×

zbMATH — the first resource for mathematics

Decay of a potential vortex in an Oldroyd-B fluid. (English) Zbl 1211.76008
Summary: Analytical expressions for the velocity field and the associated tangential tension corresponding to a potential vortex in an Oldroyd-B fluid are determined by means of the Hankel transform. The well-known solutions for a Navier-Stokes fluid as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. Finally, some comparative diagrams are presented for the circular motion of the glycerine. In each case, the velocity fields as well as the adequate tangential tensions are going to zero for \(t\) or \(r \rightarrow \infty \). Consequently, the potential vortex is damping in time and space.

MSC:
76A05 Non-Newtonian fluids
76A10 Viscoelastic fluids
76D17 Viscous vortex flows
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Rajagopal, K.R., Mechanics of non-Newtonian fluids, (), 129-162 · Zbl 0818.76003
[2] Rivlin, R.S.; Ericksen, J.L., Stress deformation relation for isotropic materials, J. ration. mech. anal., 4, 323-425, (1955) · Zbl 0064.42004
[3] 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
[4] 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
[5] Rajagopal, K.R.; Srinivasa, A.R., A thermodynamic frame work for rate type fluid models, J. non-Newtonian fluid mech., 88, 207-227, (2000) · Zbl 0960.76005
[6] Oldroyd, J.G., On the formulation of the rheological equations of state, Proc. royal soc. lond. ser. A, 200, 523-541, (1950) · Zbl 1157.76305
[7] Larson, R.G., Constitutive equations for polymer melts and solutions, (1989), Butterworts Boston-London-Singapore-Sydney-Toronto-Wellington
[8] Joseph, D.D., Fluid dynamics of viscoelastic liquids, (1990), Springer Verlag New York · Zbl 0698.76002
[9] J.M. Burgers, Mechanical considerations–model systems–phenomenological theories of relaxation and of viscosity. First report on viscosity and plasticity. Prepared by the committee for the study of viscosity of the academy of sciences at Amsterdam, second ed., Nordemann Publ, New York, 1939
[10] GuillopĂ©, C.; Saut, J.C., Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO model. math. anal. numer., 24, 369-401, (1990) · Zbl 0701.76011
[11] Fontelos, M.A.; Friedman, A., Stationary non-Newtonian fluid flows in channel-like and pipe-like domains, Arch. ration. mech. anal., 151, 1-43, (2000) · Zbl 1018.76005
[12] 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
[13] Fetecau, C., The Rayleigh-Stokes problem for an edge in an Oldroyd-B fluid, C.R. acad. sci. Paris ser. I, 335, 979-984, (2002) · Zbl 1032.76005
[14] 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
[15] Fetecau, C.; Fetecau, Corina, The first problem of Stokes for an Oldroyd-B fluid, Int. J. non-linear mech., 38, 1539-1544, (2003) · Zbl 1287.76044
[16] Georgiou, G.C., On the stability of the shear flow of a viscoelastic fluid with slip along the fixed wall, Rheol. acta, 35, 39-47, (1996)
[17] Zierep, J., Similarity laws and modeling, (1971), Marcel Dekker, Inc. New York · Zbl 0223.76017
[18] Srivastava, P.N., Non-steady helical flow of a visco-elastic liquid, Arch. mech. stos., 18, 2, 145-150, (1966)
[19] Rajagopal, K.R.; Gupta, A.S., An exact solution for the flow of a non-Newtonian fluid past an infinite porous plate, Meccanica, 19, 1948-1954, (1984) · Zbl 0552.76008
[20] Rajagopal, K.R., On the creeping flow of the second-order fluid, J. non-Newtonian fluid mech., 15, 239-246, (1984) · Zbl 0568.76015
[21] Rajagopal, K.R.; Kaloni, P.N., Some remarks on boundary conditions for flows of fluids of the differential type, () · Zbl 0706.76008
[22] Rajagopal, K.R., On boundary conditions for fluids of the differential type, (), 273-278 · Zbl 0846.35107
[23] Sneddon, I.N., Fourier transforms, (1951), McGraw Hill Book Company, Inc. New York-Toronto-London · Zbl 0099.28401
[24] Fetecau, C.; Fetecau, Corina; Zierep, J., Decay of a potential vortex and propagation of a heat wave in a second grade fluid, Int. J. non-linear mech., 37, 6, 1051-1056, (2002) · Zbl 1346.76033
[25] Fetecau, C.; Fetecau, Corina, Decay of a potential vortex in a Maxwell fluid, Int. J. non-linear mech., 38, 7, 985-990, (2003) · Zbl 1287.76043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.