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Unsteady flow of a generalized Maxwell fluid with fractional derivative due to a constantly accelerating plate. (English) Zbl 1165.76307
Summary: The velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized Maxwell fluid are determined using Fourier sine and Laplace transforms. They are presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions. The similar solutions for Maxwell and Newtonian fluids, performing the same motion, are obtained as limiting cases of our general results. Graphical illustrations show that the velocity profiles corresponding to a generalized Maxwell fluid are going to that for an ordinary Maxwell fluid if $\alpha \rightarrow 1$.

##### MSC:
 76A05 Non-Newtonian fluids 26A33 Fractional derivatives and integrals (real functions)
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##### References:
 [1] Choi, J. J.; Rusak, Z.; Tichy, J. A.: Maxwell fluid suction flow in a channel, J. non-Newton fluid mech. 85, 165-187 (1999) · Zbl 0941.76079 · doi:10.1016/S0377-0257(98)00197-9 [2] Palade, L. I.; Attane, P.; Huilgol, R. R.; Mena, B.: Anomalous stability behavior of a properly invariant constitutive equation which generalizes fractional derivative models, Internat. J. Engrg. sci. 37, 315-329 (1999) · Zbl 1210.76021 · doi:10.1016/S0020-7225(98)00080-9 [3] Rossikhin, Y. A.; Shitikova, M. V.: A new method for solving dynamic problems of fractional derivative viscoelasticity, Internat. J. Engrg. sci. 39, 149-176 (2001) · Zbl 1026.74040 [4] N. Makris, Theoretical and experimental investigation of viscous dampers in applications of seismic and vibration isolation, Ph.D. Thesis, State Univ. of New York at Buffalo, Buffalo, NY, 1991 [5] Makris, N.; Constantinou, M. C.: Fractional derivative model for viscous dampers, J. struct. Eng. ASCE 117, 2708-2724 (1991) [6] Tan, W.; Xu, M.: Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model, Acta mech. Sinica 18, 342-349 (2002) [7] Tan, W.; Pan, W.; Xu, M.: A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, Internat. J. Non-linear mech. 38, 645-650 (2003) · Zbl 05138171 [8] Hayat, T.; Nadeem, S.; Asghar, S.: Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model, Appl. math. Comput. 151, 153-161 (2004) · Zbl 1161.76436 · doi:10.1016/S0096-3003(03)00329-1 [9] Qi, H.; Jin, H.: Unsteady rotating flows of a viscoelastic fluid with the fractional Maxwell model between coaxial cylinders, Acta mech. Sinica 22, 301-305 (2006) · Zbl 1202.76016 · doi:10.1007/s10409-006-0013-x [10] Yin, Y.; Zhu, K. Q.: Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model, Appl. math. Comput. 173, 231-242 (2006) · Zbl 1105.76009 · doi:10.1016/j.amc.2005.04.001 [11] Shaowei, W.; Mingyu, X.: Exact solution on unsteady Couette flow of generalized Maxwell fluid with fractional derivative, Acta mech. 187, 103-112 (2006) · Zbl 1103.76011 · doi:10.1007/s00707-006-0332-9 [12] Qi, H.; Xu, M.: Unsteady flow of a viscoelastic fluid with fractional Maxwell model in a channel, Mech. res. Comm. 34, 210-212 (2007) · Zbl 1192.76008 · doi:10.1016/j.mechrescom.2006.09.003 [13] Tan, W.: Velocity overshoot of start-up flow for a Maxwell fluid in a porous half-space, Chin. phys. 15, 2644-2650 (2006) [14] Khan, M.; Maqbool, K.; Hayat, T.: Influence of Hall current on the flows of a generalized Oldroyd-B fluid in a porous space, Acta mech. 184, 1-13 (2006) · Zbl 1096.76061 · doi:10.1007/s00707-006-0326-7 [15] Tan, W.; Masuoka, T.: Stability analysis of a Maxwell fluid in a porous medium heated from below, Phys. lett. A 360, 454-460 (2007) · Zbl 1236.76072 [16] Khan, M.: Partial slip effects on the oscillatory flows of a fractional Jeffrey fluid in a porous medium, J. porous media. 10, 473-487 (2007) [17] Khan, M.; Hayat, T.: Some exact solutions for fractional generalized Burgers fluid in a porous space, Nonlinear anal. RWA 9, 2288-2295 (2008) · Zbl 1156.76444 · doi:10.1016/j.nonrwa.2007.06.005 [18] Shaowei, W.; Tan, W.: Stability analysis of double-diffusive convection of Maxwell fluid in a porous medium heated from below, Phys. lett. A 372, 3046-3050 (2008) · Zbl 1220.76031 · doi:10.1016/j.physleta.2008.01.024 [19] Xue, C.; Nie, J.: Exact solutions of Rayleigh-Stokes problem for heated generalized Maxwell fluid in a porous half-space, Math. prob. Eng. 2008 (2008) · Zbl 1151.76006 · doi:10.1155/2008/641431 [20] Qi, H.; Xu, M.: Stokes first problem for a viscoelastic fluid with generalized Oldroyd-B model, Acta mech. Sinica 23, 463-469 (2007) · Zbl 1202.76017 · doi:10.1007/s10409-007-0093-2 [21] Srivastava, P. N.: Non-steady helical flow of a visco-elastic liquid, Arch. mech. Stos. 18, 145-150 (1966) [22] Tong, D.; Liu, Y.: Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe, Internat. J. Engrg. sci. 43, 281-289 (2005) · Zbl 1211.76014 · doi:10.1016/j.ijengsci.2004.09.007 [23] Tong, D.; Wang, R.; Yang, H.: Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe, Sci. China ser. G physics, mechanics astronomy 48, 485-495 (2005) [24] Khan, M.; Ali, S. Hyder; Qi, H.: Some accelerated flows of a generalized Oldroyd-B fluid, Nonlinear anal. RWA (2007) [25] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008 [26] Sneddon, I. N.: Fourier transforms, (1951) · Zbl 0038.26801 [27] C.F. Lorenzo, T.T. Hartley, Generalized functions for the fractional calculus, NASA/TP-1999-209424, 1999 [28] Fetecau, C.; Prasad, S. C.; Rajagopal, K. R.: A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid, Appl. math. Modelling 31, 647-654 (2007) · Zbl 1287.76047 [29] Vieru, D.; Fetecau, Corina; Fetecau, C.: Exact solutions for the flow of an Oldroyd-B fluid due to an infinite flat plate, Z. angew. Math. phys. 59, 834-847 (2008) · Zbl 1158.76004 · doi:10.1007/s00033-007-6133-8