حل تحلیلی معادلات امواج در جریان‌ لزج دارای حباب گازی، با در نظر گرفتن انتقال حرارت

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه ریاضی، دانشکده علوم پایه دانشگاه بزرگمهر قائنات، قائن، ایران

2 دانشکده ریاضی دانشگاه مازندران، بابلسر، ایران

چکیده

ابتدا، هدف این مقاله به­دست آوردن معادلات غیرخطی برای توصیف امواج فشار در یک جریان لزج حاوی حباب­های گازی می­باشد. بدین منظور، معادلات غیرخطی مرتبه چهارم و بعضی از موارد خاص آن­ها برای توصیف تغییرات فشار امواج در ترکیب حباب­های گاز در مایع در نظر گرفته شده است. سپس، با در نظر گرفتن تاثیر لزجت و وجود انتقال حرارت، یک رابطه دیفرانسیلی بین فشار ترکیب مایع و گاز نسبت به تغییرات شعاع حباب­های گازی به­دست آمده است. نشان داده شد که معادلات برگرز، کا دی وی و کا دی وی- برگرز حالت­های خاصی برای توصیف فشار امواج در این مورد می­باشند. همچنین، یک روش بسط معرفی شد که به کمک آن می­توان جواب­های دقیق معادلات دیفرانسیل غیرخطی را بدون نیاز به شرایط اولیه و مرزی به­دست آورد. در این روش، با انتخاب ثابت­های خاص در جواب­های به­دست­آمده، جواب­های بیشتری از معادلات امواج در جریان­های لزج قابل محاسبه می­باشد. این ویژگی­ها برتری روش معرفی شده را نسبت به سایر روش­ها نشان می­دهد. علاوه­بر این، جواب­های دقیق این معادلات که کاربردهای فراوانی در علوم و مهندسی دارند، به کمک روش بسط به­دست آمده­اند.

کلیدواژه‌ها

عنوان مقاله [English]

Analytical Solution of Wave Equations in Viscous Flow with Gas Bubbles and Heat Transfer Considerations

نویسندگان [English]

  • Nematollah Kadkhoda 1
  • hossein Jafari 2
1 Department of Mathematics Faculty of Basic Sciences Bozorgmehr University Of Qaenat, Qaen, Iran
2 Department of Mathematics University of Mazandaran, Babolsar, Iran
چکیده [English]

The first purpose of this paper was to obtain non-linear equations for describing pressure waves in a liquid with gas bubbles. For this purpose, non-linear equations of fourth order and some of their special cases for describing pressure waves in a mixture of fluid and gas were considered. Then, considering the influence of heat transfer and viscosity in fluid, a differential relation between pressure and perturbation radius of gas bubbles is obtained. It is shown that the Burgers, the KdV and KdV- Burger’ equations are special cases for describing pressure waves. Also, we introduced a -expansion method, which can obtain exact solutions of non-linear differential equations without requiring the boundary and initial conditions. In this method, by choosing special constants in the obtained solutions, more solutions of these equations can be obtained. These features exhibit the superiority of the introduced approach. compared to the other methods. Furthermore, exact solutions of these equations are obtained using -expansion method, which has many applications in science and engineering.

کلیدواژه‌ها [English]

  • Non-linear Equations
  • Heat Transfer
  • Viscosity
  • Expansion Method
  • Gas Bubbles

مراجع

  1. Kudryashov, N.A. and Sinelshchikoy, D. “Extended Models of Non-linear Waves in Liquid with Gas Bubbles”, Int. J. Non-lin. Mech., Vol. 63, pp. 31–38, 2014.
  2. Jahangiri, A. and Biglari, M. “Thermo-Physical Investigation of the Explosive Rapid Boiling the Contact of a Single Water Drop with Liquid Methane”, The Inst. Mech. Eng., Part E. Vol. 229, No.4, pp. 256- 264, 2015.
  3. Korteweg, D.J. and De Vries, G. “On the Change of Form of Long Waves Advancing in a Rectangular Canal and on a New Type of Long Stationary Waves”, Philos. Mag., Vol. 39, No. 6, pp. 422–443, 1895.
  4. Van, W.L. “On the Equations of Motion for Mixtures of Liquid and Gas Bubbles”, J. Fluid. Mech., Vol. 33, No. 3, pp. 465–474, 1968.
  5. Nakoryakov, V.E., Pokusaev, B.G., and Shraiber, I.R. “Wave Dynamics of Gas and Vapor–Liquid Media”, Energoatomizdat, Moscow, Russia, 1990.
  6. Oganyan, G.G. “Increase in the Compression Wave Amplitude in a Liquid with Dissolved Gas Bubbles”, J. Fluid. Dyn. Vol. 40, No. 1, pp.95–102, 2005.
  7. Nigmatulin, R.I. and Khabeev, N.S. “Heat Transfer of Gas Bubbles in Liquid”, J. Fluid. Dyn., Vol. 9, No. 2, pp. 759–769, 1974.
  8. Aidagulov, R.R., Khabeev, N.S., and Shagapov, V.Sh. “Shock Wave Structure in a Liquid with Gas Bubbles with Allowance for Unsteady Interphase Heat Transfer”, J. Appl. Mech. Tech. Phys. Vol. 18, No. 3, pp. 334-342. 1977.
  9. Ryabov, P.N. “Exact Solutions of the Kudryashov–Sinelshchikov Equation”, Appl. Math. Comput. Vol. 217, No. 3, pp. 3585–3590, 2010.
  10. Kudryashov, N.A. and Sinelshchikov, D.I. “Non-linear Waves in Bubbly Liquids with Consideration for Viscosity and Heat Transfer”, Phys. Lett. A., Vol. 374, No. 19, pp. 2011-2016, 2010.
  11. Kudryashov, N.A. and Sinelshchikov, D.I. “An Extended Equation for the Description of Non-linear Waves in a Liquid with Gas Bubbles”, Wave. Motion. Vol. 50, No. 3, pp. 351-362, 2013.
  12. Kudryashov, N.A. and Sinelshchikov, D.I. “Analytical Solutions of the Rayleigh Equation for Empty and Gas-Filled Bubble”, J. Phys. A: Math. Theor., Vol. 47, No. 40, pp. 52-58, 2014.
  13. Klotz, A.R. “Bubble Dynamics in N-dimensions”, Phys. Fluids. Vol. 25, No. 8, pp. 082-109, 2013.
  14. Kudryashov, N.A. and Sinelshchikov, D.I. “Analytical Solutions for Problems of Bubble Dynamics”, Phys. Lett. A. Vol. 379, No. 8, pp. 798-802, 2015.
  15. Fuster, D. and Montel, F. “Mass Transfer Effects on Linear Wave Propagation in Diluted Bubbly Liquids”, J. Fluid. Mech. Vol. 779, pp. 598-621, 2015.
  16. Jafari, Y., Taebie Rahni, M., and Adamian, A. “Computational Modeling of Flow and Natural Convective Heat Transfer in Porous Media, Using LBM-Effects of Nano-Fluidicity and Domain Geometry”, Fluid. Mech. Aerodyn., Vol. 3, No. 4, pp. 17-30, 2015 (In Persian).
  17. Nakoryakov, V.E., Sobolev, V.V., and Shreiber, I.R. “Long Wave Perturbation in a Gas–Liquid Medium”, Fluid. Dyn. Vol. 7, No. 5, pp. 763–768, 1972.  
  18. Nigmatulin, R.I. “Dynamics of Multiphase Media, Part 2”, Hemisphere, New York, United States, 1991.
  19. Rayleigh, L. “On the Pressure Developed in a Liquid during the Collapse of a Spherical Void", Philos. Mag., Vol. 34, No. 200, pp. 94–98, 1917.
  20. Plesset, M.S., and Prosperetti, A. “Bubble Dynamics and Cavitation”, Annu. Rev. Fluid. Mech., Vol. 9, pp. 145–185, 1977.
  21. Wijngaarden, L.V. “One-dimensional Flow of Liquid Containing Small Gas Flow”, Annu. Rev. Fluid. Mech. Vol. 4, No. 1, pp. 369–396, 1972.
  22. Kudryashov, N.A. and Sinelshchikov, D.I. “Non-linear Evolution Equations for Describing Waves in Bubbly Liquids with Viscosity and Heat Transfer Consideration”, Appl. Math. Comput. Vol. 217, No. 1, pp. 414-421, 2010.
  23. Kudryashov, N.A. and Chernyavskii, I.L. “Non-linear Waves in Fluid Flow Through Viscoelastic Tube”, J. Fluid. Dyn., Vol. 41, No. 1, pp. 49–62, 2006.
  24. Jafari, H., Kadkhoda, N., and Biswas, A. “The G'/G-Expansion Method for Solutions of Evolution Equations from Isothermal Magnetostatic Atmospheres”, J. King. Saud. Univ. Sci. Vol. 25, No. 1, pp. 57–62, 2013.

فایل ها

سابقه مقاله

  • تاریخ دریافت: 15 خرداد 1397
  • تاریخ پذیرش: 22 دی 1397
  • تاریخ انتشار: 01 تیر 1398

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