Transient Electro-osmotic Slip Flow of an Oldroyd-B Fluid with Time-fractional Caputo-Fabrizio Derivative

Document Type: Research Paper


1 Department of Mathematics, Lahore Leads University, Lahore Pakistan

2 Abdus Salam School of Mathematical Sciences, GC University, Lahore, 54600, Pakistan

3 School of Mathematics and Statistics, Shandong University, Weihai, 264209, PR China

4 School of Civil Engineering, Shandong University, Jinan, 250061, PR China

5 FAST, University Tun Hussein Onn Malaysia, 86400, Parit Raja, Batu Pahat, Johor State, Malaysia

6 Public Authority of Applied Education and Training, College of Technological Studies, Applied Science Department, Shuwaikh, Kuwait


In this article, the electro-osmotic flow of Oldroyd-B fluid in a circular micro-channel with slip boundary condition is considered. The corresponding fractional system is represented by using a newly defined time-fractional Caputo-Fabrizio derivative without singular kernel. Closed form solutions for the velocity field are acquired by means of Laplace and finite Hankel transforms. Additionally, Stehfest’s algorithm is used for inverse Laplace transform. The solutions for fractional Maxwell, ordinary Maxwell and ordinary Newtonian fluids are obtained as limiting cases of the obtained solution. Finally, the influence of fractional and some important physical parameters on the fluid flow are spotlighted graphically.


Main Subjects

[1] Ghosal, S., Fluid mechanics of electroosmotic flow and its effect on band broadening in capillary electrophoresis, Electrophoresis, 25 (2004) 214-228.

[2] Wang, X., Cheng, C., Wang, S., Liu, S., Electroosmotic pumps and their applications in microfluidic systems, Microfluidics and Nanofluidics, 6 (2009) 145.

[3] Bhattacharyya, S., Zheng, Z., Conlisk, A.T., Electro-osmotic flow in two-dimensional charged micro- and nanochannels, Journal of Fluid Mechanics, 540 (2005) 247-267.

[4] Wang, C.Y., Liu, Y.H., Chang, C.C., Analytical solution of electroosmotic flow in a semicircular microchannel, Physics of Fluids, 20 (2008) 063105.

[5] Chang, S.-H., Electroosmotic flow in a dissimilarly charged slit microchannel containing salt-free solution, European Journal of Mechanics - B/Fluids, 34 (2012) 85-90.

[6] Das, S., Chakraborty, S., Analytical solutions for velocity, temperature and concentration distribution in electroosmotic microchannel flows of a non-Newtonian bio-fluid, Analytica Chimica Acta, 559 (2006) 15-24.

[7] Chakraborty, S., Electroosmotically driven capillary transport of typical non-Newtonian biofluids in rectangular microchannels, Analytica Chimica Acta, 605 (2007) 175-184.

[8] Tan, Z., Qi, H., Jiang, X., Electroosmotic flow of Eyring fluid in slit microchannel with slip boundary condition, Applied Mathematics and Mechanics, 35 (2014) 689-696.

[9] Ferrás, L.L., Afonso, A.M., Alves, M.A., Nóbrega, J.M., Pinho, F.T., Analytical and numerical study of the electro-osmotic annular flow of viscoelastic fluids, Journal of Colloid and Interface Science, 420 (2014) 152-157.

[10] Tang, G.H., Li, X.F., He, Y.L., Tao, W.Q., Electroosmotic flow of non-Newtonian fluid in microchannels, Journal of Non-Newtonian Fluid Mechanics, 157 (2009) 133-137.

[11] Hu, Y., Werner, C., Li, D., Electrokinetic Transport through Rough Microchannels, Analytical Chemistry, 75 (2003) 5747-5758.

[12] Sadr, R., Yoda, M., Zheng, Z., Conlisk, A.T., An experimental study of electro-osmotic flow in rectangular microchannels, Journal of Fluid Mechanics, 506 (2004) 357-367.

[13] Hsieh, S.S., Lin, H.C., Lin, C.Y., Electroosmotic flow velocity measurements in a square microchannel, Colloid and Polymer Science, 284 (2006) 1275-1286.

[14] Kulish, V.V., Lage, J.L., Application of Fractional Calculus to Fluid Mechanics, Journal of Fluids Engineering, 124 (2002) 803-806.

[15] Tenreiro Machado, J.A., Silva, M.F., Barbosa, R.S., Jesus, I.S.R., Marcos, M.G., Galhano, A.F., Some Applications of Fractional Calculus in Engineering, Mathematical Problems in Engineering, 2010 (2010) 1-34.

[16] Debnath, L., Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 2003 (2003) 3413-3442.

[17] Caputo, M., Linear models of dissipation whose Q is almost frequency independent, Part II, Geophysical Journal of the Royal Astronomical Society, 13 (1967) 529-539.

[18] Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015) 73-85.

[19] Losada, J., Nieto, J.J., Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015) 87-92.

[20] Alsaedi, A., Baleanu, D., Etemad, S., Rezapour, S., On Coupled Systems of Time-Fractional Differential Problems by Using a New Fractional Derivative, Journal of Function Spaces, 2016 (2016) 8.

[21] Baleanu, D., Agheli, B., Qurashi, M.M.A., Fractional advection differential equation within Caputo and Caputo–Fabrizio derivatives, Advances in Mechanical Engineering, 8(12) (2016) 1687814016683305.

[22] Chatterjee, A., Statistical origins of fractional derivatives in viscoelasticity, Journal of Sound and Vibration, 284 (2005) 1239-1245.

[23] Kawada, Y., Nagahama, H., Hara, H., Irreversible thermodynamic and viscoelastic model for power-law relaxation and attenuation of rocks, Tectonophysics, 427 (2006) 255-263.

[24] Shah, N.A., Khan, I., Heat transfer analysis in a second grade fluid over and oscillating vertical plate using fractional Caputo–Fabrizio derivatives, The European Physical Journal C, 76 (2016) 362.

[25] Khan, I., Shah, N.A., Mahsud, Y., Vieru, D., Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo-Fabrizio derivatives, The European Physical Journal Plus, 132 (2017) 194.

[26] Zheng, L., Liu, Y., Zhang, X., Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear Analysis: Real World Applications, 13 (2012) 513-523.

[27] Fetecau, C., Mahmood, A., Fetecau, C., Vieru, D., Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder, Computers & Mathematics with Applications, 56 (2008) 3096-3108.

[28] Qi, H., Xu, M., Stokes’ first problem for a viscoelastic fluid with the generalized Oldroyd-B model, Acta Mechanica Sinica, 23 (2007) 463-469.

[29] Jiang, Y., Qi, H., Xu, H., Jiang, X., Transient electroosmotic slip flow of fractional Oldroyd-B fluids, Microfluidics and Nanofluidics, 21 (2017) 7.

[30] Wang, S., Zhao, M., Analytical solution of the transient electro-osmotic flow of a generalized fractional Maxwell fluid in a straight pipe with a circular cross-section. European Journal of Mechanics - B/Fluids, 54 (2015), 82–86.

[31] Guo, X., Qi, H., Analytical Solution of Electro-Osmotic Peristalsis of Fractional Jeffreys Fluid in a Micro-Channel, Micromachines (Basel), 8(12) (2017), 341.

[32] Wang, X., Qi, H., Yu, B., Xiong, Z., Xu, H., Analytical and numerical study of electroosmotic slip flows of fractional second grade fluids. Communications in Nonlinear Science and Numerical Simulation, 50 (2017) 77-87.

[33] Awan, A.U., Hisham, M.D., Raz, N., The effect of slip on electroosmotic flow of a second grade fluid between two plates with Caputo-Fabrizio time fractional derivatives. Canadian Journal of Physics, 50 (2018) doi: 10.1139/cjp-2018-0406.