Fractional Navier–Stokes Equation from Fractional Velocity Arguments and Its Implications in Fluid Flows and Microfilaments

References

[1] M. S. Miller and B. Ross, An introduction to the fractional integrals and derivatives-theory and application, Wiley, New York, 1993.Search in Google Scholar

[2] I. Podlubny, Fractional differential equations, Academic, New York, 1999.Search in Google Scholar

[3] R. Herrmann, Fractional calculus: An introduction for physicists, World Scientific Publishing Company, Singapore, 2011.10.1142/8072Search in Google Scholar

[4] R. Hilfer, Applications of fractional calculus in physics, World Scientific Publishing, River Edge, NJ, USA, 2000.10.1142/3779Search in Google Scholar

[5] X. J. Yang, H. M. Srivastava and C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equation, Rom. Rep. Phys. 67 (2015), 752–761.Search in Google Scholar

[6] M. Safari, D. D. Ganji and M. Moslemi, M: Application of he’s variational iteration method and Adomian’s decomposition method to the fractional KdV-Burgers-Kuramoto equation, Comput. Math. Appl. 58 (2009), 2091–2097.10.1016/j.camwa.2009.03.043Search in Google Scholar

[7] Y. F. Zakariya, Y. O. Afolabi, R. I. Nuruddeen and I. O. Sarumi, Analytical solutions to fractional fluid flows and oscillatory process models, Fractal Fract. 2 (18) (2018), 1–12.10.3390/fractalfract2020018Search in Google Scholar

[8] F. Gao and X.-J. Yang, Fractional Maxwell fluid with fractional derivative without singular kernel, Thermal Sci. 20 (2016), S871–S877.10.2298/TSCI16S3871GSearch in Google Scholar

[9] Y. Bai, Y. Jiang, F. Liu and Y. Zhang, Numerical analysis of fractional MHD Maxwell fluid with the effects of convection heat transfer condition and viscous dissipation, AIP Adv. 7 (2017), 126309–126314.10.1063/1.5011789Search in Google Scholar

[10] B. H. Yan, L. Yu and Y. H. Yang, Study of oscillating flow in rolling motion with the fractional derivative Maxwell model, Prog. Nucl. Energy. 53 (2011), 132–138.10.1016/j.pnucene.2010.07.009Search in Google Scholar

[11] D. Tripathi, Peristaltic transport of fractional Maxwell fluids in uniform tubes: Applications in endoscopy, Comp. Math. Appl. 62 (2011), 1116–1126.10.1016/j.camwa.2011.03.038Search in Google Scholar

[12] L. C. Zheng, K. N. Wang and Y. T. Gao, Unsteady flow and heat transfer of a generalized Maxwell fluid due to a hyperbolic sine accelerating plate, Comp. Math. Appl. 61 (2011), 2209–2212.10.1016/j.camwa.2010.09.017Search in Google Scholar

[13] M. Athar, A. U. Awan, C. Fetecau and M. Rana, Unsteady flow of a Maxwell fluid with fractional derivatives in a circular cylinder moving with a nonlinear velocity, Quest. Math. 37 (2014), 139–156.10.2989/16073606.2014.871445Search in Google Scholar

[14] M. Jamil and N. A. Khan, Slip effects in fractional viscoelastic fluids, Int. J. Diff. Equa. Article ID193813. 2011 (2011), 19.10.1155/2011/193813Search in Google Scholar

[15] S. Wang and M. Xu, Axial Couette flow of two kinds of fractional viscoelastic fluids in an annulus, Nonlinear Anal: Real World Appl. 10 (2009), 1087–1096.10.1016/j.nonrwa.2007.11.027Search in Google Scholar

[16] A. Heibig and L. I. Palade, On the rest state stability of an objective fractional derivative viscoelastic fluid model, J. Math. Phys. 49 (2008), 043101.10.1063/1.2907578Search in Google Scholar

[17] W. Tan and M. Xu, Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model, Acta Mech. Sinica. 18 (2002), 342–349.10.1007/BF02487786Search in Google Scholar

[18] T. Hayat, S. Zaib, C. Fetecau and C. Fetecau, Flows in a fractional generalized Burgers’ fluid, J. Porous Med. 13 (2010), 725–739.10.1615/JPorMedia.v13.i8.40Search in Google Scholar

[19] A. J. Nazari, A. F. Nasiry and S. Honma, Effect of fractional flow curves on the recovery of different types of oil in petroleum reservoirs, Proc. Schl. Eng. Tokai Univ., Ser. E41 (2016), 53–58.Search in Google Scholar

[20] K. Wang and S. Liu, Analytical study of time fractional Navier-Stokes equation by using transform methods, Adv Diff. Equa. 61 (2016) (2016), 1–12.10.1186/s13662-016-0783-9Search in Google Scholar

[21] N. A. Khan, Analytical study of Navier-Stokes equation with fractional orders using He’s homotopy perturbation and variational iteration methods, Int. J. Nonlinear Sci. Numer. Simul. 10 (9) (2009), 1127–1134.10.1515/IJNSNS.2009.10.9.1127Search in Google Scholar

[22] Y. Zhou and L. Peng, On the time-fractional Navier-Stokes equations, Comput. Math. Appl. 73 (2017), 874–891.10.1016/j.camwa.2016.03.026Search in Google Scholar

[23] Y. Zhou and L. Peng, Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Comput. Math. Appl. 73 (2017), 1016–1027.10.1016/j.camwa.2016.07.007Search in Google Scholar

[24] D. Kumar, J. Singh and S. Kumar, A fractional model of Navier-Stokes equation arising in unsteady flow of a viscous fluid, J. Ass. Arab. Univ. Basic Appl. Sci. 17 (2015), 14–19.10.1016/j.jaubas.2014.01.001Search in Google Scholar

[25] X. Li, X. Yang and Y. Zhang, Error estimates of mixed finite element methods for time fractional Navier-Stokes equations, J. Sci. Comput. 70 (2017), 500–515.10.1007/s10915-016-0252-3Search in Google Scholar

[26] S. Momani and Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method, Appl. Math. Comput. 177 (2006), 488–494.10.1016/j.amc.2005.11.025Search in Google Scholar

[27] A. Yildirim, Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method, Int. J. Numer. Meth. Heat & Fluid Flow. 20 (2010), 186–200.10.1108/09615531011016957Search in Google Scholar

[28] H. Zhang, X. Jiang and X. Yang, A time-space spectral method for the time-space fractional Fokker-Planck equation and its inverse problem, Appl. Math. Comput. 320 (2018), 302–318.10.1016/j.amc.2017.09.040Search in Google Scholar

[29] B. Yu, X. Jiang and H. Xu, A novel compact numerical method for solving the two dimensional nonlinear fractional reaction-subdiffusion equation, Numer. Algorithms. 68 (2015), 923–950.10.1007/s11075-014-9877-1Search in Google Scholar

[30] A. Karcı, A new approach for fractional order derivative and its applications, Univ. J. Eng. Sci. 1 (2013), 110–117.10.13189/ujes.2013.010306Search in Google Scholar

[31] A. Karcı, The physical and geometrical interpretation of fractional order derivatives, Univ. J. Eng. Sci. 3 (2015), 53–63.10.13189/ujes.2015.030401Search in Google Scholar

[32] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Frac. Diff. Appl. 1 (2015), 73–85.Search in Google Scholar

[33] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular Kernel, Prog. Frac. Diff. Appl. 1 (2015), 87–92.Search in Google Scholar

[34] A. Atangana, On the new fractional derivative and application to nonlinear Fisher’s reaction-diffusion equation, Appl. Math. Comp. 273 (2016), 948–956.10.1016/j.amc.2015.10.021Search in Google Scholar

[35] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), 1–15.Search in Google Scholar

[36] A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernel, Therm. Sci. 20 (2016), 757–763.10.2298/TSCI160111018ASearch in Google Scholar

[37] D. Prodanov, Regularization of derivatives on non-differentiable points, J, Phys.: Conf. Ser. 701 (2016), 012031.10.1088/1742-6596/701/1/012031Search in Google Scholar

[38] D. Prodanov, Some applications of fractional velocities, Fract. Calc. Appl. Anal. 19 (2016), 173–187.10.1515/fca-2016-0010Search in Google Scholar

[39] D. Prodanov, Conditions for continuity of fractional velocity and existence of fractional Taylor expansions, Chaos Solitons Fractals. 102 (2017), 236–244.10.1016/j.chaos.2017.05.014Search in Google Scholar

[40] D. Prodanov, Fractional velocity as a tool for the study of non-linear problems, Fractals Fract. 2 (2018), 1–23.10.3390/fractalfract2010004Search in Google Scholar

[41] D. Prodanov, On the conditions for existence and continuity of fractional velocity, Chaos Solitons Fractals. 102 (2017), 236–244.10.1016/j.chaos.2017.05.014Search in Google Scholar

[42] A. S. Balankin, The concept of multifractal elasticity, Phys. Lett. A210 (1996), 51–59.10.1016/0375-9601(95)00874-8Search in Google Scholar

[43] A. S. Balankin and B. E. Elizarraraz, Map of fluid flow in fractal porous medium into fractal continuum flow, Phys. Rev. E85 (2012), 056314.10.1103/PhysRevE.85.056314Search in Google Scholar PubMed

[44] A. S. Balankin and B. Espinoza, Hydrodynamics of fractal continuum flow, Phys. Rev. E. 85 (2012), 025302(R.10.1103/PhysRevE.85.025302Search in Google Scholar PubMed

[45] A. S. Balankin, Steady laminar flow of fractal fluids, Phys. Lett. A381 (2017), 623–628.10.1016/j.physleta.2016.12.007Search in Google Scholar

[46] A. K. Golmankhaneh and D. Baleanu, Diffraction from fractal grating Cantor sets, J. Mod. Optic. 63 (2016), 1364–1369.10.1080/09500340.2016.1148209Search in Google Scholar

[47] A. K. Golmankhaneh and D. Baleanu, About Schrödinger equation on fractals curves imbedding in, Int. J. Theor. Phys. 54 (2015), 1275–1282.10.1007/s10773-014-2325-0Search in Google Scholar

[48] A. K. Golmankhaneh and D. Baleanu, Fractal calculus involving Gauge function, Commun. Nonlinear Sci. 37 (2016), 125–130.10.1016/j.cnsns.2016.01.007Search in Google Scholar

[49] A. K. Golmankhaneh and D. Baleanu, Non-local integrals and derivatives on fractal sets with applications, Open Phys. 14 (2016), 542–548.10.1515/phys-2016-0062Search in Google Scholar

[50] A. K. Golmankhaneh and C. Tunc, On the Lipschitz condition in the fractal calculus, Chaos Solitons Fractals. 95 (2017), 140–147.10.1016/j.chaos.2016.12.001Search in Google Scholar

[51] Q. A. Naqvi and M. Zubair, On cylindrical model of electrostatic potential in fractional dimensional space, Optik: Int. J. Light Electron Opt, USSR, 127 (2016), 3243–3247.10.1016/j.ijleo.2015.12.019Search in Google Scholar

[52] M. Zubair, M. J. Mughal and Q. A. Naqvi, The wave equation and general plane wave solutions in fractional space, Prog. Electromagnet. Res. Lett. 19 (2010), 137–146.10.2528/PIERL10102103Search in Google Scholar

[53] A. Karci and A. Karadoğan, Fractional order derivative and relationship between derivative and complex functions, IECMSA-2013: 2nd International Eurasian Conference on Mathematical Sciences and Applications, Sarajevo, Bosnia and Herzogovina, 2013.Search in Google Scholar

[54] A. Karci and A. Karadoğan, Fractional order derivative and relationship between derivative and complex functions, Math. Sci. Appl. E-Notes. 2 (2014), 44–54.Search in Google Scholar

[55] L. D. Landau and E. M. Lifshitz, Fluid mechanics, course of theoretical physics, 2nd, Pergamon Press, 1987.Search in Google Scholar

[56] T. D. Placek and F. R. Notes, Lectures given at the chemical engineering department, Auburn University, Alabama, 2013.Search in Google Scholar

[57] M. Zubair, M. J. Mughal and Q. A. Naqvi, On electromagnetic wave propagation in fractional space, Nonlinear Anal: Real World Appl. 12 (2011), 2844–2850.10.1016/j.nonrwa.2011.04.010Search in Google Scholar

[58] M. Zubair, M. J. Mughal and Q. A. Naqvi, An exact solution of spherical wave in D-dimensional fractional space, J. Electrom. Waves Appl. 25 (2011), 1481–1491.10.1163/156939311796351605Search in Google Scholar

[59] M. Zubair, M. J. Mughal and Q. A. Naqvi, Electromagnetic wave propagation in fractional space, in: Electromagnetic fields and waves in fractional dimensional space, springer briefs in applied sciences and technology, Springer, Berlin, Heidelberg, 2012.10.1007/978-3-642-25358-4Search in Google Scholar

[60] F. H. Stillinger, Axiomatic basis for spaces with noninteger dimension, J. Math. Phys. 18 (1977), 1224–1234.10.1063/1.523395Search in Google Scholar

[61] R. D. Vale, The molecular motor toolbox for intracellular transport, Cell. 112 (2003), 467–480.10.1016/S0092-8674(03)00111-9Search in Google Scholar

[62] I. M. Cheeseman and A. Desai, Molecular architecture of the kinetochore-microtubule interface, Nature Rev. Mol. Cell Biol. 9 (2008), 33–46.10.1038/nrm2310Search in Google Scholar PubMed

[63] I. M. Kulic, A. E. X. Brown, H. Kim, C. Kural, B. Blehm, P. R. Selvin, P. C. Nelson and V. I. Gelfand, The role of microtubule movement in bidirectional organelle transport, Pnas. 105 (2008), 10011–10016.10.1073/pnas.0800031105Search in Google Scholar PubMed PubMed Central

[64] C. Cai, Q. Sun and I. D. Boyd, Gas flows in microchannels and microtubes, J. Fluid Mech. 589 (2007), 305–315.10.1017/S0022112007008178Search in Google Scholar

[65] K. C. Pong, C. M. Ho and J. Q. Liu, Nonlinear pressure distribution in uniform microchannels. In Application of Microfabrication to Fluid Mechanics, ASME Winter Annual Meeting. Chicago. pp. 51–56, 1994.Search in Google Scholar

[66] L. Brandt, Fluid mechanics, lectures given at department of mechanics, Royal Institute of Technology (KTH) SE-100 44, Stockholm Sweden, (2015).Search in Google Scholar

[67] J. A. K. Suykens, Extending Newton’s law from nonlocal-in-time kinetic energy, Phys. Lett. A373 (2009), 1201–1211.10.1016/j.physleta.2009.01.065Search in Google Scholar

[68] Z.-Y. Li, J.-L. Fu and L.-Q. Chen, Euler-Lagrange equation from nonlocal-in-time kinetic energy of nonconservative system, Phys. Lett. A374 (2009), 106–109.10.1016/j.physleta.2009.10.080Search in Google Scholar