Artículo
Abstract:
Under assumptions that are not too restrictive it is possible to reduce the equations that describe steady viscous gravity flows of a power-law liquid on an inclined plane to an equivalent problem consisting of an unsteady one-dimensional nonlinear diffusion process. In a paper dealing with the steady spreading flow of a Herschel-Buckley liquid, Wilson and Burgess ["The steady, spreading flow of a rivulet of mud," J. Non-Newtonian Fluid Mech. 79, 77 (1998)] noticed a formal analogy between the steady, two-dimensional viscous gravity flows of a power-law liquid on an incline and a one-dimensional time-dependent nonlinear diffusion phenomena; however, they did not pursue the matter further. Here we develop the analogy and show how it can be used to find a large number of exact solutions representing steady two-dimensional flows of power-law liquids, based on the available knowledge concerning nonlinear diffusion. We describe flows whose widths stay constant until a certain distance from the source, which are analogous to the well-known waiting-time solutions of nonlinear diffusion. We then introduce a phase-plane formalism that allows us to find self-similar solutions and we give as examples three different currents limited laterally by a wall that ends abruptly and currents on an inclined stripe. Finally we describe the two-dimensional currents that are analogous to the traveling wave solutions of the nonlinear diffusion equation. The approximations involved in the analogy are essentially equivalent to those of the lubrication theory, so that they do not impose restrictions more severe than those usually present in problems of this type. The present theory does not include surface tension effects, which implies that the appropriate Bond number must be large. © 2005 American Institute of Physics.
Registro:
| Documento: | Artículo |
| Título: | Exact solutions for two-dimensional steady flows of a power-law liquid on an incline |
| Autor: | Perazzo, C.A.; Gratton, J. |
| Filiación: | Universidad Favaloro, Solís 453, 1078 Buenos Aires, Argentina INFIP CONICET, Departamento de Física, Universidad de Buenos Aires, Pab. I, Buenos Aires, Argentina |
| Palabras clave: | Diffusion; Nonlinear equations; Nonlinear systems; Viscous flow; Nonlinear diffusion; Power-law liquids; Viscous gravity flows; Steady flow; liquid flow; mathematical analysis; power law fluid; steady flow; two-dimensional flow |
| Año: | 2005 |
| Volumen: | 17 |
| Número: | 1 |
| Página de inicio: | 013102 |
| Página de fin: | 013102-8 |
| DOI: | http://dx.doi.org/10.1063/1.1829625 |
| Título revista: | Physics of Fluids |
| Título revista abreviado: | Phys. Fluids |
| ISSN: | 10706631 |
| CODEN: | PHFLE |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_10706631_v17_n1_p013102_Perazzo |
Referencias:
- Mei, C.C., "Nonlinear gravity waves in a thin sheet of viscous fluid" (1966) J. Math. Phys., 45, p. 266. , JMAPAQ0022-2488
- Buckmaster, J., "Viscous sheets advancing over dry beds" (1977) Fluid Mech., 81, p. 735. , JFLSA70022-1120
- Kondic, L., Diez, J., "Pattern formation in the flow of thin films down an incline: Constant flux configuration" (2001) Phys. Fluids, 13, p. 3168. , PHFLE61070-663110.1063/1.1409965
- Smith, P.C., "A similarity solution for slow viscous flow down an inclined plane" (1973) J. Fluid Mech., 58, p. 257. , JFLSA70022-1120
- Wilson, S.D.R., Burgess, S.L., "The steady, spreading flow of a rivulet of mud" (1998) J. Non-Newtonian Fluid Mech., 79, p. 77. , JNFMDI03-025710.1016/S03-0257(98)000-2
- Perazzo, C.A., Gratton, J., "Thin film of non-Newtonian fluid on an incline" (2003) Phys. Rev. E, 67, p. 016307. , PLEEE81063-651X10.1103/PhysRevE
- Peletier, L.A., "The porous media equation," (1981) Application of Nonlinear Analysis in the Physical Sciences, pp. 229-240. , edited by H. Amman, N. Bazley, and K. Kirchgaessner (Pitman, London), Chap. 11
- Pert, G.J., "A class of similar solutions of the non-linear diffusion equation" (1977) Phys. A, 10, p. 583. , JPHAC50305-44710.1088/0305-4470//4/020
- Grundy, R.E., "Similarity solutions of the non-linear diffusion equation" (1979) Q. Appl. Math., 37, p. 259. , QAMAAY0033-569X
- Diez, J.A., Gratton, J., Minotti, F., "Self-similar solutions of the second kind of non-linear diffusion-type equations" (1992) Q. Appl. Math., 50, p. 401. , QAMAAY0033-569X
- Perazzo, C.A., Vigo, C.L.M., Gratton, J., "A two-parameter study of waiting-time solutions in nonlinear diffusion" (2003) Int. Math., 13, p. 3. , ZZZZZZ0129-167X10.1016/0167-896(95)00018-6
- Higuera, F.J., Weidman, P.D., "Natural convection beneath a downward facing heated plate in a porous medium" (1995) Eur. Mech. B/Fluids, 14, p. 29. , EJBFEV0997-7546
- Huppert, H.E., "The propagation of two-dimensional viscous gravity currents over a rigid horizontal surface" (1982) J. Fluid Mech., 121, p. 43. , JFLSA70022-1120
- Huppert, H.E., "Flow and instability of a viscous current down a slope" (1982) Nature (London), 300, p. 427. , NATUAS0028-083610.1038/a0
- Diez, J.A., Gratton, R., Gratton, J., "Self-similar solution of the second kind for a convergent viscous gravity current" (1992) Phys. Fluids A, 4, p. 118. , PFADEB0899-821310.1063/1.858233
- Marino, B.M., Thomas, L.P., Gratton, R., Diez, J.A., Betelú, S., Gratton, J., "Waiting-time solutions of a nonlinear diffusion equation: Experimental study of a creeping flow near a waiting front" (1996) Phys. Rev. E, 54, p. 2628. , PLEEE81063-651X10.1103/PhysRevE
- Perazzo, C.A., Gratton, J., "Navier-Stokes solutions for parallel flow in rivulets on an inclined plane" (2004) Fluid Mech., 507, p. 367. , JFLSA70022-112010.1017/S002211008791
- Barenblatt, G.I., "One some unsteady motions of fluids and gases in a porous medium" (1952) Prikl. Mat. Mekh., 16, p. 67. , PMAMAF0032-8235
- Zel'dovich, Ya.B., Kompaneets, A.S., "On the propagation of heat for nonlinear heat conduction" (1959) Izdat. Akad. Nauk SSSR, , in Collection Dedicated to the Seventieth Birthday of Academician F. Joffee, edited by P. I. Lukirskii, Moscow
- Pattle, R.E., "Diffusion from an instantaneous point source with a concentration-dependent coefficient" (1959) Q. J. Mech. Appl. Math., 12, p. 407. , QJMMAV0033-5614
- Schwartz, L.W., Michaelides, E.E., "Gravity flow of a viscous liquid down a slope with injection" (1988) Phys. Fluids, 31, p. 2739. , PFLDAS0031-917110.1063/1.866977
- Gratton, J., Vigo C.L., M., "Evolution of self-similarity, and other properties of waiting-time solutions of the porous medium equation: The case of viscous gravity currents" (1998) Eur. Appl. Math., 9, p. 327. , ZZZZZZ056-725
- Gratton, J., Minotti, F., Self-similar viscous gravity currents: Phase-plane formalism (1990) Fluid Mech., 210, p. 155. , JFLSA70022-1120
- Perazzo, C.A., "Soluciones con tiempo de espera de la ecuación de difusión no lineal" (2002), Ph.D. thesis, University of Buenos Aires, Buenos Aires; Barenblatt, G.I., (1979) Similarity, Self-Similarity, and Intermediate Asymptotics, , (Consultants Bureau, New York)
- Gratton, J., Minotti, F., Mahajan, S., "Theory of creeping gravity currents of a non-Newtonian liquid" (1999) Phys. Rev. E, 60, p. 69. , PLEEE81063-651X10.1103/PhysRevE
Citas:
---------- APA ----------
Perazzo, C.A. & Gratton, J. (2005) . Exact solutions for two-dimensional steady flows of a power-law liquid on an incline. Physics of Fluids, 17(1), 013102-013102-8.http://dx.doi.org/10.1063/1.1829625
---------- CHICAGO ----------
Perazzo, C.A., Gratton, J. "Exact solutions for two-dimensional steady flows of a power-law liquid on an incline" . Physics of Fluids 17, no. 1 (2005) : 013102-013102-8.http://dx.doi.org/10.1063/1.1829625
---------- MLA ----------
Perazzo, C.A., Gratton, J. "Exact solutions for two-dimensional steady flows of a power-law liquid on an incline" . Physics of Fluids, vol. 17, no. 1, 2005, pp. 013102-013102-8.http://dx.doi.org/10.1063/1.1829625
---------- VANCOUVER ----------
Perazzo, C.A., Gratton, J. Exact solutions for two-dimensional steady flows of a power-law liquid on an incline. Phys. Fluids. 2005;17(1):013102-013102-8.http://dx.doi.org/10.1063/1.1829625
