Analysis of Shear Layers in a Fluid with Temperature-Dependent Viscosity
The presence of viscosity normally has a stabilizing effect on the flow of a fluid. However, experiments show that the flow of a fluid in which viscosity decreases as temperature increases tends to form shear layers, narrow regions in which the velocity of the fluid changes sharply. In general, adia...
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| Published in | Journal of computational physics Vol. 173; no. 1; pp. 17 - 60 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
10.10.2001
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0021-9991 1090-2716 |
| DOI | 10.1006/jcph.2001.6837 |
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| Abstract | The presence of viscosity normally has a stabilizing effect on the flow of a fluid. However, experiments show that the flow of a fluid in which viscosity decreases as temperature increases tends to form shear layers, narrow regions in which the velocity of the fluid changes sharply. In general, adiabatic shear layers are observed not only in fluids but also in thermo-plastic materials subject to shear at a high-strain rate and in combustion and there is widespread interest in modeling their formation. In this paper, we investigate a well-known model representing a basic system of conservation laws for a one-dimensional flow with temperature-dependent viscosity using a combination of analytical andnumerical tools. We present results to substantiate the claim that the formation of shear layers can only occur in solutions of the model when the viscosity decreases sufficiently quickly as temperature increases and we further analyze the structure and stability properties of the layers. |
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| AbstractList | The presence of viscosity normally has a stabilizing effect on the flow of a fluid. However, experiments show that the flow of a fluid in which viscosity decreases as temperature increases tends to form shear layers, narrow regions in which the velocity of the fluid changes sharply. In general, adiabatic shear layers are observed not only in fluids but also in thermo-plastic materials subject to shear at a high-strain rate and in combustion and there is widespread interest in modeling their formation. In this paper, we investigate a well-known model representing a basic system of conservation laws for a 1D flow with temperature-dependent viscosity using a combination of analytical andnumerical tools. We present results to substantiate the claim that the formation of shear layers can only occur in solutions of the model when the viscosity decreases sufficiently quickly as temperature increases and we further analyze the structure and stability properties of the layers. (Author) The presence of viscosity normally has a stabilizing effect on the flow of a fluid. However, experiments show that the flow of a fluid in which viscosity decreases as temperature increases tends to form shear layers, narrow regions in which the velocity of the fluid changes sharply. In general, adiabatic shear layers are observed not only in fluids but also in thermo-plastic materials subject to shear at a high-strain rate and in combustion and there is widespread interest in modeling their formation. In this paper, we investigate a well-known model representing a basic system of conservation laws for a one-dimensional flow with temperature-dependent viscosity using a combination of analytical andnumerical tools. We present results to substantiate the claim that the formation of shear layers can only occur in solutions of the model when the viscosity decreases sufficiently quickly as temperature increases and we further analyze the structure and stability properties of the layers. |
| Author | Estep, Donald J. Williams, Roy D. Verduyn Lunel, Sjoerd M. |
| Author_xml | – sequence: 1 givenname: Donald J. surname: Estep fullname: Estep, Donald J. organization: Department of Mathematics, Colorado State University, Fort Collins, Colorado, 80523, f1estep@math.colostate.eduf1 – sequence: 2 givenname: Sjoerd M. surname: Verduyn Lunel fullname: Verduyn Lunel, Sjoerd M. organization: Mathematisch Instituut, Universiteit Leiden, 9512, RA Leiden, 2300, The Netherlandsf2verduyn@math.leidenuniv.nlf2 – sequence: 3 givenname: Roy D. surname: Williams fullname: Williams, Roy D. organization: Center for Advanced Computing Research, California Institute of Technology, Pasadena, California, 91125, f3roy@willow.caltech.eduf3 |
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| CitedBy_id | crossref_primary_10_1007_s00205_016_1071_2 crossref_primary_10_1016_j_jmps_2016_08_015 crossref_primary_10_1016_j_euromechflu_2005_02_002 crossref_primary_10_1137_21M1458247 crossref_primary_10_1142_S0218202510004295 crossref_primary_10_1137_080727919 |
| Cites_doi | 10.1137/0522021 10.1137/S0036139997322419 10.1007/BF00282051 10.1016/B978-0-12-309770-5.50007-2 10.1115/1.3176046 10.1115/1.3121382 10.1088/0965-0393/2/5/001 10.1512/iumj.1977.26.26029 10.1016/0749-6419(92)90023-6 10.1016/0020-7683(87)90008-4 10.1016/0045-7825(94)90110-4 10.1007/978-1-4684-0152-3 10.1016/S0022-5096(96)00098-1 10.1016/0020-7683(92)90193-W 10.1016/0362-546X(82)90058-X 10.1090/qam/700660 10.1090/qam/860899 10.1137/S0036139995289593 10.1016/0022-5096(90)90012-S 10.1016/0022-5096(87)90008-1 10.1002/(SICI)1098-2426(199605)12:3<393::AID-NUM5>3.0.CO;2-J 10.1007/BF00363993 10.1016/0036-9748(84)90418-6 10.1090/qam/840438 10.1216/jiea/1181075606 10.1016/0022-5096(87)90051-2 10.1016/0022-5096(95)00066-6 |
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| Keywords | blow-up, conservation laws adaptive error control plane Couette flow reaction–diffusion equations shear layers temperature-dependent viscosity finite element methods fluid flow invariant rectangles thermal diffusion residual errors a posteriori error estimates |
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| References | Bertsch, Peletier, Verduyn Lunel (RF6) 1991; 22 Drew, Flaherty (RF14) 1984; 52 Chueh, Conley, Smoller (RF7) 1977; 26 French (RF18) 1996; 12 Dafermos, Hsiao (RF9) 1982; 6 Needleman (RF21) 1989; 56 Walter (RF26) 1992; 8 Bayliss, Belytshko, Kulkarnia, Lott-Crumbler (RF5) 1994 Estep, Larson, Williams (RF16) 2000; 146 Maddocks, Malek-Madani (RF20) 1992; 29 Edwards, French (RF15) 1998; 59 Wright, Walter (RF29) 1987; 35 Tzavaras (RF25) 1991; 3 Wright (RF27) 1987; 35 Tzavaras (RF24) 1987; 99 French, Garcia (RF19) 1994; 118 Batra (RF2) 1987; 23 D. J. Estep, S. M. Verduyn Lunel, and, R. D. Williams, An adaptive finite element method for the approximation of a shear flow with temperature-dependent viscosity, in preparation. Bai, Dodd (RF1) 1992 Wright (RF28) 1990; 38 Dafermos, Hsiao (RF10) 1983; XLI Dafermos, Hsiao (RF11) 1986; XLIV Clifton, Duffy, Hartley, Shawki (RF8) 1984; 18 Wright, Walter (RF30) 1996; 44 Tzavaras (RF23) 1986; 44 DiLellio, Olmstead (RF12) 1997; 57 Batra (RF3) 1992; 45 Batra, Ko (RF4) 1992; 10 DiLellio, Olmstead (RF13) 1997; 45 Smoller (RF22) 1983 Dafermos (10.1006/jcph.2001.6837_RF11) 1986; XLIV Batra (10.1006/jcph.2001.6837_RF2) 1987; 23 Estep (10.1006/jcph.2001.6837_RF16) 2000; 146 Batra (10.1006/jcph.2001.6837_RF4) 1992; 10 Needleman (10.1006/jcph.2001.6837_RF21) 1989; 56 10.1006/jcph.2001.6837_RF17 Maddocks (10.1006/jcph.2001.6837_RF20) 1992; 29 Wright (10.1006/jcph.2001.6837_RF28) 1990; 38 Dafermos (10.1006/jcph.2001.6837_RF10) 1983; XLI Dafermos (10.1006/jcph.2001.6837_RF9) 1982; 6 Walter (10.1006/jcph.2001.6837_RF26) 1992; 8 Bai (10.1006/jcph.2001.6837_RF1) 1992 Batra (10.1006/jcph.2001.6837_RF3) 1992; 45 Smoller (10.1006/jcph.2001.6837_RF22) 1983 Chueh (10.1006/jcph.2001.6837_RF7) 1977; 26 Tzavaras (10.1006/jcph.2001.6837_RF25) 1991; 3 Clifton (10.1006/jcph.2001.6837_RF8) 1984; 18 Tzavaras (10.1006/jcph.2001.6837_RF23) 1986; 44 Bayliss (10.1006/jcph.2001.6837_RF5) 1994 Bertsch (10.1006/jcph.2001.6837_RF6) 1991; 22 DiLellio (10.1006/jcph.2001.6837_RF13) 1997; 45 Wright (10.1006/jcph.2001.6837_RF29) 1987; 35 Wright (10.1006/jcph.2001.6837_RF30) 1996; 44 Wright (10.1006/jcph.2001.6837_RF27) 1987; 35 DiLellio (10.1006/jcph.2001.6837_RF12) 1997; 57 French (10.1006/jcph.2001.6837_RF19) 1994; 118 Edwards (10.1006/jcph.2001.6837_RF15) 1998; 59 Drew (10.1006/jcph.2001.6837_RF14) 1984; 52 French (10.1006/jcph.2001.6837_RF18) 1996; 12 Tzavaras (10.1006/jcph.2001.6837_RF24) 1987; 99 |
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| Title | Analysis of Shear Layers in a Fluid with Temperature-Dependent Viscosity |
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