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Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time-dependent shear stresses. (English) Zbl 1273.76041
Summary: Exact solutions corresponding to the unsteady helical flow of an Oldroyd-B fluid due to an infinite circular cylinder subject to torsional and longitudinal time-dependent shear stresses are established using Hankel transforms. These solutions, presented under series form in terms of Bessel functions \(J _{0}(\cdot ), \, J _{1}(\cdot )\) and \(J _{2}(\cdot )\), can be easily specialized to give the similar solutions for Maxwell, Second grade and Newtonian fluids performing the same motion. Some characteristics of the motion, as well as the influence of pertinent parameters on the velocity profiles, are underlined by graphical illustrations.

MSC:
76A10 Viscoelastic fluids
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[1] Stokes G.G.: On the Effect of the Rotation of Cylinders and Spheres About Their Axis in Increasing the Logarithmic Decrement of the Arc of Vibration. Cambridge Universitry Press, Cambridge (1886)
[2] Taylor G.I.: Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. A 223, 289–298 (1923) · JFM 49.0607.01 · doi:10.1098/rsta.1923.0008
[3] Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967) · Zbl 0152.44402
[4] Ting T.W.: Certain non-steady flows of second-order fluids. Arch. Rational Mech. Anal. 14, 1–23 (1963) · Zbl 0139.20105 · doi:10.1007/BF00250690
[5] Srivastava P.N.: Non-steady helical flow of a viscoelastic liquid. Arch. Mech. Stos. 18, 145–150 (1966)
[6] Waters N.D., King M.J.: The unsteady flow of an elastico-viscous liquid in a straight pipe of circular cross section. J. Phys. D Appl. Phys. 4, 204–211 (1971) · doi:10.1088/0022-3727/4/2/304
[7] Rajagopal K.R., Bhatnagar R.K.: Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech. 113, 233–239 (1995) · Zbl 0858.76010 · doi:10.1007/BF01212645
[8] Wood W.P.: Transient viscoelastic helical flows in pipes of circular and annular cross-section. J. Non-Newton. Fluid Mech. 100, 115–126 (2001) · Zbl 1014.76004 · doi:10.1016/S0377-0257(01)00130-6
[9] Fetecau C.: Analytical solutions for non-Newtonian fluid flows in pipe-like domains. Int. J. Non-Linear Mech. 39, 225–231 (2004) · Zbl 1287.76036 · doi:10.1016/S0020-7462(02)00170-1
[10] Hayat T., Khan M., Wang T.: Non-Newtonian flow between concentric cylinders. Comm. Non-Linear Sci. Numer. Simm. 11, 297–305 (2006) · Zbl 1078.35091 · doi:10.1016/j.cnsns.2004.11.007
[11] Fetecau C., Corina Fetecau, Vieru, D.: On some helical flows of Oldroyd-B fluids. Acta Mech. 189, 53–63 (2007) · Zbl 1108.76008
[12] Corina Fetecau, Hayat T., Fetecau C.: Starting solutions for oscilating motions of Oldroyd-B fluids in cylindrical domains. J. Non-Newton. Fluid Mech. 153, 191–201 (2008) · Zbl 1273.76040 · doi:10.1016/j.jnnfm.2008.02.005
[13] Bandelli R., Rajagopal K.R.: Start-up flows of second grade fluids in domains with one finite dimension. Int. J. Non-Linear Mech. 30, 817–839 (1995) · Zbl 0866.76004 · doi:10.1016/0020-7462(95)00035-6
[14] Bandelli R., Rajagopal K.R., Galdi G.P.: On some unsteady motions of fluids of second grade. Arch. Mech. 47, 661–676 (1995) · Zbl 0835.76002
[15] Siddique I.: Exact solution of some helical flows of Newtonian fluids. Proc. World Acad. Sci. Eng. Technol. 33, 664–667 (2008)
[16] Nazar, M., Athar, M., Akhtar, W.: Axial Couette flow of second grade fluid due to a longitudinal time dependent shear stress. In: Proceedings of the Tenth International Conference Zaragoza-Pau on Applied Mathematics and Statistics Jaca, September 15–17, Spain (2008) · Zbl 1317.76010
[17] Khan M., Nadeem S., Hayat T., Siddiqui A.M.: Unsteady motions of a generalized second grade fluid. Math. Comput. Model 43, 16–29 (2006) · Zbl 1163.34006 · doi:10.1016/j.mcm.2005.05.017
[18] Tong D., Liu Y.: Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe. Int. J. Eng. Sci. 43, 281–289 (2005) · Zbl 1211.76014 · doi:10.1016/j.ijengsci.2004.09.007
[19] Tong D., Ruihe Y., Heshan W.: Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe. Sci. China Ser. G Phys. Mech. Astron. 48, 485–495 (2005) · doi:10.1360/04yw0105
[20] Debnath L., Bhatta D.: Integral Transforms and their Applications, 2nd edn. Chapman and Hall, CRC Press, Boca-Raton, London (2007) · Zbl 1113.44001
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