}As, received from a liquid phase epitaxy.

The investigated structure was grown by means of liquid phase epitaxy. The GaAs plate was used as a substrate. The Al_{0.28}Ga_{0.72}As base layer was grown according to the cascade method of refrigeration. Thin emitter layers Al_{0.1}Ga_{0.9}As and a wide bandgap window Al_{0.9}Ga_{0.1}As were produced at lower temperatures (600 °C), from a thin layer of molten aluminum gallium arsenic (0.5 mm thick).
The modelling was performed by using AFORS-HET program. During the simulation the following parameters of semiconductor layers were changed: composition of the ternary solution, thickness of the layers, doping level of the layers. Characteristics of the layers for different Al_{x}Ga_{1-x}As compositions were calculated by means of MATLAB 7 application program package. No defects were supposed to be present in the structure, the temperature of the photoelectric converter and the environment was equal to 300 K, multiplicity of radiation was equal to 1, and the conditions of sunlight lighting were equal to AM 1,5.

The article shows that an optimum composition of the solid solution for the base layer Al_{0.28}Ga_{0.72}As, is the following: the thickness of layer is 100 microns and the concentration of the acceptor impurity is 10^{19} cm^{-3}. For the emitter layer Al_{0.1}Ga_{0.9}As optimum layer thickness is 50 nm, optimum concentration of the donor impurity is 10^{16} cm^{-3} and the wide bandgap window Al_{0.9}Ga_{0.1}As optimum thickness is 50 nm, optimum concentration of impurity is 2.5$\cdot$ 10^{17} cm^{-3}. The conversion efficiency in this case can reach 40.99 %.

*Keywords*: solar energetics, photoelectric converters, liquid-phase epitaxy, modeling.

^{1}Platov South-Russian State Polytechnic University (NPI), Novocherkassk, 346428, Russia

^{2}Southern Scientific Center RAS, Rostov-on-Don, 344006, Russia

Corresponding author’s e-mail: galeriandavid@gmail.com

- Andreev V.M., Khvostikov V.P., Larionov V.R., Rumiantcev V.D., Paleeva E.V., Shvarts M.Z. Vysokoehffektivnye koncentratornye (2500 solnc) AlGaAs/GaAs-solnechnye ehlementy [Highly efficient concentrator (2500 suns) AlGaAs/GaAs solar cells].
*Fizika i tekhnika poluprovodnikov*[Physics and technics of Semiconductor], 1999, vol. 33, no. 9, pp. 1070-1072. (In Russian) - Bogatov N.M., Matveyakin M.P., Pershin N.V., Rodomanov R.R. Opredelenie vremeni zahvata neravnovesnogo poverhnostnogo zaryada v poluprovodnikovyh strukturah po spadu fotoehds [Determination of the time the capture of nonequilibrium surface charge in semiconductor structures from the decay of the photo-emf].
*Ehkologicheskij vestnik nauchnyh centrov Chernomorskogo ehkonomicheskogo sotrudnichestva*[Ecological Bulletin of research centers of the Black Sea economic cooperation], 2008, no. 2, pp. 57-61. (In Russian) - Bogatov N.M., Korneev A.I., Matveyakin M.P., Rodomanov R.R. Vliyanie neravnovesnogo zaryada granicy SiO2-Si na nestacionarnost’ spektral’noj harakteristiki solnechnyh ehlementov s submikronnym r-n-perekhodom [Influence of nonequilibrium charge SiO2-Si boundaries for the nonstationarity of the spectral characteristics of solar cells with submicron p-n-junction].
*Ehkologicheskij vestnik nauchnyh centrov Chernomorskogo ehkonomicheskogo sotrudnichestva*[Ecological Bulletin of research centers of the Black Sea economic cooperation], 2006, no. 4, pp. 63-67. (In Russian) - Vlasov A.S., Khvostikov V.P., Karlina L.B., Sorokina S.V., Potapovich N.S., Shvarts M.Z., Timoshina N.H., Lantratov V.M., Mintairov S.A., Kalyuzhnyj N.A., Maruhina E.P., Andreev V.M. Koncentratornye fotoehlektricheskie moduli so spektral’nym rasshchepleniem sveta s solnechnymi ehlementami na osnove struktur AlGaAs/GaAs/GaSb i GaInP/InGaAs(P) [Concentrator photovoltaic modules with a spectral splitting of the light with solar cell structures based on AlGaAs/GaAs/GaSb and GaInP/InGaAs(P)].
*Zhurnal tekhnicheskoj fiziki*[J. of Technical Physics], 2013, vol. 83, no. 7, pp. 106-110. (In Russian) - Lozovskij V.N., Lozovskij S.V., CHebotarev S.N. Issledovanie kraevogo temperaturnogo ehffekta pri zonnoj sublimacionnoj perekristallizacii [A study of the boundary temperature effect in zone sublimation recrystallization].
*Izvestiya vysshih uchebnyh zavedenij. Severo-Kavkazskij region. Seriya: Tekhnicheskie nauki*[Proc. of the higher educational institutions. North-Caucasian region. Series: Engineering], 2007, no. 5, pp. 52-56. (In Russian) - Lozovskij V.N., Lozovskij S.V., Chebotarev S.N., Irkha V.A. Osazhdenie tugoplavkih metallov na rel’efnye podlozhki metodom zonnoj sublimacionnoj perekristallizacii [The deposition of refractory metals on the relief substrate by zone sublimation recrystallization].
*Izvestiya vysshih uchebnyh zavedenij. Severo-Kavkazskij region. Seriya: Tekhnicheskie nauki*[Proceedings of the higher educational institutions. North-Caucasian region. Series: Engineering], 2007, no. 4, pp. 68-70. (In Russian) - Lunin L.S., Sysoev I.A., Chebotarev S.N., Pashchenko A.S. Formirovanie kvantovyh tochek InAs na podlozhkah GaAs metodom ionno-luchevogo osazhdeniya [Formation of InAs quantum dots on GaAs substrates by ion beam deposition].
*Nauka Yuga Rossii*[Science of Southern Russia], 2010, vol. 6, no. 4, pp. 46-49. (In Russian) - Lozovskij V.N., Irkha V.A., Chebotarev S.N. Metodika polucheniya nanometok i ih primenenie dlya pozicionirovaniya v skaniruyushchej zondovoj mikroskopii [A method of production of nano-markers and their use for positioning in scanning probe microscopy].
*Zavodskaya laboratoriya. Diagnostika materialov*[Factory Laboratory. Diagnosis materials], 2012, vol. 78, no. 9, pp. 33-36. (In Russian) - Gudovskikh A.S., Kaluzhniy N.A., Lantratov V.M., Mintairov S.A., Shvarts M.Z., Andreev V.M. Numerical modelling of GaInP solar cells with AlInP and AlGaAs windows.
*Thin Solid Films*, 2008, vol. 516, iss. 20, pp. 6739-6743. - Chebotarev S.N., Pashchenko A.S., Lunina M.L. Modelirovanie zavisimostej funkcional’nyh harakteristik kremnievyh solnechnyh ehlementov, poluchennyh metodom ionno-luchevogo osazhdeniya, ot tolshchiny i urovnya legirovaniya frontal’nogo sloya [Modeling dependencies functional characteristics of silicon solar cells, grown by ion-beam deposition, on the thickness and doping level of the front layer].
*Nauka Yuga Rossii*[Sc. of Southern Russia], 2011, vol. 7, no. 4, pp. 25-30. (In Russian) - Chebotarev S.N., Pashchenko A.S., Lunin L.S., Irkha V.A. Modelirovanie kremnievyh tonkoplenochnyh trekhkaskadnyh solnechnyh ehlementov α–Si:H/μC–Si:O/μC–Si:H [Modeling of the thin-film three cascade silicon solar cell α–Si:H/μC–Si:O/μC–Si:H].
*Nauka Yuga Rossii*[Science of Southern Russia], 2013, vol. 9. no. 4, pp. 18-25. (In Russian) - Arustamyan D.A., Chebotarev S.N., Lunina M.L., Sysoev I.A., Pashchenko A.S., Kazakova A.E., Yatsenko A.N. Zavisimost’ harakteristik solnechnyh ehlementov na osnove AlGaAs ot tolshchiny i urovnya legirovaniya bazy [The dependence of the characteristics of the solar cells based on the AlGaAs on the thickness and doping level of the base].
*Vestnik Severo-Kavkazskogo federal’nogo universiteta*[Bulletin of the North Caucasus Federal University], 2016, no. 4 (55), pp. 7-12. (In Russian) - Musalinov S.B., Bychkov I.V., Anzulevich A.P., Gudovskikh A.S. Modelirovanie dvuh i trekhslojnyh prosvetlyayushchih pokrytij dlya geterostrukturnyh solnechnyh ehlementov [Modeling of two and three-layer antireflection coatings for heterostructure solar cells].
*Vestnik Chelyabinskogo gosudarstvennogo universiteta*[Bulletin of the Chelyabinsk State University], 2015, no. 7 (362), pp. 60-63. (In Russian) - Goldberg Yu.A.
*Handbook Series on Semiconductor Parameters*, vol. 2. World Scientific, London, 1999, pp. 1-36. - Arsentev I.N., Lunin L.S., Kuznecov V.V., Ratushnyj V.I., Shishkov M.V., Ulin B.P., Khristenko A.E., Smolin A.Yu. Poluchenie tverdyh rastvorov gainasp na podlozhkah poristogo fosfida indiya metodom zhidkofaznoj ehpitaksii [Formation of the solid solutions GaInAsP on substrates of porous indium phosphide LPE].
*Poverhnost’. Rentgenovskie, sinhrotronnye i nejtronnye issledovaniya*[Surface. X-ray, synchrotron and neutron investigations], 2006, no. 1, pp. 14-20. (In Russian)

*Subjects*: physics

**Babenko I.D.**^{1}, **Galutskiy V.V.**^{1}, **Ivashko S.S.**^{1}, **Stroganova E.V.**^{1}

**Temperature dependence of spectral-kinetic properties of crystals Er _{3+}, Yb_{3+}:LiNbO^{3}**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 13-19.

The subject of this study was to explore the temperature dependence of spectral-kinetic properties of gradient-activated crystals Er^{3+},Yb^{3+}:LiNbO^{3} and Er^{3+}:LiNbO_{3} with optical concentration profiles of the impurities.

The algorithm of the registration mode: 1) set the initial wavelength registration (λ_{1}); 2) inclusion of heating of a sample while running the check in automatic mode kinetic luminescence on the first channel of the oscilloscope. Check the array temperature data is up to 150 °C; 3) displacement monochromator for λ_{2} and repeat step 2) algorithm. This approach required repeated heating and cooling of the sample, but allowed to save the time of the experiment. You can reduce the number of heating cycles of the investigated sample by omitting stabilization of the parameters (wavelength or temperature), because the assumed lifetime of the luminescence is much less than the time of warm-up or rewinding of a monochromator. With this approach the registration scheme includes three signal channels in the oscilloscope: the intensity, temperature and the channel through which wavelength of the monochromator is transmitted. In this case, the number of cycles of heating/cooling the sample in the experiments was within the order of a dozen.

It was found that the luminescence intensity increased by 15-20% in the gradient-activated crystals Er^{3},Yb^{3+}:LiNbO_{3} in the temperature range from 25 °C to 150 °C and decreased in the intensity of 1,5 μm luminescence 30% in the gradient-activated crystals Er^{3+}:LiNbO_{3} in the same temperature range.

*Keywords*: erbium, ytterbium, gradient crystal, lithium niobate.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: stroganova@phys.kubsu.ru

- Zhang Peixiong, Yin Jigang, Zhang LianHan, Liu Youchen, Hong Jiaqi, Ning Kaijie, Chen Zhi, Wang Xiangyong, Shi Chunjun, Hang Yin. Efficient enhanced 1.54 μm emission in Er/Yb: LiNbO
_{3}crystal codoped with Mg^{2+}ions.*Optical Materials*, 2014, vol. 36, pp. 1986-1990. - Denker B., Galagan B., Osiko V., Sverchkov S., Balbashov A.M., Hellstrom J.E., Pasiskevicius V, Laurell F. Yb
^{3+},Er^{3+}:YAG at high temperatures: Energy transfer and spectroscopic properties.*Optics Communications*, 2007, vol. 271, pp. 142-147. - Galagan B.I., Denker B.I., Osiko V.V., Sverchkov S.E. Spektral’no-kineticheskie svoystva kristallov Er
^{3+}, b^{3+}:Y_{3}Al_{5}O_{12}pri vysokikh temperaturakh [Spectral and kinetic properties of Er^{3+}, Yb^{3+}:Yb_{3}Al_{5}O_{12}crystals at high temperatures].*Kvantovaya elektronika*[Quantum Electronic], 2006, vol. 36, pp. 595-600. (In Russian) - Galagan B.I., Denker B.I., Osiko V.V., Sverchkov S.E. Effektivnost’ zaseleniya urovnya
^{4}I_{13/2}iona Er$^{3+}$ i vozmozhnost’ generatsii izlucheniya s dlinoy volny 1,5 mkm v IAG:Yb, Er pri vysokikh temperaturakh [Efficiency of population of the^{4}I_{13/2}level of the Er$^{3+}$ ion and the possibility of lasing at 1.5 μm in Yb, Er:YAG at high temperatures].*Kvantovaya elektronika*[Quantum Electronics], 2007, vol. 37, pp. 971-973. (In Russian) - Stroganova E.V., Galucci V.V., Nalbantov N.N., Kozin A.S. Spectral and luminescent characteristics of the gradient activated lithium niobate crystals with concentration profiles of ions Yb
^{3+}and Er^{3+}.*Sbornik materialov mezhdunarodnoy nauchnoy konferentsii SibOptika-2016*[Proc. of Inter-Expo Geo-Siberia Conf.], 2016, no. 2, pp. 9-15. - Galutskiy V.V., Vatlina M.I., Stroganova E.V. Growth of single crystal with a gradient of concentration of impurities by the Czochralski method using additional liquid charging.
*Journal of Crystal Growth*, 2009, no. 311, pp. 1190-1194. - Galutskiy V.V., Stroganova E.V., Yakovenko N.A. Spektral’noe razdelenie opticheskikh tsentrov Cr
^{3+}v stekhiometricheskikh kristallakh niobata litiya s magniem [Spectral separation of Cr^{3+}optical centers in stoichiometric magnesium-doped lithium niobate crystals].*Optika i spektroskopiya*[Optics and Spectroscopy], 2011, vol. 110, pp. 401-407. (In Russian) - Sidorov N.V., Volk T.R., Mavrin B.N., Kalinnikov V.T. \textit{Niobat litiya: defecti, fotorefrakciya, kolebatelniy spektr, polyaritoni [Niobate of lithium: defects, photorefraction, vibration spectrum, polaritons]. Moscow, Nauka Publ., 2003, 250 p. (In Russian)
- Tsuboi T., Kaczmarek S., Boulon G. Spectral properties of Yb
^{3+}ions in LiNbO_{3}single crystals: influences of other rare-earth ions, OH ions, and γ-irradiation.*Journal of Alloys and Compounds*, 2004, vol. 380, pp. 196-200.

*Subjects*: physics

**Bocharova O.V.**^{1,2}, **Sedov A.V.**^{1}, **Andjikovich I.E.**^{2}, **Kalinchuk V.V.**^{1,2}

**On efficient method of signal processing in problems of low-frequency defetoscopy**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 20-25.

A method for low-frequency diagnostics of internal inhomogeneities based on the analysis of the surface wave field parameters, created by impact action is proposed. Multifunctional measuring complex, allowing carrying out research, comparing signals, and constructing spectral characteristics by using sensors of various types has been created to conduct a series of experimental investigations. For the processing of the recorded signal, a bispectral method is applied based on using of optimal orthogonal expansions of signal in the basis, adaptively customized on the training sample. A series of experimental studies to investigate the possibility of using bispectral method to identify the size and depth of the cracks has been carried out. The experimental results showed that using of the proposed method provides a clear recognition of the size and depth of the cracks in the diagnostic space of images.

*Keywords*: erbium, ytterbium, gradient crystal, lithium niobate.

^{1}Southern Scientific Center, Rostov-on Don, 344006, Russia

^{2}Southern Federal University, Rostov-on Don, 344006, Russia

Corresponding author’s e-mail: olga.v.bocharova@gmail.com

- Sharp R. (ed.)
*Metody nerazrushayushchikh ispytaniy*[Methods of non-destructive testing]. Moscow, Mir Pub., 1972, 494 p. (In Russian) - Kershenbaum V.Ya. (ed.)
*Nerazrushayushchie metody kontrolya*[Non-destructive testing methods]. Moscow, Nauka i tekhnika Pub., 1992, 254 p. (In Russian) - Kalinchuk V.V., Polyakova I.B. O vibratsii shtampa na poverkhnosti predvaritel’no napryazhennogo poluprostranstva [About the vibration die on the surface of the half-space prestressed].
*Prikladnaya mekhanika*[J. of Applied Mechanics], 1982, vol. 18, no. 6, pp. 22-27. (In Russian) - Belyankova T.I., Kalinchuk V.V. O vzaimodeystvii ostsilliruyushchego shtampa s predvaritel’no napryazhennym poluprostranstvom [On the interaction of the oscillating die with prestressed half].
*Prikladnaya matematika i mekhanika*[Applied Mathematics and Mechanics], 1993, vol. 57, no. 4, pp. 123-134. (In Russian) - Belyankova T.I., Kalinchuk V.V. Dinamika massivnogo tela, vzaimodeystvuyushchego s predvaritel’no napryazhennym poluprostranstvom [Dynamics of a massive body interacting with a half-space prestressed].
*Izvestiya Rossiyskoy akademii nauk. Mekhanika tverdogo tela*[Proc. of the Russian Academy of Sciences. Mechanics of rigid body], 1994, no. 6, pp. 83-94. (In Russian) - Akhtyamov A.M., Karimov A.R. Diagnostirovanie polosti v prizmaticheskoy balke [Diagnosing a cavity in a prismatic beam].
*Defektoskopiya*[Defectoscopy], 2013, no. 3, pp. 15-20. (In Russian) - Bocharova O.V. On inverse problems of identification the cavity in the elastic cylinder.
*J. of Physics: Conference Series*, 2014, vol. 490, no. 1, p. 012057. doi: 10.1088/1742-6596/490/1/012057. - Esipov Yu.V., Mukhortov V.M., Kalinchuk V.V. Ispytatel’naya ustanovka dlya analiza deformatsii modeley trekhmernykh konstruktsiy [The test setup for the analysis of deformation of three-dimensional structural models].
*Izmeritel’naya tekhnika*[Measuring equipment], 2008, no. 10, pp. 39-42. (In Russian) - Akop’yan V.A., Cherpakov A.V., Rozhkov E.V., Solov’ev A.N. Integral’nyy diagnosticheskiy priznak identifikatsii povrezhdeniy v elementakh sterzhnevykh konstruktsiy [Integrated diagnostic feature in the identification of core damage structural elements].
*Kontrol’. Diagnostika*[Control. Diagnostics], 2012, no. 7, pp. 50-56. (In Russian) - Bocharova O.V., Lyzhov V.A., Andzhikovich I.E. Nekotorye osobennosti volnovykh poley na poverkhnosti tel, oslablennykh nalichiem defektov [Some features of wave fields on the surface of the body, weakened by the presence of defects].
*Vestnik Yuzhnogo nauchnogo tsentra RAN*[Bull. of the Southern Research Center of the RAS], 2013, vol. 9, no. 2, pp. 11-15. (In Russian) - Sedov A.V.
*Modelirovanie ob”ektov s diskretno-raspredelennymi parametrami: dekompozitsionnyy podkhod*[Modeling objects with discrete distributed parameters: decomposition approach]. Moscow, Nauka Pub., 2010, 438 p. (In Russian) - Bocharova O.V., Andzhikovich I.E., Vorovich E.I. K probleme issledovaniya poverkhnostnykh volnovykh poley s pomoshch’yu tonkoplenochnykh segnetoelektricheskikh datchikov [To study the problem of surface wave fields using ferroelectric thin film sensors].
*Vestnik Yuzhnogo nauchnogo tsentra RAN*[Bull. of the Southern Research Center of the RAS], 2015, vol. 11, no. 1, pp. 30-35. (In Russian) - Bocharova O.V., Sedov A.V., Andzhikovich I.E., Kalinchuk V.V. O modelirovanii poverkhnostnykh volnovykh poley v sredakh s neodnorodnostyami [On the modeling of surface wave fields in media with inhomogeneities].
*Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2015, no. 4, pp. 33-36. (In Russian) - Bocharova O.V., Sedov A. V., Andzhikovich I.E., Kalinchuk V.V. Ob odnom metode identifikatsii defektov, osnovannom na kontrole struktury i osobennostey poverkhnostnykh volnovykh poley [A method for identifying defects, based on the control of the structure and characteristics of the surface wave fields].
*Defektoskopiya*[Defectoscopy], 2016, vol. 52, no. 7, pp. 21-28. (In Russian)

*Subjects*: physics

**Gorbacheva E.V.**^{1}, **Kalaydin E.N.**^{1}

**On efficient method of signal processing in problems of low-frequency defetoscopy**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 26-34.

The present paper considers a two-phase micro/nanoflow system of conductive (electrolyte) and non-conductive (dielectric) viscous liquids bounded by two solid walls in an external electric field. The charge near the solid body is immobile, but the surface charge is mobile.

Electrostatic attraction then creates an excess of counter ions within the electrolyte solution next to the solid surface or interface, thereby forming electric Debye layers near both surfaces. We study both the micro- and nanoscale electrolyte layers. In the latter case Debye layers in the electrolyte aren’t overlapped.

Related two-layer Couette-Poiseuille flow of viscous liquids has been thoroughly studied using asymptotic and numerical analysis of the Orr-Sommerfeld equation. These studies revealed existence of two types of instabilities: short and long-wave instabilities related to inertia and viscous effects.

In this work the problem is described by the Nernst-Planck-Poisson-Stokes system in the liquid-electrolyte phase; the Laplace-Stokes system in the liquid-dielectric phase; and appropriate boundary conditions on the solid-electrolyte, the solid-dielectric, and the liquid-liquid interfaces. The problem has 1D steady-state answer: equilibrium between solution and a plug-like velocity profile.

*Keywords*: liquid film, mobile surface charge, free interface, instability, electrolyte, Nernst-Planck-Poisson-Stokes equations, double ion layer.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: katya1911@list.ru

- Chang H.C., Yossifon G., Demekhin E.A. Nanoscale electrokinetics and microvortices: How microhydrodynamics affects nanofluidic ion flux.
*Annu. Rev. Fluid Mech.*, 2012, no. 44, pp. 401. - Lee J.S., Li D. Electro-osmotic flow at a liquid-air interface.
*Microfluid. Nanofluidics*, 2006, no. 2, pp. 361. - Gao Y., Wang T.N., Yang C. Transient two-liquid electro-osmotic flow with electric charges at the interface.
*Colloids Surfaces A.*, 2005, no. 266, pp. 117. - Gao Y., Wang T.N., Yang C., Ooi K.T. Two-fluid electro-osmotic flow in microchannals.
*J. Colloid Interface Sci.*, 2005, no. 284, pp. 306. - Landau L. D., Lifshitz E.M., Pitaevskii L.P. Electrodynamics of Continuous Media.
*Butterworth-Heinemann*, 1984, no. 8, pp. 2. - Ganchenko G.S., Demekhin E.A., Mayur M., Amiroudine S. Electrokinetic instability of liquid micro- and nanofilms with a mobile charge.
*Physics of Fluids*, 2015, no. 27, pp. 062002. - Navarkar A., Amiroudine S., Mayur M., Demekhin E.A. Long-wave interface instabilities of a two-liquid DC electroosmotic system for thin films.
*Microfluid Nanofluid*, 2015, no. 19, pp. 813.

*Subjects*: physics

**Gusev A.A.**^{1}, **Shvetsova N.A.**^{1}, **Voloshin A.E.**^{2}, **Yakovenko N.A.**^{1}

**The study of the applicability of the tool for fuzzy inductive reasoning to the problem of solar activity prediction**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 35-38.

The accumulation of large amounts of data in a variety of domains creates a demand for the development of the new tools for data processing and forecasting for decision support. The article is devoted to the developed by the authors tool for fuzzy inductive models construction. The tool is a cross-platform, standalone program which implements the fuzzy inductive reasoning methodology (FIR) for model creation. Authors explored the applicability of the tool for prediction nature of the 22-year cycle of solar activity. The input data for the inductive model construction were the Wolf number time series. Authors created the model with the tool; estimated the model against the control sample and got the forecast for the cycle’s nature till 2030. The forecast matches the theory of age-long cycles of solar activity. Further study will be focused on elaboration of the constructed model with the final aim of obtaining, with the using of the tool, a set of inductive models for application in different domains.

*Keywords*: general systems theory, decision support, forecasting, data mining, fuzzy inductive reasoning, solar activity, Hale cycle.

^{1}Kuban State University, Krasnodar, 350040, Russia

^{2}Construction Bureau “Selena”, Krasnodar, 350072, Russia

Corresponding author’s e-mail: gusev@ftf.kubsu.ru

- Klir G.J., Elias D.
*Architecture of systems problem solving*, 2nd ed. New York, Plenum Publishers, 2003, 349 p. - Cellier F.E.
*Publications related to Fuzzy Inductive Reasoning*. Available at: https://www.inf.ethz.ch/personal/cellier/Pubs/FIR/Pubs\_fir\_index\_engl.html - Escobet A., Nebot A., Mugica F.Fuzzy fault diagnosis in fuel cell power systems
*Engineering Applications of Artificial Intelligence*, 2014, no. 36, pp. 40-53. - Nebot A., Mugica F. Fuzzy Inductive Reasoning: a consolidated approach to data-driven construction of complex dynamical systems (Invited overview paper)
*International Journal of General Systems*, 2012, no. 41, iss. 7, pp. 645-665. - Gusev A.A., Shvetsova N.A.
*Program for management decision-making “AimDSS”*, Certificate of State Registration for Computer Program no. 2016610899 Russian Federation. - Sunspot Index and Long-term Solar Observations. WDC-SILSO, Royal Observatory of Belgium, Brussels. Available at: http://www.sidc.be/silso/datafiles.

*Subjects*: physics

**Demekhin E.A.**^{1}, **Morshneva I.V.**^{2}, **Kalaydin E.N.**^{3}

**Mathematical modeling of electrodynamics of bipolar membranes with water dissociation and chemical reaction of ionizable groups**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 39-46.

The microscale electrolyte behavior both near and inside the bipolar ion-selective electric membrane in microscales under an external normal to the membrane surface electric field is scrutinized. Bipolar membrane is a combination of cation-exchange and anion-exchange membranes. Strong electric field in the junction between membranes leads to more intensive than in the monopolar membranes water dissociation process, that is why the bipolar membranes are widely used in the chemical industry. For investigation of aforementioned phenomena the three-layer system electrolyte-membrane-electrolyte is considered. The base of the mathematical model is the Nernst-Planck-Poisson system of nonlinear equations and an extra transport equations for ions of dissociated water with a source terms are added to the basic system of equations. It is found numerically that the maximal dissociation takes place within the junction between membranes. The flux of water ions not only enhances the total electric current through the system, but also leads to an exaltation effect. Taking into account the second Wien effect allows to explain the transition to the overlimiting mode in the system, which has been observed during the experiments.

*Keywords*: microfluidics, bipolar membrane, Nernst-Plank-Poisson system, second Wien effect, numerical solution.

^{1}Financial University under the Government of the Russian Federation, Krasnodar, 350051, Russia

^{2}Southern Federal University, Rostov-on-Don, 344090, Russia

^{3}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: enkalaydin@fa.ru

- Frilette V.J. Preparation and characterization of bipolar ion-exchange membranes.
*J. Phys. Chem.*, 1956, vol. 60, no. 4, pp. 435-439. - Greben’ V.P., Pivovarov N.Ja., Kovarskij N.Ja., Nefedova G.Z. Vlijanie prirody ionita na fiziko-himicheskie svojstva bipoljarnyh ionoobmennyh membran [Influence of the nature of the resin on the physicochemical properties of the bipolar ion-exchange membranes].
*Zhurn. fiz. himii*[J. of Physical Chemistry A], 1978, vol. 52, no. 10, pp. 2641. (In Russian) - Simons R. A novel method preparing bipolar membranes.
*Electrochim. Acta.*, 1986, vol. 31, pp. 1175-1177. - Timashev S.F., Kirganova E.V. O mehanizme jelektroliticheskogo razlozhenija molekul vody v bipoljarnyh membranah [On the mechanism of electrolytic decomposition of water molecules in the bipolar membranes].
*Elektrokhimiya*[Electrochemistry], 1981, vol. 17, no. 3, pp. 440-443. (In Russian) - Mafe S., Ramirez P., Alcaraz A. Electric field-assisted proton transfer and water dissociation at the junction of a fixed-charge bipolar membrane.
*Chem. Phys. Lett.*, 1998, vol. 294, no. 4-5, pp. 406-412. - Zabolotsky V.I., Gnusin N.P., Sheldeshov N.V. Vol’tampernye kharakteristiki perekhodnoy oblasti bipolyarnoy membrany MB-1 [Current-voltage characteristics of the transition region of the bipolar membrane MB-1].
*Elektrokhimiya*[Electrochemistry], 1984, vol. 20, no. 10, pp. 1340-1345. (In Russian) - Conroy D.T., Craster R.V., Matar O.K., Cheng L.-J., and Chang H.-C. Nonequilibrium hysteresis and Wien effect water dissociation at a bipolar membrane.
*Phys. Rev. E.*, 2012, no. 86, pp. 056104. - Grabowski A., Zhang G., Strathmann H., Eigenberger G. Production of high-purity water by continuous electrodeionization with bipolar membranes: Influence of concentrate and protection compartment.
*Sep. Purif. Technol*, 2008, vol. 60, pp. 86-95. - Schiffbauer J., Leibowitz N., Yossifon G. Extended space charge near nonideally selective membranes and nanochannels.
*Phys. Rev. E.*, 2015, vol. 92, pp. 013002. - Demekhin E.A., Nikitin N.V., Shelistov V.S. Direct numerical simulation of electrokinetic instability and transition to chaotic motion.
*Physics of Fluids*, 2013, vol. 25, pp. 122001. - Danielsson C.-O., Dahlkild A., Velin A., Behm M.A. Model for the enhanced water dissociation on monopolar membranes.
*Electrochimica Acta*, 2009, vol. 54, pp. 2983-2991. - Simons R. Electric field effects on proton transfer between ionizable groups and water in ion exchange membranes [Electric field effects on proton transfer between ionizable groups and water in ion exchange membranes].
*Electrochimica Acta*[Electrochimica Acta], 1984, vol. 29, pp. 151-158. - Sheldeshov N.V., Zabolotsky V.I., Pismenskaya N.D., Gnusin N.P. Kataliz reaktsii dissotsiatsii vody fosfornokislotnymi gruppami bipoliarnoi membrany MB-3 [Catalysis of the water dissociation reaction bipolar membrane phosphoric acid group MB-3].
*Elektrokhimiya*[Electrochemistry], 1986, vol. 22, no. 6, pp. 791-795. (In Russian) - Sheldeshov N.V., Zabolotsky V.I., Lebedev K.A., Alpatova N.V., Kovalev N.V. Stroenie oblasti prostranstvennogo zaryada na bipolyarnoy granitse i dissotsiatsiya molekul vody v bipolyarnoy membrane modifitsirovannoy soedineniem khroma(III) [Structure of the space charge region at bipolar junction and dissociation of water molecules in bipolar membrane modified by chromium (III) compound].
*Politematicheskiy setevoy elektronnyy nauchnyy zhurnal Kubanskogo gosudarstvennogo agrarnogo universiteta*[Polythematic online scientific journal of Kuban State Agrarian University], 2014, no. 10, pp. 990-1009. (In Russian) - Zabolotsky V.I., Nikonenko V.V.
*Perenos ionov v membranakh*[Ion transport in membranes]. Moscow, Nauka Pub., 1996, pp. 392. (In Russian)

*Subjects*: physics

**Drobotenko M.I.**^{1}, **Volchenko N.N.**^{1}, **Samkov A.A.**^{1}, **Svidlov A.A.**^{1}

**Mathematical modeling of electrodynamics of bipolar membranes with water dissociation and chemical reaction of ionizable groups**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 47-51.

An important problem that limits use of the microbial fuel cell (MFC) is a scaling problem, e.g. the problem of choosing the optimal size. The main objective of the paper is to develop a mathematical model of the processes in the MFC, so that it allows solving the scaling problem analytically and by means of numerical experiments. The proposed in the paper MFC model includes the electrodiffusion processes equation for positive and negative ions in the anode chamber, the positive ions dynamic equation in the cathode chamber, the biological processes kinetics equation in the anode chamber, initial and boundary conditions. The model takes into account the residual charge dynamics in the anode chamber. The paper contains the comparison of the amperage dynamics in the MFC electrical circuit obtained from mathematical modeling and natural experiments conducted to confirm the correctness of the model. Qualitative agreement of the results of the natural and numerical experiments has been obtained for MFCs with different geometrical characteristics. In the natural and numerical experiments two surges of amperage have been revealed. The first surge of amperage does not depend on the geometry of the MFC chambers, it can be explained by the existence of the residual charge in the hinge of the sediments placed in MFS anode chamber. The second surge of amperage is caused by microbial processes in the MFC anode chamber; its amplitude is much larger than the amplitude of the first surge.

*Keywords*: microbial fuel cell, model waste water, anaerobic heterotrophic microflora, facultative anaerobic heterotrophic microflora, math modeling, geometry of anode chamber, dynamics of current-voltage characteristics.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: mdrobotenko@mail.ru

- Oliveira V.B., Simoes M., Melo L.F., Pinto A.M.F.R. A 1D mathematical model for a microbial fuel cell.
*Energy*, 2013, no. 61, pp. 463-471. - Picioreanua C., Head I.M., Katuri K.P., Van Loosdrecht M.C.M., Scott K. A computational model for biofilm-based microbial fuel cells.
*Water research*, 2007, no. 41, pp. 2921-2940. - Pinto R.P. , Srinivasan B. , Manuel M.-F., Tartakovsky B. A two-population bio-electrochemical model of a microbial fuel cell.
*Bioresource Technology*, 2010, no. 101, pp. 5256-5265. - Picioreanu C., Van Loosdrecht M.C.M., Curtis T.P., Scott K. Model based evaluation of the effect of pH and electrode geometry on microbial fuel cell performance.
*Bioelectrochemistry*, 2010, no. 78, pp. 8-24. - Logan B.E.
*Microbial fuel cells*. New Jersey, John Wiley & Sons, Inc., 2008, 200 p. - Ghangrekar M.M., Shinde V.B. Performance of membrane-less microbial fuel cell treating wastewater and effect of electrode distance and area on electricity production.
*Bioresource Technology*, 2007, no. 98, pp. 2879-2885.

*Subjects*: physics

**Kamalyan S.R.**^{1}, **Kamalyan R.Z.**^{2}

**To the dynamics of saturated soils**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 52-60.

There are considered the deformation processes in the soil body, resulting under the pulse action. Saturated soils play an important role in the study of such processes. This feature is related to short-term action. As a result of short-term action of excessive pressure, redistribution of water and air in the soil does not have time to occur and deformation is realized due to the air compression (reduction of water-ratio) and partially elastic compression of water and mineral matrix.

In the low moisture soils mineral matrix receives most of the dynamic load, with considerable friction between the particles that prevents their mutual displacement and repacking, and hence the development of volumetric deformation. Increasing humidity results in improved particles repacking and increased density. However, at high rates of soil moisture, water receives dynamic load, which deformation is elastic and soil compaction reduces. Therefore, there is optimum moisture rate under dynamic loading, which achieves the highest packing. From the analysis of experimental curves it shows that this value is approximately equal to the optimum moisture content of 20%.

The ability to achieve maximum compaction of soil at the optimum moisture content is at the core of the explosive method of soil compaction. The paper shows the results of soil compaction on the experimental plots. We describe the details of the research. It confirmed the effectiveness of the method.

*Keywords*: water-saturated ground, the shock wave, distortion, dynamic loading, thinning, compaction, humidity.

^{1}Kuban State University, Krasnodar, 350040, Russia

^{2}Academy of marketing and socially-information technologies, Krasnodar, 350010, Russia

Corresponding author’s e-mail: kasarub@gmail.com

- Grigoryan S.S. Ob udarnom uplotnenii lessovykh gruntov [Shock compaction loess soils].
*Izvestiya AN SSSR. Otdelenie tekhnicheskikh nauk. Mekhanika i mashinostroenie*[News of the Academy of Sciences of the USSR. Department of Technical Sciences. Mechanics and Mechanical Engineering], 1964, no. 6, pp. 127-130. (In Russian) - Kamalyan R.Z. Ob uplotnenii prosadochnykh gruntov energiey vzryva [On the subsidence of soil compaction energy explosion].
*Izvestiya vuzov. Severo-Kavkazskiy region. Tekhnicheskie nauki*[Proc. of the Universities. North-Caucasian region. Technical science], 1994, no. 1-2, pp. 137-142. (In Russian) - Litvinov I.M.
*Ukreplenie i uplotnenie prosadochnykh gruntov v zhilishchnom i promyshlennom stroitel’stve*[Strengthen and seal collapsible soils in the residential and industrial construction]. Kiev, Budivel’nik Pub., 1977, 288 p. (In Russian) - Ivanov P.L.
*Uplotnenie malosvyazannykh gruntov vzryvami*[Seal loosely soils by explosions]. Moscow, Nedra Pub., 1983, 230 p. (In Russian) - Sadovskiy M.A. (ed.)
*Mekhanicheskiy effekt podzemnogo vzryva*[The mechanical effect of an underground explosion]. Moscow, Nedra Pub., 1971, 224 p. (In Russian) - Adushkin V.V., Pernik L.M. Proval’nye voronki pri podzemnykh vzryvakh v slabosvyazannykh gruntakh [The failed funnel in underground explosions in weakly soils].
*Fizika goreniya i vzryva*[Physics of combustion and explosion], 1978, no. 3, pp. 97-104. (In Russian) - Adushkin V.V., Kostyuchenko V.N., Nikolaevskiy V.N., Tsvetkov V.M. Mekhanika podzemnogo vzryva [Mechanics underground explosion].
*Itogi nauki i tekhniki. Ser.: Mekhanika tverdogo deformiruemogo tela*[Results of science and technology. Series: Mechanics of solid deformable body], 1977, vol. 7, pp. 87-197. (In Russian) - Luchko I.A. (ed.)
*Mekhanicheskiy effekt vzryva v gruntakh*[The mechanical effect of the explosion in the ground]. Kiev, Naukova dumka Pub., 1989, 232 p. (In Russian) - Lyakhov G.M.
*Osnovy dinamiki vzryvnykh voln v gruntakh i gornykh porodakh*[Fundamentals of dynamics of shock waves in the soils and rocks]. Moscow, Nedra Pub., 1974, 192 p. (In Russian) - Lyakhov G.M.
*Volny v gruntakh i poristykh mnogokomponentnykh sredakh*[Waves in soils and porous multicomponent media]. Moscow, Nauka Pub., 1982, 286 p. (In Russian) - Vovk A.A., Smirnov A.G., Kravets V.G.
*Dinamika vodonasyshchennykh gruntov*[Dynamics of saturated soils]. Kiev, Naukova dumka Pub., 1975, 302 p. (In Russian) - Kul’chitskiy K.V., Us’yarov O.G.
*Fiziko-khimicheskie osnovy formirovaniya svoystv glinistykh porod*[Physical and chemical bases of formation properties of clay rocks]. Moscow, Nedra Pub., 1981, 178 p. (In Russian) - Koste Zh., Sanglera G.
*Mekhanika gruntov*[Soil mechanics]. Moscow, Stroyizdat Pub., 1981, 455 p. (In Russian) - Meschyan S.R.
*Mekhanicheskie svoystva gruntov i laboratornye metody ikh opredeleniya*[Mechanical properties of soils and laboratory methods of their definition]. Moscow, Nedra Pub., 1974, 192 p. (In Russian) - Mikhalyuk A.V. Ob”emnoe deformirovanie gruntov pri dinamicheskom nagruzhenii i ego energoemkost’ [Volumetric deformation of soil under dynamic loading and its energy intensity]. In
*Vzryvnoe delo*[Explosion technology], 81/38, Moscow, Nedra Pub., 1979, pp. 98-106. (In Russian) - Van der Kog H. Wave phenomena.
*Computers and geotechnics*, 1987, vol. 3, no. 1, pp. 21-28. - Yegian M.K., Whitman R.V. Risk analisis for ground failure by liquefaction.
*Journal of the Geotechnical Engineering Division*, 1978, July, pp. 921-937. - Kamalyan R.Z., Kamalyan S.R. O razzhizhenii gruntov pri dinamicheskikh vozdeystviyakh [About soil liquefaction by the dynamic impacts].
*Obozrenie prikladnoy i promyshlennoy matematiki*[Review of Applied and Industrial Mathematics], 2001, vol. 8, iss. 2, pp. 602-603. (In Russian) - Kamalyan S.R. O probleme razzhizheniya gruntov pri podzemnykh vzryvakh [On the problem of soil liquefaction in underground explosions].
*Priroda. Obshchestvo. Chelovek. Estestvennye nauki*[Nature. Society. Human. Natural Sciences], 2012, no. 1(14), pp. 120. (In Russian) - Stanyukovich K.P. (ed.)
*Fizika vzryva*[Explosion physics]. Moscow, Nauka Pub., 1975, 704 p. (In Russian) - Arutyunov O.A., Kitaer B.I., Kamalyan R.Z. Effektivnost’ uplotneniya prosadochnykh gruntov vzryvom [The effectiveness of compaction of soil subsidence explosion].
*Stroitel’stvo i arkhitektura Uzbekistana*[Construction and architecture of Uzbekistan], 1979, no. 5, pp. 34-35. (In Russian) - Kamalyan R.Z., Kamalyan S.R. K dinamike gruntovykh mass [Dynamics of groundwater masses].
*Vestnik IMSITa*[Bulletin of Academy of marketing and socially-information technologies], 2015, no. 2, pp. 65-68. (In Russian)

*Subjects*: physics

**Kovalenko A.V.**^{1}

**Mathematical classification of electroconvection in electro-membrane systems**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 61-68.

The article shows that the cause of electroconvection is vortex nature of electric power, which has a significant value. The mathematical classification of phenomena electroconvection (electroosmosis) is proposed in the article, based on an analysis of the rotor of the electric force. The connection of the rotor of the electric force and the known classification Dukhin-Mishchuk and Rubinstein-Zaltzman is showed. The causes and mechanism of occurrence electroconvection in the flow-through membrane channel is analyzed and it is shown, that the forced flow of solution in the channel desalination has a significant influence on the development of electroconvection. It is shown that although electroconvection in a flow-through membrane channel has features as electroosmosis of the first kind Dukhin-Mishchuk and also unsteady electroosmosis of the second kind Rubinstein-Zaltzman, nevertheless it is a qualitatively new type of electroconvection. It is shown that electroconvection in the presence of forced fluid flow is the qualitatively new type of electroconvection associated with the rotor of the electric force.

*Keywords*: electroconvection, electroosmosis, forced fluid flow, equations of Nernst-Planck-Poisson-Navier-Stokes, the rotor of the electric force.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: savanna-05@mail.ru

- Urtenov M.K., Uzdenova A.M., Kovalenko A.V., Nikonenko V.V., Pismenskaya N.D., Vasil’eva V.I., Sistat P., Pourcelly G. Basic mathematical model of overlimiting transfer in electrodialysis membrane systems enhanced by electroconvection.
*J. of Membrane Science*, 2013, vol. 447, pp. 190-202. DOI: 10.1016/j.memsci.2013.07.033 - Kwak R., Guan G., Peng W.K., Han J. Microscale electrodialysis: concentration profiling and vortex visualization.
*Desalination*, 2012, vol. 308, pp. 138-146. - Nikonenko V., Kovalenko A., Urtenov M., Pismenskaya N., Han J., Sistat P., Pourcelly G. Desalination at overlimiting currents: State-of-the-art and perspectives.
*Desalination*, 2014, vol. 342, pp. 85-106. Available at: http://www.sciencedirect.com/science/article/ pii/S001191641400023X (accessed 25.01.2015). - Kovalenko A.V., Urtenov M.Kh., Geryugova A.A. Elektroosmos v mikro- i nanokanalakh. Chast’ 1. vyvod ierarkhicheskoy sistemy matematicheskikh modeley s ispol’zovaniem metoda dekompozitsii [The electroosmosis in micro- and nanochannel. Part 1. conclusion a hierarchical system of mathematical models using decomposition method].
*Politematicheskiy setevoy elektronnyy nauchnyy zhurnal Kubanskogo gosudarstvennogo agrarnogo universiteta*[Polythematic network electronic scientific journal of the Kuban state agrarian University], 2015, no. 114, pp. 370-387. (In Russian) - Nikonenko V.V., Vasil’eva V.I., Akberova E.M., Uzdenova A.M., Urtenov M.K., Kovalenko A.V., Pismenskaya N.P., Mareev S.A., Pourcelly G. Competition between diffusion and electroconvection at an ion-selective surface in intensive current regimes.
*Advances in Colloid and Interface Science*, 2016, no. 235, pp. 233-246. DOI: http://dx.doi.org/10.1016/j.cis.2016.06.014 - Kovalenko A.V. Vliyanie dissotsiatsii vody na razvitie elektrokonvektsii v membrannykh sistemakh [Influence of water dissociation on electroconvection development in membrane systems].
*Kondensirovannye sredy i mezhfaznye granitsy*[Condensed substance and phase boundary], 2014, vol. 16, no. 3, pp. 288-293. (In Russian) - De Jong J., Lammertink R.G.H., Wessling M. Membranes and microfluidics: a review.
*Lab on a Chip-Miniaturisation for Chemistry and Biology*, 2006, vol. 6, no. 9, pp. 1125-1139. - Dukhin S.S., Mishchuk H.A., Zholkovskiy E.K. Kontsentratsionnaya polyarizatsiya dvoynogo sloya dispersnoy chastitsy pri bol’shikh chislakh Pekle [The concentration polarization of the double layer of disperse particles at large Peclet numbers].
*Kolloidnyy zhurnal*[Colloid J.], 1987, vol. 49, no. 5, pp. 865-874. (In Russian) - Dukhin S.S. Electrokinetic phenomena of the 2nd kind and their applications.
*Adv. Colloid Interface Sci.*, 1991, vol. 35, pp. 173-196. - Rubinstein I., Zaltzman B. Electroosmotically induced convection at a permselective membrane.
*Physical Review E*, 2000, vol. 62, pp. 2238-2251. - Kovalenko A.V., Zabolotskiy V.I., Nikonenko V.V., Urtenov M.Kh. Matematicheskoe modelirovanie vliyaniya morfologii poverkhnosti geterogennykh ionoobmennykh membran na elektrokonvektsiyu [Mathematical modeling of the influence of the morphology of the heterogeneous ion-exchange membranes on electroconvection].
*Politematicheskiy setevoy elektronnyy nauchnyy zhurnal KubGAU*[Polythematic network electronic scientific journal of the Kuban State Agrarian University], 2014, no. 10. Available at: http://ej.kubagro.ru/2014/10/pdf/46.pdf (accessed 20.01.2015). (In Russian) - Mishchuk N.A. Concentration polarization of interface and non-linear electrokinetic phenomena.
*Adv. in Colloid and Interface Science*, 2010, vol. 160, pp. 16-39. - Mishchuk N.A., Takhistov P.V. Electroosmosis of the second kind.
*Colloids Surf. A*, 1995, vol. 95, no. 2, pp. 119-131. - Rubinstein I., Shtilman L. Voltage against current curves of cation exchange membranes.
*J. Chem. Soc.*, 1979, vol. 75, pp. 231-246. - Kovalenko A.V., Urtenov M.Kh., Nikonenko V.V., Loyko V.I. Fizicheskiy smysl nekotorykh kriteriev podobiya protsessa perenosa v kanale obessolivaniya elektrodializnogo apparata s uchetom elektrokonvektsii [The physical meaning of some similarity criteria for the process of transfer in the desalination channel of the electrodialytic apparatus, taking into account electroconvective].
*Politematicheskiy setevoy elektronnyy nauchnyy zhurnal KubGAU. Elektronnyy zhurnal*[Polythematic network electronic scientific journal of the Kuban State Agrarian University. Electron. J.], 2015, no. 1. pp. 846-865. Available at: http://ej.kubagro.ru/2015/01/pdf/51.pdf (accessed 29.09.2015). (In Russian) - Davidson S.M., Wessling M., Mani A. On the Dynamical Regimes of Pattern-Accelerated Electroconvection.
*Scientific Reports*, 2016, no. 6, p. 22505. DOI: 10.1038/srep22505 - Kovalenko A.V., Uzdenova A.M., Urtenov M.A.Kh., Nikonenko V.V. Kriterial’nye chisla obrazovaniya nestabil’nykh elektrokonvektivnykh vikhrey v kanale obessolivaniya elektrodializnogo apparata [Criterion number education electroconvective unstable vortices in the desalination channel of the electrodialytic device].
*Sorbtsionnye i khromatograficheskie protsessy*[Sorption and chromatographic processes], 2014, vol. 14, no. 2, pp. 260-269. (In Russian) - Kovalenko A.V., Uzdenova A.M., Urtenov M.Kh., Nikonenko V.V. Kriterial’nye chisla vozniknoveniya elektrokonvektsii v kamere obessolivaniya elektrodializatora [The criterion of the number of occurrence of electroconvection in the desalination chamber of electrodialyzer].
*Kondensirovannye sredy i mezhfaznye granitsy*[Condensed matter and interphase boundaries], 2013, vol. 15, no. 4, pp. 404-412. (In Russian) - Kovalenko A.V., Pis’menskiy A.V., Urtenov M.Kh. Teoriya podobiya elektromembrannykh sistem s uchetom vynuzhdennoy, gravitatsionnoy i elektrokonvektsii [Similarity theory electro-membrane systems with consideration of forced, gravitational and electroconvection].
*Politematicheskiy setevoy elektronnyy nauchnyy zhurnal Kubanskogo gosudarstvennogo agrarnogo universiteta*[Polythematic network electronic scientific journal of the Kuban state agrarian University], 2015, no. 105, pp. 866-887. (In Russian) - Kovalenko A.V. 2D modelirovanie perenosa proizvol’nogo binarnogo elektrolita v elektromembrannykh sistemakh pri vypolnenii usloviya elektroneytral’nosti [2D modeling of the transfer of arbitrary binary electrolyte in electromembrane systems under the condition of electroneutrality].
*Fundamental’nye issledovaniya*[Fundamental Research], 2015, no. 11, pt. 2, pp. 257-266. (In Russian) - Kovalenko A.V. Chislennyy analiz 2D modeli ZOM perenosa simmetrichnogo binarnogo elektrolita [Numerical analysis of the 2D model ZOOM transfer a symmetric binary electrolyte].
*Fundamental’nye issledovaniya*[Fundamental Research], 2015, no. 11, pt. 1, pp. 59-65. (In Russian) - Pismenskiy A.V., Urtenov M.K., Kovalenko A.V., Mareev S.V. Electrodialysis desalination process in conditions of mixed convection.
*Desalination and Water Treatment*, 2014, no. 1-3. DOI: 10.1080/19443994.2014.981407 - Pis’menskiy A.V., Kovalenko A.V., Urtenov M.Kh. Matematicheskoe modelirovanie protsessov massoperenosa v elektromembrannykh sistemakh v usloviyakh odnovremennogo deystviya vynuzhdennoy, gravitatsionnoy i elektrokonvektsii. Zavisimost’ ot nachal’noy kontsentratsii [Mathematical modeling of mass transfer processes in electromembrane systems under the simultaneous action of forced, gravitational and electroconvection. Dependence on the initial concentration].
*Ekologicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2014, no. 3, pp. 59-68. (In Russian) - Kovalenko A.V., Evdochenko E.N., Urtenov M.Kh. Raschet i analiz vremennykh kharakteristik elektrokonvektsii v membrannykh sistemakh [Calculation and analysis of the temporal characteristics of electroconvection in membrane systems].
*Politematicheskiy setevoy elektronnyy nauchnyy zhurnal KubGAU*[Polythematic network electronic scientific journal of the Kuban state agrarian University], 2015, no. 109, pp. 958-970. Available at: http://ej.kubagro.ru/2015/05/pdf/66.pdf (accessed 20.11.2016)

*Subjects*: physics

**Kochergin V.S.**^{1}, **Kochergin S.V.**^{1}

**The parameters of identification algorithm of instant point source pollution in the Azov Sea on the basis of the method of adjoint equations**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 69-74.

The ecological state of the Azov Sea is required the creation of reliable environmental monitoring systems that allow you effectively to assess the situation in the areas subject to technological impact, especially in the areas of the intensive shipping and the construction of communication systems of a different nature. For such evaluations of the examined object it is effective to use highly productive computer technologies and approaches allowing implementing parallelization of calculations. Solving the problems of the pollution spreading of a different nature in the sea is possible on the basis of the methods of the mathematical modeling and the methods for the solving inverse problems, when according to the measurement data due to their assimilation occurs the identification of the certain parameters of the transport model. Recently, variational methods of assimilation and the method of the adjoint equations have been actively developed and used to solve the oceanographic problems. The algorithms of the measurement data adoption are based generally on the minimization of a quadratic functional prediction quality that characterizes the deviation of the model solutions from the measurement data. The transport model of the passive admixture acts as a limit to the variations of the input parameters. In this work, the method of the adjoint equations is applied that allows searching for the location of the source of pollution. The Identification of the source power was produced by using the variational filtering. The numerical experiments were conducted using a hydrodynamic model of the Azov Sea. The resultant flow fields were used in the modeling the transport of the passive admixture. The numerical experiments have shown that the result of the identification significantly depends on the location of measurement points. The most accurate reproduction of the true value of the power source of pollution is obtained in the case, where measurements are carried out in the region of maximum concentration values of the field, which leads to a better conditionality of the solving problem. In general, the carried out numerical experiments have shown the reliable operation of the power of the algorithm identifying the source of pollution, related to the model of passive admixture transport in the Azov Sea.

*Keywords*: method of the adjoint equations, identification of input parameters, passive admixture, transport model, Azov Sea, spreading of pollution, assimilation of the data measurements.

^{1}Marine Hydrophysical Institute, Sevastopol, 299011, Russia

Corresponding author’s e-mail: vskocher@gmail.com

- Ivanov V.A.. Fomin V.V.
*Matematicheskoye modelirovaniye dinamicheskikh protsessov v zone more - susha*[Mathematical modeling of dynamic processes in the sea-land]. Sevastopol, EKOSI-gidrofizika Pub., 2008, 363 p. (In Russian) - Marchuk G.I., Penenko V.V. Application of optimization methods to the problem of mathematical simulation of atmospheric processes and environment.
*Modelling and Optimization of Complex Systems Proc. of the IFIP-TC7 Working conf.*, New York, Springer, 1978, pp. 240-252. - Kochergin V.S., Kochergin S.V. Ispol’zovanie variacionnyh princi-pov i resheniya sopryazhennoj zadachi pri identifikacii vhodnyh parametrov modeli perenosa passivnoj primesi [The use of variational principles and the solution of the adjoint problem, identification of input parameters for models of transport of passive tracer].
*Ekologicheskaya bezopasnost’ pribrezhnoj i shel’fovoj zon i kompleksnoe ispol’zovanie resursov shel’fa*[Ecological safety of coastal and shelf zones and complex use of shelf resources]. Sevastopol’, EKOSI-gidrofizika Pub., 2010, no. 22, pp. 240-244. (In Russian) - Penenko V.V.
*Metody chislennogo modelirovaniya atmosfernykh protsessov*[Methods for numerical modeling of atmospheric processes]. Leningrad, Gidrometeoizdat, 1981, 350 p. (In Russian) - Agoshkov V.I., Parmuzin E.I., SHutyaev V.P. Assimilyaciya dannyh nablyudenij v zadache cirkulyacii Chernogo morya i analiz chuvstvitel’nosti eyo resheniya [Data assimilation of observations in the problem of Black sea circulation and sensitivity analysis of its solution].
*Izv. RAN, Fizika atmosfery i okeana*[Bulletin of the Russian Academy of Sciences. Physics of atmosphere and ocean], 2013, vol. 49, no. 6, pp. 643-654. (In Russian) - Shutyaev V.P., Le Dime F., Agoshkov V.I., Parmuzin E.I. Chuvstvitel’nost’ funkcionalov zadach’ variacionnogo usvoeniya dannyh nablyu-deni [The sensitivity of functionals of the challenges of variational data assimilation].
*Izv. RAN, Fizika atmosfery i okeana*[Bulletin of the Russian Academy of Sciences. Physics of atmosphere and ocean], 2015, vol. 51, no. 3, pp. 392-400. (In Russian) - Ryabcev Yu. N., Shapiro N.B. Opredelenie nachal’nogo polozheniya obnaruzhennyh v otkrytoj chasti morya poverhnostnyh linz ponizhennoj solenosti primesi [Specify a start position detected in the open part of the sea surface lens of low salinity impurities]. In
*Ekologicheskaya bezopasnost’ pribrezhnoj i shel’fovoj zon i kompleksnoe ispol’zovanie resursov shel’fa*[Ecological safety of coastal and shelf zones and complex use of shelf resources]. Sevastopol’, EKOSI-gidrofizika Pub., 2009, no. 18, pp. 141-157. (In Russian) - Kochergin V.S., Kochergin S.V., Identifikaciya moshchnosti istochnika zagryazneniya v Kazantipskom zalive na osnove primeneniya variacionnogo algoritma [Identification of power source of pollution in the Kazantipsky Gulf through the application of a variational algorithm].
*Morskoj gidrofizicheskij zhurnal*[Marine hydrophysical journal], 2015, no. 2, pp. 79-88. (In Russian) - Marchuk G.I.
*Matematicheskoye modelirovaniye v probleme okruzhayushchey sredy*[Mathematical modeling in the environmental problem]. Moscow, Nauka Pub., 1982, 320 p. (In Russian) - Kochergin V.S. Opredelenie polya koncentracii passivnoj primesi po nachal’nym dannym na osnove resheniya sopryazhennyh zadach [Determination of the concentration field of a passive impurity in the initial data based on the solution of conjugate objectives]. In
*Ekologicheskaya bezopasnost’ pribrezhnoj i shel’fovoj zon i kompleksnoe ispol’zovanie resursov shel’fa*[Ecological safety of coastal and shelf zones and complex use of shelf resources], MGI NANU, Sevastopol’ 2011, no. 25, vol. 2, pp. 270-376. (In Russian) - Strahov V.N. Metod fil’tracii sistem linejnyh algebraicheskih uravnenij - osnova dlya resheniya linejnyh zadach gravimetrii i magnitometrii [Filtering method systems of linear algebraic equations - the basis for solving linear problems of gravimetry and magnetometry].
*Dokl. AN SSSR*[Rep. of the Academy of Sciences of the USSR], 1991, vol. 320, no. 3, pp. 595-599. (In Russian) - Eremeyev V.N., Kochergin V.P., Kochergin S.V., Sklyar S.N.
*Matematicheskoye modelirovaniye gidrodinamiki glubokovdnykh basseynov*[Mathematical modeling of hydrodynamics of deep-water basins]. Sevastopol, EKOSI-Gidrofizika Pub., 2002, 238 p. (In Russian)

*Subjects*: mechanics

**Rubtsov S.E.**^{1}, **Pavlova A.V.**^{1}

**To the study of the mixed dynamic problems for a limited volume of fluid on an elastic foundation**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 75-81.

The paper presents the results of analytical studies of the distribution of contact stresses at the interface between a limited pool of liquid and an elastic foundation. A limited amount of an ideal compressible fluid, located on deformable foundation, is considered. An elastic layer and an elastic half-space rigidly coupled with non-deformable base are considered as the latter. In the system “elastic medium - liquid” the oscillations are excited by surface vibrator.

The velocity potential that satisfies the wave equation as the characteristic of the wave field in the liquid is being considered. It is assumed that the hydrodynamic pressure on the upper liquid surface is absent. The condition of impermeability is given on vertical borders, on bottom surface the liquid is affected by an elastic foundation. The displacement vector points of the elastic base satisfy the system of Lame differential equations. The interaction of the liquid and elastic medium is determined by the continuity of the vertical speed component in the contact area. It is believed that the system vibrations are of steady character.

In this work the integral equation of the first kind with kernel is obtained and solved, depending on both difference and sum of the arguments, the function is also built, that describes the distribution of contact stresses in the area of contact between the liquid and the elastic media with consideration of physical and frequency factors.

The relevance of research of dynamic interaction of hydraulic structures with deformable foundation is determined by the high requirements for reliability of their exploitation and the degree of certainty of the forecast of consequences in case of vibroseismic interactions. The results of this study may serve as a foundation for the further development of methods for solving dynamic contact problems of joint oscillations of elastic and liquid mediums.

*Keywords*: limited pool of liquid, elastic foundation, harmonic oscillations, the integral factorization method.

^{1}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: kmm@fpm.kubsu.ru

- Korenev B.G. Dejstvie impul’sa na cilindricheskie i prizmaticheskie rezervuary, napolnennye zhidkost’ju [Pulse action on the cylindrical and prismatic tanks filled with liquid]. In
*Stroitel’naja mehanika*[Structural mechanics]. Moscow, Strojizdat Pub., 1966, pp. 213-266. (In Russian) - Klimov M.A. Opredelenie prisoedinennoj massy zhidkosti v sluchae neosesimmetrichnyh kolebanij dnishh rezervuarov [Defining the associated mass of liquid in the case of axisymmetric vibrations of tank bottoms]. In
*Dinamicheskie naprjazhenija i deformacii v jelementah jenergeticheskogo oborudovanija*[Dynamic stresses and strains in power equipment elements]. Moscow, Nauka Publ., 1977, pp. 76-83. (In Russian) - Sejmov V.M., Ostroverh B.N., Ermolenko A.I.
*Dinamika i sejsmostojkost’ gidrotehnicheskih sooruzhenij*[The dynamics and earthquake resistance of hydraulic structures]. Kiev, Nauk. dumka Pub., 1983, 320 p. (In Russian) - Sejmov V.M., Trofimchuk A.N., Savickij O.A.
*Kolebanija i volny v sloistyh sredah*[Oscillations and waves in layered media]. Kiev, Nauk. dumka Pub., 1990, 224 p. - Pavlova A.V., Rubtsov S.E., Telyatnikov I.S., Zaretskaja M.V. Issledovanie naprjazhennogo sostojanija sloistoj sredy s zhidkim vkljucheniem [Investigation of the stress state of layered medium with liquid inclusion].
*Jekologicheskij vestnik nauchnyh centrov Chernomorskogo jekonomicheskogo sotrudnichestva*[Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2016, no. 1, pp. 71-78. (In Russian) - Vorovich I.I., Aleksandrov V.M., Babeshko V.A.
*Neklassicheskie smeshannye zadachi teorii uprugosti*[Non-classical mixed problem of elasticity theory]. Moscow, Nauka Publ., 1974, 455 p. (In Russian). - Vorovich I.I., Babeshko V.A.
*Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskih oblastei*[Dynamic mixed problem of elasticity theory for nonclassical domains]. Moscow, Nauka Publ., 1979, 319 p. (In Russian) - Rubtsov S.E. Issledovanie ustanovivshihsja kolebanij ogranichennogo ob#ema zhidkosti na uprugom sloe [The study stationary vibrations limited volume of fluid in the elastic layer].
*Izvestija vysshih uchebnyh zavedenij, Severo-Kavkazskij region, estestvennye nauki*[Proc. of the Universities. North-Caucasian region. Natural Sciences], 2000, no. 1, pp. 49-51. (In Russian) - Babeshko V.A.
*Obobshchennyi metod faktorizacii v prostranstvennykh dinamicheskikh smeshannykh zadachakh teorii uprugosti*[Generalized factorization method in spatial dynamic mixed problems of elasticity theory]. Moscow, Nauka Pub., 1984, 265 p. (In Russian) - Nobl B.
*Metod Vinera-Hopfa*[Wiener-Hopf method]. Moscow, Zurubezhnaya lit. Pub., 1962, 280 p. (In Russian)

*Subjects*: mechanics

**Syromyatnikov P.V.**^{1}

**The simulation of surface disturbances elastic semi-infinite medium caused by moving oscillating source**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 82-91.

This paper considers the problem of the motion at a constant speed over the surface of the elastic layer of the mobile source which oscillates. The problem is solved by means of Fourier integral transformations. The homogeneous boundary value problem is being considered in the moving coordinate system, connected with the source. It is assumed that there is a steady-harmonic oscillation. By using the direct method of contour integration, which is described in detail in the work, the elastic surface perturbation obtained by numerical integration of two-dimensional Fourier integrals. This method makes it possible to calculate the wave processes in the same manner as for the fixed sources. Due to its simplicity, the method of integration can be considered as engineering method, although it can be successfully used for scientific purposes. Examples of calculation of the surface perturbations of isotropic elastic layer in flat and three-dimensional problem are given, caused by moving source in a speed range from 0.5 values of Rayleigh waves velocity up to 1.16 of the longitudinal wave velocity, both in the presence of the oscillations and in the absence of oscillations. The method can be applied without any additional modifications for multilayer anisotropic and isotropic materials, as cushion base.

*Keywords*: elastic layer, surface moving oscillating source, surface perturbation, numerical integration.

^{1}Southern Scientific Centre of the Russian Academy of Sciences, Krasnodar Branch, Krasnodar, 350040, Russia

^{2}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: syromyatnikov_pv@mail.ru

- Pflanz G., Garcia J., Schmid G. Vibrations due to loads moving with sub-critical and super-critical velocities on rigid track.
*Proc. Intern. Workshop WAVE2000 Moving Load - Wave Propagation - Vibration Reduction*. Rotterdam, Balkema, 2000, pp. 131-148. - Babeshko V.A, Zinchenko J.F, Glushkov E.V.
*Dinamika neodnorodnih lineino-uprugih sred*[The dynamics of inhomogeneous linear-elastic media]. Moscow, Nauka Pub., 1989, 344 p. (In Russian) - Kalinchuk V.V., Belyankova T.I.
*Dinamika poverhnosti neodnorodnih sred*[The dynamics of the surface of inhomogeneous media]. Moscow, FIZMATLIT Pub., 2009, 240 p. (In Russian) - Belokon A.V., Nasedkin A.V. Vzaimodeistvie dvijushihsya shtampov s uprugimi i vyazkouprugimi telami [Interaction moving punches with elastic and viscoelastic bodies]. In
*Mehhanika kontaktnih vzaimodeistvii*[Contact mechanics]. Moscow, FIZMATLIT Pub., 2001, 672 p. (In Russian) - Kirillova E., Syromyatnikov P., Didenko A. Wave fields generated by an oscillating mechanical source moving on the surface of an elastic semibounded medium.
*Proc. of IC-SCCE - 6th Int. Conf. from Scientific Computing to Computational Engineering, Athens, Greece, 9-12 July*, 2014, vol. 2, pp. 536-544. - D01AKF Subroutine. NAG Fortran Library. Available at: http://www.nag.co.uk/numeric/ FL/FLdescription.asp. (accessed 09.12.2016)

*Subjects*: mechanics

**Fomenko S.I.**^{1}, **Alexandrov A.A.**^{1}

**Wavefields and band-gaps in layered piezoelectric phononic crystals**

Ecological bulettin of research centers of the Black Sea Economic Cooperation, 2016, no. 4, pp. 92-99.

The paper describes a mathematical model of plane wave propagation in elastic and piezoelectric layered phononic crystals with a finite number of unit-cells. Wave motion is excited by an incident plane P- or SV- wave coming from an outer elastic half-space. The wavefield is obtained by using the transfer matrix (T-matrix) method. The methods to construct a T-matrix of a piezoelectric layer and T-matrix of the periodic structure consisted of given number of unit-cells are developed. The transmitted wave field is expressed in terms of expansion in eigenvalues of the T-matrix unit-cell. A classification of band-gaps and pass-bands in layered phononic crystals is proposed. The classification relies on the analysis of the eigenvalues of T-matrix for a unit-cell and the asymptotics for the transmission coefficient, when the number of the unit cells tends to infinity. Two kinds of band-gaps, where the transmission coefficient decays exponentially with the number of unit-cells, are specified. The so-called low transmission pass-bands (LTPB) are introduced in order to identify frequency ranges where the wave transmission is very low, but it does not tend to zero exponentially. Very week conversion effects of modes are observed in the LTPB. The types of band-gaps and their transformation with a change of the incident angles are discussed with numerical examples. The influence of piezoelectric and dielectric constants on width and location of band-gaps is studied.

*Keywords*: piezoelectric phononic crystal, T-matrix method, band-gap, low transmission pass-band.

^{1}Southern Scientific Centre of the Russian Academy of Sciences, Krasnodar Branch, Krasnodar, 350040, Russia

^{2}Kuban State University, Krasnodar, 350040, Russia

Corresponding author’s e-mail: syromyatnikov_pv@mail.ru

- Golenischev-Kutuzov A.V., Golenischev-Kutuzov V.A., Kalimullin R.I.
*Fotonnyie i fononnyie kristallyi. Formirovanie i primenenie v opto- i akustoelektronike*[Photonic and phononic crystals. Their engineering and application in opto- and acoustoelectronics], Moscow, Fiz.-Mat. lit. Publ., 2010. 160 p. (In Russian) - Lucklum R., Li J., Zubtsov M. 1D and 2D Phononic Crystal Sensors.
*Procedia Engineering*, 2010, vol. 5. P. 436–-439. - Deymier P.A.
*Acoustic metamaterials and phononic crystals*. Springer, 2013, 387 p. - Khelif A., Adibi A.
*Phononic Crystals Fundamentals and Applications*. Springer-Verlag, New York, 2016. 245 p. - Yankin S.S., Talbi A., Gerbedoen ZH.-K., Preobrazhenskij V.L., Perno F., Matar O. Bu Rasprostranenie poverhnostnoj akusticheskoj volny v dvumernom fononnom kristalle na p’ezoehlektricheskoj podlozhke [Surface acoustic wave propagation in two-dimensional phononic crystal on a piezoelectric substrate].
*Izvestiya Saratovskogo universiteta. Novaya Seriya. Seriya Fizika*[News of Saratov University. New series. Series Physics], 2014, vol. 14, no. 2, pp. 5-12. (In Russian) - Vorovich I.I., Babeshko V.A., Pryahina O.D.
*Dinamika massivnyih tel i rezonansnyie yavleniya v deformiruemyih sredah*[The dynamics of massive objects and resonance phenomena in deformable media]. Moscow: Nauchnyiy mir Publ., 1999, 246 p. (In Russian) - Glushkov E., Glushkova N., and Zhang C. Surface and pseudo-surface acoustic waves piezoelectically excited in diamond-based structures.
*J. Appl. Phys.*, 2012, no. 112, pp. 064911. - Laude V., Wilm M., Benchabane S., Khelif A. Full band gap for surface acoustic waves in a piezoelectric phononic crystal.
*Physical Review E.*, 2005, vol. 71, no. 3, pp. 036607. - Ponge M.-F., Dubus B., Granger Ch., Vasseur J., Thi M.Ph., Hladky-Hennion A.-Ch. Optimization of a tunable piezoelectric resonator using phononic crystals with periodic electrical boundary conditions.
*Physics Procedia*, 2015, vol. 70, pp. 258-261. - Darinskii A., Shuvalov A., Poncelet O., Kutsenko A. Bulk longitudinal wave reflection/transmission in periodic piezoelectric structures with metallized interfaces.
*Ultrasonics*, 2015, vol. 63, pp. 118-125. - Piliposyan D.G., Ghazaryan K.B., Piliposian G.T. Magneto-electro-elastic polariton coupling in a periodic structure.
*Journal of Physics D: Applied Physics*, 2015, vol. 48, no. 17, pp. 175501. - Golub M.V., Fomenko S.I., Bui T.Q., Zhang Ch., Wang Y.-S. Transmission and band gaps of elastic SH waves in functionally graded periodic laminates.
*Int. J. of Solids and Structures*, 2012, vol. 49, no. 2, pp. 344-354. - Fomenko S.I. Volnovyie polya i zapreschennyie zonyi v kvaziperiodicheskih sloistyih kompozitah [Wave fields and restricted areas in the quasi-layered composites].
*Ekologicheskiy vestnik nauchnyih tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva*, 2013, No. 4, Vol. 1, pp. 120-126. (In Russian) - Fomenko S.I., Golub M.V., Bui T.Q., Zhang Ch., Wang Y.-S. In-plane elastic wave propagation and band-gaps in layered functionally graded phononic crystals.
*Int. J. of Solids and Structures*, 2014, vol. {51}, no. 13), pp. 2491-2503.