Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders
 A. Mahmood,
 C. Fetecau,
 N. A. Khan,
 M. Jamil
 … show all 4 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and at t = 0^{+} both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies Ω_{1} and Ω_{2}. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.
 Caputo M., Mainardi F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, 134–147 (1971) CrossRef
 Slonimsky G.L.: On the law of deformation of highly elastic polymeric bodies. Dokl. Akad. Nauk BSSR 140, 343–346 (1961)
 Stiassnie M.: On the application of fractional calculus for the formulation of viscoelastic models. Appl. Math. Modell. 3, 300–302 (1979) CrossRef
 Mainardi F.: Applications of fractional calculus in mechanics. In: Rusev, P., Dimovschi, I., Kiryakova, V. (eds) Transform Methods and Special Functions, Varna’96, pp. 309–334. Bulgarian Academy of Sciences, Sofia (1998)
 Bagley R.L., Torvik P.J.: A theoretical basis for the application of fractional calculus to viscoelastisity. J. Rheol. 27, 201–210 (1983) CrossRef
 Bagley R.L., Torvik P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, 133–155 (1986) CrossRef
 Rogers L.: Operators and fractional derivatives for viscoelastic constitutive equations. J. Rheol. 27, 351–372 (1983) CrossRef
 Koeller R.C.: Applications of fractional calculus to the theory of viscoelasticity. Trans. ASME J. Appl. Mech. 51, 299–307 (1984) CrossRef
 Xu M., Tan W.: Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion. Sci. China Ser. A 44, 1387–1399 (2001) CrossRef
 Xu M., Tan W.: The representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions. Sci. China Ser. G 46, 145–157 (2003) CrossRef
 Debnath L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 54, 3413–3442 (2003) CrossRef
 Podlubny I.: Fractional Differential Equations. Academic Press, San Diego (1999)
 Rouse P.E.: The theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21, 1272–1280 (1953) CrossRef
 Ferry J.D., Landel R.F., Williams M.L.: Extensions of the Rouse theory of viscoelastic properties to undiluted linear polymers. J. Appl. Phys. 26, 359–362 (1955) CrossRef
 Stokes G.G.: On the Effect of the Rotation of Cylinders and Spheres About Their Own Axes in Increasing the Logarithmic Decrement of the Arc of Vibration, pp. 204–217. Cambridge University Press, Cambridge (1886)
 Casarella M.J., Laura P.A.: Drag on oscillating rod with longitudinal and torsional motion. J. Hydronaut. 3, 180–183 (1969) CrossRef
 Rajagopal K.R.: Longitudinal and torsional oscillations of a rod in a nonNewtonian fluid. Acta Mech. 49, 281–285 (1983) CrossRef
 Rajagopal K.R., Bhatnagar R.K.: Exact solutions for some simple flows of an OldroydB fluid. Acta Mech. 113, 233–239 (1995) CrossRef
 Khan, M., Asghar, S., Hayat, T.: Oscillating flow of a Burgers’ fluid in a pipe. The Abdus Salam International Center for Theoretical Physics, IC/2005/071
 Fetecau C., Fetecau C.: Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder. Int. J. Eng. Sci. 44, 788–796 (2006) CrossRef
 Mahmood A., Parveen S., Ara A. et al.: Exact analytic solutions for the unsteady flow of a nonNewtonian fluid between two cylinders with fractional derivative model. Commun. Nonlinear Sci. Numer. Simulat. 14, 3309–3319 (2009) CrossRef
 Vieru D., Akhtar W., Fetecau C. et al.: Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains. Meccanica 42, 573–583 (2007) CrossRef
 Fetecau C., Hayat T., Fetecau C.: Starting solutions for oscillating motions of OldroydB fluids in cylindrical domains. J. NonNewtonian Fluid Mech. 153, 191–201 (2008) CrossRef
 Massoudi M., Phuoc T.X.: On the motion of a second grade fluid due to longitudinal and torsional oscillations of a cylinder: a numerical study. Appl. Math. Comput. 203(4), 471–481 (2008) CrossRef
 Khan M., Ali S.H., Qi H.: Exact solutions of starting flows for a fractional Burgers’ fluid between coaxial cylinders. Nonlinear Anal. Real World Appl. 10(3), 1775–1783 (2009) CrossRef
 Tong D., Wang R., Yang H.: Exact solutions for the flow of nonNewtonian fluid with fractional derivative in an annular pipe. Sci. China Ser. G 48, 485–495 (2005) CrossRef
 Tan W.C., Xu M.Y.: The impulsive motion of flat plate in a generalized second grade fluid. Mech. Res. Commun. 29, 3–9 (2002) CrossRef
 Shen F., Tan W.C., Zhao Y. et al.: The Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative model. Nonlinear Anal. Real World Appl. 7, 1072–1080 (2006) CrossRef
 Fetecau C., Mahmood A., Fetecau C. et al.: Some exact solutions for the helical flow of a generalized OldroydB fluid in a circular cylinder. Comput. Math. Appl. 56, 3096–3108 (2008) CrossRef
 Sneddon I.N.: Functional Analysis in: Encyclopedia of Physics, vol. II. Springer, Berlin (1955)
 Lorenzo. C.F., Hartley, T.T.: Generalized functions for the fractional calculus. NASA/TP1999209424/Rev1 (1999)
 Debnath L., Bhatta D.: Integral Transforms and Their Applications, 2nd edn. Chapman & Hall/CRC, London (2007)
 Title
 Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders
 Journal

Acta Mechanica Sinica
Volume 26, Issue 4 , pp 541550
 Cover Date
 20100801
 DOI
 10.1007/s1040901003534
 Print ISSN
 05677718
 Online ISSN
 16143116
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Generalized second grade fluid
 Velocity field
 Shear stress
 Longitudinal oscillatory flow
 Laplace and Hankel transforms
 Industry Sectors
 Authors

 A. Mahmood ^{(1)}
 C. Fetecau ^{(2)}
 N. A. Khan ^{(3)}
 M. Jamil ^{(1)} ^{(4)}
 Author Affiliations

 1. Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan
 2. Department of Mathematics, Technical University of Iasi, Iasi, 700050, Romania
 3. Department of Mathematics, University of Karachi, Karachi, 75270, Pakistan
 4. Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan