On nonlocal fractal laminar steady and unsteady flows. (English) Zbl 1484.76025
Summary: In this study, we join the concept of fractality introduced by J. Li and M. Ostoja-Starzewski [Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 378, No. 2172, Article ID 20190288, 16 p. (2020; Zbl 1462.82031)] with the concept of nonlocality to produce a new set of nonlocal fractal fluid equations of motion. Both the unsteady and steady laminar flows are discussed. It is revealed that a damped wave equation emerges from the nonlocal fractal Navier-Stokes equation, a result which could lead to a better understanding of fluids turbulence.
Citations:
Zbl 1462.82031References:
| [1] | Sreenivasan, KR; Meneveau, C., The fractal facets of turbulence, J. Fluid Mech., 173, 357-386 (1986) |
| [2] | Falconer, KF, The Geometry of Fractal Sets (1985), Cambridge: Cambridge University Press, Cambridge · Zbl 0587.28004 |
| [3] | Feder, J., Fractals (1988), New York: Plenum Press, New York · Zbl 0648.28006 |
| [4] | Carpinteri, A.; Mainardi, G., Fractals and Fractional Calculus in Continuum Mechanics (1997), New York: Springer, New York · Zbl 0917.73004 |
| [5] | Tarasov, VE, Flow of fractal fluid in pipes: non-integer dimensional space approach, Chaos Solitons Fractals, 67, 26-37 (2014) · Zbl 1349.76066 |
| [6] | Lazopoulos, KA; Lazopoulos, AK, Fractional vector calculus and fluid mechanics, J. Mech. Behav. Mater., 26, 43-54 (2017) |
| [7] | Li, J.; Ostoja-Starzewski, M., Thermo-poromechanics of fractal media, Phil. Trans. R. Soc. A, 378, 20190288 (2020) · Zbl 1462.82031 |
| [8] | Balankin, AS; Elizarraraz, BE, Map of fluid flow in fractal porous medium into fractal continuum flow, Phys. Rev. E, 85, 056314 (2012) |
| [9] | Butera, S.; di Paola, M., A physically based connection between fractional calculus and fractal geometry, Ann. Phys., 350, 146-158 (2014) · Zbl 1344.26001 |
| [10] | Balankin, AS; Nena, B.; Susarrey, O.; Samayoa, D., Steady laminar flow of fractal fluids, Phys. Lett. A, 381, 623-638 (2017) · Zbl 1372.76006 |
| [11] | Tarasov, VE, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (2010), Berlin: Springer, Berlin · Zbl 1214.81004 |
| [12] | Gibson, RL; Toksoz, MN, Permeability estimation from velocity anisotropy in fractured rock, J. Geophys. Res., 95, 15643-15655 (1990) |
| [13] | Huh, K.; Oh, D.; Son, SY, Laminar flow assisted anisotropic bacteria absorption for chemotaxis delivery of bacteria-attached microparticle, Micro and Nano Syst. Lett., 4, 1 (2016) |
| [14] | Galindo-Torres, SA; Scheuermann, A.; Li, L., Numerical study on the permeability in a tensorial form for laminar flow in anisotropic porous media, Phys. Rev. E, 86, 046306 (2012) |
| [15] | Khalil, R.; Horani, M.; Yousef, A.; Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65-70 (2014) · Zbl 1297.26013 |
| [16] | Chen, W., Time-space fabric underlying anomalous diffusion, Chaos Solitons Fractals, 28, 923-929 (2006) · Zbl 1098.60078 |
| [17] | Goddard, JD, A weakly nonlocal anisotropic fluid model for inhomogeneous Stokesian suspensions, Phys. Fluids, 20, 040601 (2008) · Zbl 1182.76273 |
| [18] | Viallat, A.; Abkarian, M., Dynamics of Blood Cell Suspensions in Microflows (2019), Boca Raton: CRC Press, Boca Raton |
| [19] | Cemal Eringen, A., On nonlocal microfluid mechanics, Int. J. Eng. Sci., 11, 291-306 (1973) · Zbl 0256.76004 |
| [20] | El-Nabulsi, RA, Dynamics of pulsatile flows through microtubes from nonlocality, Mech. Res. Commun., 86, 18-26 (2017) |
| [21] | Adams, DJ, Particle trajectories in classical fluid are not anomalously fractal, Mol. Phys., 59, 1277-1281 (1986) |
| [22] | Eringen, AC, Linear theory of nonlocal elasticity and dispersion of plane waves, Int. J. Eng. Sci., 10, 561-575 (1972) · Zbl 0241.73005 |
| [23] | Speziale, CG; Eringen, AC, Nonlocal fluid mechanics description of wall turbulence, Comput. Math. Appl., 7, 27-41 (1981) · Zbl 0464.76046 |
| [24] | Hansen, JS; Daivis, PJ; Travis, KP; Todd, BP, Parameterization of the nonlocal viscosity kernel for an atomic fluid, Phys. Rev. E, 76, 041121 (2007) |
| [25] | Eidelman, SD; Ivasyshen, SD; Kochubei, AN, Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Volume 152 of Operator Theory: Advances and Applications (2004), Basel: Birkhäuser Verlag, Basel · Zbl 1062.35003 |
| [26] | Suykens, JAK, Extending Newton’s law from nonlocal-in-time kinetic energy, Phys. Lett. A, 373, 1201-1211 (2009) · Zbl 1228.70004 |
| [27] | Kamalov, TF, Classical and quantum-mechanical axioms with the higher time derivative formalism, J. Phys. Conf. Ser., 442, 012051 (2013) |
| [28] | Kamalov, TF, Model of extended mechanics and non-local hidden variables for quantum theory, J. Russ. Laser Res., 30, 466-471 (2009) |
| [29] | Kamalov, TF, Quantum corrections of Newton’s law of motion, Symmetry, 12, 63 (2020) |
| [30] | El-Nabulsi, RA, Free variable mass nonlocal systems, jerks and snaps, and their implications in rotating fluids in rockets, Acta Mech. (2020) · Zbl 1458.70018 · doi:10.1007/s00707-020-02843-z |
| [31] | El-Nabulsi, RA; Moaaz, O.; Bazighifan, O., New results for oscillatory behavior of fourth-order differential equations, Symmetry, 12, 136 (2020) |
| [32] | Bazighifan, P.; Moaaz, O.; El-Nabulsi, RA; Muhib, A., Some new oscillation results for fourth-order neutral differential equations with delay argument, Symmetry, 12, 1248 (2020) |
| [33] | Moaaz, O.; El-Nabulsi, RA; Bazighifan, O., Oscillatory behavior of fourth-order differential equations with neutral delay, Symmetry, 12, 371 (2020) |
| [34] | Moaaz, O.; El-Nabulsi, RA; Bazighifan, O., Behavior of non-oscillatory solutions of fourth-order neutral differential equations, Symmetry, 12, 477 (2020) |
| [35] | El-Nabulsi, RA, Quantum LC-circuit satisfying the Schrödinger-Fisher-Kolmogorov equation and quantization of DC-dumper Josephson parametric amplifier, Phys. E Low Dimens. Syst. Microstruct., 112, 115-120 (2019) |
| [36] | El-Nabulsi, RA, Fourth-order Ginzburg-Landau differential equation a la Fisher-Kolmogorov and its implications in superconductivity, Phys. C Supercond. Appl., 567, 1353545 (2019) |
| [37] | El-Nabulsi, RA, Nonlocal-in-time kinetic energy description of superconductivity, Phys. C Supercond. Appl., 577, 1353716 (2020) |
| [38] | El-Nabulsi, RA, On nonlocal complex Maxwell equations and wave motion in electrodynamics and dielectric media, Opt. Quantum Electron., 50, 170 (2018) |
| [39] | Dong, B-Q; Chen, Z-M, On upper and lower bounds of higher-order derivatives for solutions to the 2D micropolar fluid equations, J. Math. Anal. Appl., 334, 1386-1399 (2007) · Zbl 1158.35074 |
| [40] | Khalil, N.; Garz, V.; Santos, A., Hydrodynamic Burnett equations for inelastic Maxwell models of granular gases, Phys. Rev. E, 89, 052201 (2014) |
| [41] | Gorban, AN; Karlin, IV, Beyond Navier-Stokes equations: capillarity of ideal gas, Contemp. Phys., 58, 70-90 (2017) |
| [42] | Feynman, RP, Space-time approach to relativistic quantum mechanics, Rev. Mod. Phys., 20, 367-387 (1948) · Zbl 1371.81126 |
| [43] | Nelson, E., Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150, 1079-1085 (1996) |
| [44] | Li, Z-Y; Fu, J-L; Chen, L-Q, Euler-Lagrange equation from nonlocal-in-time kinetic energy of nonconservative system, Phys. Lett. A, 374, 106-109 (2009) · Zbl 1235.70035 |
| [45] | El-Nabulsi, RA, Non-standard non-local-in-time Lagrangians in classical mechanics, Qual. Theor. Dyn. Syst., 13, 149-160 (2014) · Zbl 1305.70043 |
| [46] | El-Nabulsi, RA, Complex backward-forward derivative operator in non-local-in-time Lagrangians mechanics, Qual. Theor. Dyn. Syst., 16, 223-234 (2016) · Zbl 1499.70021 |
| [47] | El-Nabulsi, RA, Nonlocal-in-time kinetic energy in nonconservative fractional systems, disordered dynamics, jerk and snap and oscillatory motions in the rotating fluid tube, Int. J. Nonlinear Mech., 93, 65-81 (2017) |
| [48] | El-Nabulsi, RA, Orbital dynamics satisfying the 4th order stationary extended Fisher-Kolmogorov equation, Astrodyn., 4, 31-39 (2020) |
| [49] | El-Nabulsi, RA, Fractional nonlocal Newton’s law of motion and emergence of Bagley-Torvik equation, J. Peridyn. Nonlocal Model., 2, 50-58 (2020) |
| [50] | El-Nabulsi, RA, Nonlocal wave equations from a non-local complex backward-forward derivative operator, Waves Complex Rand. Med. (2019) · doi:10.1080/17455030.2019.16 |
| [51] | El-Nabulsi, RA, Jerk in Planetary systems and rotational dynamics, nonlocal motion relative to earth and nonlocal fluid dynamics in rotating earth frame, Earth Moon Planet., 122, 15-41 (2018) |
| [52] | El-Nabulsi, RA, Time-nonlocal kinetic equations, jerk and hyperjerk in plasmas and solar physics, Adv. Space Res., 61, 2914-2931 (2018) |
| [53] | Sreenivasan, KR, Fractals in fluid mechanics, Fractals, 2, 253-263 (1994) · Zbl 0968.76576 |
| [54] | Sreenivasan, KR, Fractals and multifractals in fluid turbulence, Ann. Rev. Fluid Mech., 23, 539-600 (1991) |
| [55] | Kerr, RM, Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence, J. Fluid Mech., 153, 31-58 (1985) · Zbl 0587.76080 |
| [56] | Agrawal, RK; Abdallah, NB, BGK-Burnett equations: a new set of second-order hydrodynamic equations for flows in continuum-transition regime, Transport in Transition Regimes. The IMA Volumes in Mathematics and Its Applications (2004), New York: Springer, New York · Zbl 1116.76438 |
| [57] | Agrawal, RK; Yun, K-Y, Burnett equations for simulation of transitional flows, Appl. Mech. Rev., 55, 219-240 (2000) |
| [58] | Pnueli, D.; Gutfinger, C., Fluid Mechanics (1992), Cambridge: Cambridge University Press, Cambridge · Zbl 0784.76001 |
| [59] | Miranda, AIP; Oliveira, PJ; Pinho, FT, Steady and unsteady laminar flows of Newtonian and generalized Newtonian fluids in a planar T-junction, Int. J. Numer. Methods Fluids, 57, 295-328 (2008) · Zbl 1241.76122 |
| [60] | Ramana Murthy, CV; Kulkarni, SB, Unsteady laminar flow of visco-elastic fluid of second order type between two parallel plates, Ind. J. Eng. Mater. Sci., 12, 51-57 (2005) |
| [61] | Park, JK; Park, SO; Hyun, JM, Flow regimes of unsteady laminar flow past a slender elliptic cylinder at incidence, Int. J. Heat Fluid Flow, 10, 311-317 (1989) |
| [62] | Das, D.; Arakeri, JH, Unsteady laminar duct flow with a given volume rate variation, J. Appl. Mech., 67, 274-281 (2000) · Zbl 1110.74405 |
| [63] | Gupta, PC, Unsteady, laminar flow of a viscous incompressible fluid through porous media in a hexagonal channel, Current Sci., 49, 295-297 (1980) |
| [64] | Shapiro, A., An analytical solution of the Navier-Stokes equations for unsteady backward stagnation-point flow with injection or suction, Z. Angew. Math. Mech., 86, 281-290 (2006) · Zbl 1094.35095 |
| [65] | Cannarsa, P.; de Prato, F.; Zolesio, J-P, The damped wave equation in a moving domain, J. Differ. Equ., 85, 1-16 (1990) · Zbl 0786.35089 |
| [66] | Selvadurai, APS, Partial Differential Equations in Mechanics 1 (2000), Berlin: Springer-Verlag, Berlin · Zbl 0967.35001 |
| [67] | Che, T-C; Duan, H-F, Evaluation of plane wave assumption in transient laminar pipe flow modeling and utilization, Proc. Eng., 154, 959-966 (2016) |
| [68] | Tikoja, H., Auriemma, F., Lavrentjev, J.: Damping of acoustic waves in straight ducts and turbulent flow conditions, SAE Technical Paper 2016-01-1816, (2016) |
| [69] | Boonen, R., Sas, P., Vandenbulck, E.: Determination of the acoustic damping characteristics of an annular tail pipe. In: Proceedings of ISMA2010 Including USD2010 Conference. ISMA2010, Leuven, 20-22 September 2010 (art.nr. 684) (pp. 47-57) |
| [70] | Ogawa, A.; Tokiwa, S.; Mutou, M., Damped oscillation of liquid column in vertical U-tube for Newtonian and non-Newtonian liquids, J. Therm. Sci., 16, 289-300 (2007) |
| [71] | Borodulin, V.I., Kachanov, Y.S., Roschektayev, P.: Experimental detection of deterministic turbulence. J. Turbul. 12, Article N23 (2011) |
| [72] | Mahady, K.; Afkhami, S.; Diez, S.; Kondic, L., Comparison of Navier-Stokes simulations with long-wave theory: study of wetting and dewetting, Phys. Fluids, 25, 112103 (2013) · Zbl 1320.76032 |
| [73] | Li, J.; Ostoja-Starzewski, M., Fractal solids, product measures and fractional wave equations, Proc. R. Soc. A, 465, 2521-2536 (2009) · Zbl 1186.74011 |
| [74] | Ostoja-Starzewski, M., On turbulence in fractal porous media, ZAMP, 59, 1111-1117 (2008) · Zbl 1304.76024 |
| [75] | Li, J.; Ostoja-Starzewski, M., Micropolar continuum mechanics of fractal media, Int. J. Eng. Sci., 49, 1302-1310 (2011) · Zbl 1423.74040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
