The one-dimensional flow of a fluid with limited strain-rate
Authors:
A. Farina, A. Fasano, L. Fusi and K. R. Rajagopal
Journal:
Quart. Appl. Math. 69 (2011), 549-568
MSC (2010):
Primary 76A05, 74D10, 35R35, 35K10
DOI:
https://doi.org/10.1090/S0033-569X-2011-01249-7
Published electronically:
May 9, 2011
MathSciNet review:
2850745
Full-text PDF Free Access
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Additional Information
Abstract: We present a model for a continuum in which the strain rate depends linearly on the stress, as long as the latter is below a fixed threshold, but it is frozen to a constant value when the stress exceeds such a threshold. The constitutive equation is given in an implicit form as the stress is a multi-valued function of the strain rate. We derive the model in a general 3D setting and we study the one-dimensional case of a pressure-driven flow between two parallel plates. We prove some analytical results and describe a procedure to determine the main physical parameters (stress threshold and viscosity) by means of a rotational viscometer. Finally we show that the model can be obtained as the limit case of a piecewise linear viscous model.
References
- S. Bair and P. Kottke, Pressure-viscosity relationships for elastohydrodynamics, Tribology Transactions, 46, 289–295, (2003).
- P. W. Bridgman, The Physics of High Pressure, New York, MacMillan, 1931.
- M. Bulíček, J. Málek, and K. R. Rajagopal, Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries, SIAM J. Math. Anal. 41 (2009), no. 2, 665–707. MR 2515781, DOI https://doi.org/10.1137/07069540X
- J. Coirier, Mécanique des milieux continus, Dunod, Paris, 1997.
- Elena Comparini, A one-dimensional Bingham flow, J. Math. Anal. Appl. 169 (1992), no. 1, 127–139. MR 1180677, DOI https://doi.org/10.1016/0022-247X%2892%2990107-O
- Elena Comparini and Elena De Angelis, Flow of a Bingham fluid in a concentric cylinder viscometer, Adv. Math. Sci. Appl. 6 (1996), no. 1, 97–116. MR 1385761
- E. Comparini, Regularization procedures of singular free boundary problems in rotational Bingham flows, Z. Angew. Math. Mech. 77 (1997), no. 7, 543–554. MR 1466444, DOI https://doi.org/10.1002/zamm.19970770715
- D. Dowson and G. R. Higginson, Elastohydrodynamic Lubrication, the Fundamentals of Roller and Gear Lubrication, Pergamon, Oxford, UK (1996).
- A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. I, J. Math. Anal. Appl. 57 (1977), no. 3, 694–723. MR 487016, DOI https://doi.org/10.1016/0022-247X%2877%2990256-6
- A. Farina, A. Fasano, L. Fusi, and K. R. Rajagopal, Modeling materials with a stretching threshold, Math. Models Methods Appl. Sci. 17 (2007), no. 11, 1799–1847. MR 2372339, DOI https://doi.org/10.1142/S0218202507002480
- A. Farina, A. Fasano, L. Fusi, and K. R. Rajagopal, On the dynamics of an elastic-rigid material, Adv. Math. Sci. Appl. 20 (2010), no. 1, 193–217. MR 2760725
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- W. H. Herschel and R. Bulkley, Konsistenzmessungen von Gummi-Benzollosungen, Colloid Polym. Sci. 39(4), 291–300 (1926).
- J. Malek, V. Prusa and K. R. Rajagopal, Generalization of the Navier-Stokes fluid from a new perspective, submitted for publication.
- J. G. Oldroyd, A rational formulation of the equations of plastic flow for a Bingham solid, Proc. Cambridge Philos. Soc. 43 (1947), 100–105. MR 18095
- Alfred Schatz, Free boundary problems of Stephan type with prescribed flux, J. Math. Anal. Appl. 28 (1969), 569–580. MR 267285, DOI https://doi.org/10.1016/0022-247X%2869%2990009-2
- A. Cemal Eringen (ed.), Continuum physics. Vol II, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Continuum mechanics of single-substance bodies. MR 0468444
- K. R. Rajagopal, On implicit constitutive theories, Appl. Math. 48 (2003), no. 4, 279–319. MR 1994378, DOI https://doi.org/10.1023/A%3A1026062615145
- K. R. Rajagopal, On implicit constitutive theories for fluids, J. Fluid Mech. 550 (2006), 243–249. MR 2263984, DOI https://doi.org/10.1017/S0022112005008025
- K. R. Rajagopal, The elasticity of elasticity, Z. Angew. Math. Phys. 58 (2007), no. 2, 309–317. MR 2305717, DOI https://doi.org/10.1007/s00033-006-6084-5
- K. R. Rajagopal and A. R. Srinivasa, On the response of non-dissipative solids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463 (2007), no. 2078, 357–367. MR 2288826, DOI https://doi.org/10.1098/rspa.2006.1760
- K.R. Rajagopal and G. Saccomandi, The mechanics and mathematics of the effect of pressure on the shear modulus of elastomers, Proc. Roy. Soc. London A 465, 3859-3874 (2009).
- K.R. Rajagopal, Non-linear elastic bodies exhibiting limiting small strain, Mathematics and Mechanics of Solids, 2010, doi:10.1177/1081286509357272.
- C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd ed., Springer-Verlag, Berlin, 1992. MR 1215940
- N.J. Wagner and J.F. Brady, Shear thickening in colloidal dispersion, Physics Today, 62, 27-32 (2009).
References
- S. Bair and P. Kottke, Pressure-viscosity relationships for elastohydrodynamics, Tribology Transactions, 46, 289–295, (2003).
- P. W. Bridgman, The Physics of High Pressure, New York, MacMillan, 1931.
- M. Bulicek, J. Malek and K. R. Rajagopal, Mathematical analysis of unsteady flows of fluids with pressure, shear-rate, and temperature dependent material moduli that slip at solid boundaries, SIAM, 41, 665-707 (2009). MR 2515781 (2010f:76021)
- J. Coirier, Mécanique des milieux continus, Dunod, Paris, 1997.
- E. Comparini, A one-dimesional Bingham flow, J. Math. Anal. Appl., 169, 127-139, (1992). MR 1180677 (93j:76008)
- E. Comparini and E. De Angelis, Flow of a Bingham fluid in a concentric cylinder viscosimeter, Adv. Math. Sci. Appl., 6, N. 1, 97-116, (1996). MR 1385761 (97e:76005)
- E. Comparini, Regularization of singular free boundary problems in rotational Bingham flows, Z. Angew. Math. Mech., 77, 543-554, (1997). MR 1466444 (98f:76005)
- D. Dowson and G. R. Higginson, Elastohydrodynamic Lubrication, the Fundamentals of Roller and Gear Lubrication, Pergamon, Oxford, UK (1996).
- A. Fasano and M. Primicerio, General free boundary problems for heat equation, I, J. Math. Anal. Appl., 57, 694-723, 1977. MR 0487016 (58:6695a)
- A. Farina, A. Fasano, L. Fusi and K.R. Rajagopal , Modelling materials with a stretching threshold, Mathematical Models and Methods in Applied Sciences (M3AS), Vol. 17, Issue 11, (2007), 1799-1846. MR 2372339 (2009g:74013)
- A. Farina, A. Fasano, L. Fusi, and K.R. Rajagopal, On the dynamics of an elastic-rigid material, Adv. Math. Sci. Appl. 20 (2010), 193–217. MR 2760725
- A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, 1964. MR 0181836 (31:6062)
- W. H. Herschel and R. Bulkley, Konsistenzmessungen von Gummi-Benzollosungen, Colloid Polym. Sci. 39(4), 291–300 (1926).
- J. Malek, V. Prusa and K. R. Rajagopal, Generalization of the Navier-Stokes fluid from a new perspective, submitted for publication.
- J.G. Oldroyd, A rational formulation of the equations of plastic flow for a Bingham solid, Math. Proc. Camb. Philos. Soc. 43(01), 100–105 (1947). MR 0018095 (8:240j)
- A. Schatz, Free boundary problems of Stephan type with prescribed flux, J. Math. Anal. Appl., 28, 569-580, 1969. MR 0267285 (42:2187)
- A.J.M. Spencer, Theory of invariants, in Continuum Physics, Vol. 3, A.C. Eringen Ed., Academic Press, New York, 1975. MR 0468444 (57:8277b)
- K.R. Rajagopal, On implicit constitutive theories, Appl. Math., 48 (2003), 279–319. MR 1994378 (2004j:74006)
- K.R. Rajagopal, On implicit constitutive theories for fluids, J. Fluid Mech., 550 (2006), 243–249. MR 2263984 (2007e:76004)
- K.R. Rajagopal, The elasticity of elasticity, Zeitschrift für Angewandte Mathematik und Physik, 58, 309-417 (2007). MR 2305717 (2008a:74003)
- K.R. Rajagopal and A.R. Srinivasa, On the response of non-dissipative solids, Proc. Roy. Soc. London A463, 357-367 (2007). MR 2288826 (2007j:74001)
- K.R. Rajagopal and G. Saccomandi, The mechanics and mathematics of the effect of pressure on the shear modulus of elastomers, Proc. Roy. Soc. London A 465, 3859-3874 (2009).
- K.R. Rajagopal, Non-linear elastic bodies exhibiting limiting small strain, Mathematics and Mechanics of Solids, 2010, doi:10.1177/1081286509357272.
- C. Truesdell and W. Noll, The nonlinear field theories of mechanics, 2nd edn., Springer-Verlag, 1992. MR 1215940 (94c:73002)
- N.J. Wagner and J.F. Brady, Shear thickening in colloidal dispersion, Physics Today, 62, 27-32 (2009).
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Additional Information
A. Farina
Affiliation:
Università degli Studi di Firenze, Dipartimento di Matematica “U. Dini”, Viale Morgagni 67/a, 50134 Firenze, Italy
Email:
farina@math.unifi.it
A. Fasano
Affiliation:
Università degli Studi di Firenze, Dipartimento di Matematica “U. Dini”, Viale Morgagni 67/a, 50134 Firenze, Italy
Email:
fasano@math.unifi.it
L. Fusi
Affiliation:
Università degli Studi di Firenze, Dipartimento di Matematica “U. Dini”, Viale Morgagni 67/a, 50134 Firenze, Italy
Email:
fusi@math.unifi.it
K. R. Rajagopal
Affiliation:
Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77845
Email:
krajagopal@mengr-tamu.org
Keywords:
Non-Newtonian fluids,
implicit constitutive relations,
free boundary problems,
parabolic equations.
Received by editor(s):
March 24, 2010
Published electronically:
May 9, 2011
Article copyright:
© Copyright 2011
Brown University