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Starting solutions for some unsteady unidirectional flows of a second grade fluid. (English) Zbl 1211.76032
Summary: Exact solutions corresponding to the motions of a second grade fluid, due to the cosine and sine oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined by means of the Fourier sine transforms. These solutions, presented as sum of the steady-state and transient solutions, satisfy both the governing equations and all associate initial and boundary conditions. In the special case when \(\alpha _{1} \rightarrow 0\), they reduce to those for a Navier-Stokes fluid.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76A05 Non-Newtonian fluids
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[1] Schlichting, H., Boundary layer theory, (1968), McGraw-Hill New York
[2] Tokuda, N., On the impulsive motion of a flat plate in a viscous fluid, J. fluid mech., 33, 657-672, (1968) · Zbl 0167.55203
[3] Penton, R., The transient for stokes’s oscillating plane: a solution in terms of tabulated functions, J. fluid mech., 31, 819-825, (1968) · Zbl 0193.56302
[4] Rajagopal, K.R., A note on unsteady unidirectional flows of a non-Newtonian fluid, Int. J. non-linear mech., 17, 5-6, 369-373, (1982) · Zbl 0527.76003
[5] Hayat, T.; Asghar, S.; Siddiqui, A.M., Periodic unsteady flows of a non-Newtonian fluid, Acta mech., 131, 169-175, (1998) · Zbl 0939.76002
[6] Siddiqui, A.M.; Hayat, T.; Asghar, S., Periodic flows of a non-Newtonian fluid between two parallel plates, Int. J. non-linear mech., 34, 895-899, (1999) · Zbl 1006.76004
[7] Hayat, T.; Asghar, S.; Siddiqui, A.M., Some unsteady unidirectional flows of non-Newtonian fluid, Int. J. eng. sci., 38, 337-346, (2000) · Zbl 1210.76015
[8] Rajagopal, K.R., Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid, Acta mech., 49, 281-285, (1983) · Zbl 0539.76005
[9] 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
[10] 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
[11] Fosdick, R.L.; Rajagopal, K.R., Anomalous features in the model of second-order fluid, Arch. rational mech. anal., 70, 145-152, (1979) · Zbl 0427.76006
[12] Asghar, S.; Hayat, T.; Siddiqui, A.M., Moving boundary in a non-Newtonian fluid, Int. J. non-linear mech., 37, 75-80, (2002) · Zbl 1116.76310
[13] Dunn, J.E.; Rajagopal, K.R., Fluids of differential type: critical review and thermodynamic analysis, Int. J. eng. sci., 33, 5, 689-729, (1995) · Zbl 0899.76062
[14] Bandelli, R.; Rajagopal, K.R.; Galdi, G.P., On some unsteady motions of fluids of second grade, Arch. mech., 47, 4, 661-676, (1995) · Zbl 0835.76002
[15] R. Bandelli, Unsteady flows of non-Newtonian fluids, Ph.D. Dissertation, University of Pittsburgh, 1995. · Zbl 0837.76004
[16] Hayat, T.; Hutter, K., Rotating flow of a second order fluid on a porous plate, Int. J. non-linear mech., 39, 767-777, (2004) · Zbl 1348.76149
[17] Rajagopal, K.R., On boundary conditions for fluids of the differential type, (), 273-278 · Zbl 0846.35107
[18] Hayat, T.; Wang, Y.; Hutter, K., Hall effects on the unsteady hydromagnetic oscillatory flow of a second grade fluid, Int. J. non-linear mech., 39, 1027-1037, (2004) · Zbl 1348.76187
[19] Sneddon, I.N., Fourier transforms, (1951), McGraw Hill Book Company New York, Toronto, London · Zbl 0099.28401
[20] Sneddon, I.N., Functional analysis, ()
[21] I.S. Grandshteyn, I.M. Ryzhik, in: Alan Jeffrey (Ed.), Tables of Integrals, Series and Products, fifth ed., Academic Press, San Diego, New York, Boston, London, Sydney, Toronto, 1994 (translated from Russian).
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