Exact solutions for the flow of second-grade fluid in annulus between torsionally oscillating cylinders. (English) Zbl 1270.76006
Summary: The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time \(t = 0\), the fluid and both the cylinders are at rest and at \(t = 0^+\), cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies \(\omega_1\) and \(\omega_2\). The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The 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 Newtonian fluid are also obtained as limiting cases of our general solutions.
MSC:
| 76A05 | Non-Newtonian fluids |
Keywords:
second grade fluid; velocity field; shear stress; longitudinal oscillatory flow; Laplace and Hankel transformsReferences:
| [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 University Press, Cambridge, 1886 |
| [2] | Casarella, M.J., Laura, P.A.: Drag on oscillating rod with longitudinal and torsional motion. J. Hydronautics 3, 180–183 (1969) · doi:10.2514/3.62823 |
| [3] | Rajagopal, K.R.: Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid. Acta Mech. 49, 281–285 (1983) · Zbl 0539.76005 · doi:10.1007/BF01236358 |
| [4] | 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 |
| [5] | 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 |
| [6] | Erdogan, M.E.: A note on an unsteady flow of a viscous fluid due to an oscillating plane wall. Int. J. Non-Linear Mech. 35, 1–6 (2000) · Zbl 1006.76028 · doi:10.1016/S0020-7462(99)00019-0 |
| [7] | Fetecau, C., Fetecau, Corina: Starting solutions for some unsteady unidirectional flows of a second grade fluid. Int. J. Eng. Sci. 43, 781–789 (2005) · Zbl 1211.76032 · doi:10.1016/j.ijengsci.2004.12.009 |
| [8] | Hayat, T., Siddiqui, A.M., Asghar, S.: Some simple flows of an Oldroyd-B fluid. Int. J. Eng. Sci. 39, 135–147 (2001) · doi:10.1016/S0020-7225(00)00026-4 |
| [9] | Aksel, N., Fetecau, C., Scholle, M.: Starting solutions for some unsteady unidirectional flows of Oldroyd-B fluids. ZAMP 57, 815–831 (2006) · Zbl 1101.76006 · doi:10.1007/s00033-006-0063-8 |
| [10] | 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) · Zbl 1213.76014 · doi:10.1016/j.ijengsci.2006.04.010 |
| [11] | 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) · Zbl 1163.76321 · doi:10.1007/s11012-007-9081-7 |
| [12] | Mahmood, A., Parveen, S., Ara, A., et al.: Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model. Commun. Nonlinear Sci. Numer. Simulat. 14(8), 3309–3319 (2009) · Zbl 1221.76018 · doi:10.1016/j.cnsns.2009.01.017 |
| [13] | Mahmood, A., Fetecau, C., Khan, N.A., et al.: Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders. Acta Mech. Sin. 26(4), 541–550 (2010) · Zbl 1269.76011 · doi:10.1007/s10409-010-0353-4 |
| [14] | Mahmood, A., Khan, N.A., Fetecau, C., et al.: Exact analytic solutions for the flow of second grade fluid between two longitudinally oscillating cylinders. J. Prime Res. Math. 5, 192–204 (2009) · Zbl 1287.76039 |
| [15] | Rajagopal, K.R.: A note on unsteady unidirectional flows of a non-Newtonian fluid. Int. J. Non-Linear Mech. 17, 369–373 (1982) · Zbl 0527.76003 · doi:10.1016/0020-7462(82)90006-3 |
| [16] | Hayat, T., Khan, M., Siddiqui, A.M., et al.: Transient flows of a second grade fluid. Int. J. Non-Linear Mech. 39, 1621–1633 (2004) · Zbl 1348.76017 · doi:10.1016/j.ijnonlinmec.2002.12.001 |
| [17] | Dunn, J.E., Rajagopal, K.R.: Fluids of differential type: critical review and thermodynamic analysis. Int. J. Eng. Sci. 33, 689–729 (1995) · Zbl 0899.76062 · doi:10.1016/0020-7225(94)00078-X |
| [18] | Mahmood, A., Saifullah, Bolat, G.: Some exact solutions for the rotational flow of a generalized second grade fluid between two circular cylinders. Arch. Mech. 60(5), 385–401 (2008) · Zbl 1273.76024 |
| [19] | Fetecau, C., Mahmood, A., Fetecau, C., et al.: Some exact solutions for the helical flow of a generalized Oldroyd-B fluid in a circular cylinder. Comput. Math. Appl. 56, 3096–3108 (2008) · Zbl 1165.76306 · doi:10.1016/j.camwa.2008.07.003 |
| [20] | Mahmood, A., Saifullah, Rubab, Q.: Exact solutions for a rotational flow of generalized second grade fluids through a cylin der. Buletinul Academiei de Stiinte a Republicii Moldova. Matematica 58(3), 9–17 (2008) · Zbl 1166.76005 |
| [21] | 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 |
| [22] | Sneddon, I.N.: Functional Analysis in: Encyclopedia of Physics, Vol. II. Springer, Berlin, Göttingen, Heidelberg, 1955 |
| [23] | Debnath, L., Bhatta, D.: Integral Transforms and Their Applications (2nd edn.), Chapman & Hall/CRC (2007) · Zbl 1113.44001 |
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