A generalization of the classical Euler and Korteweg fluids

Kumbakonam R. Rajagopal

Applications of Mathematics (2023)

  • Volume: 68, Issue: 4, page 485-497
  • ISSN: 0862-7940

Abstract

top
The aim of this short paper is threefold. First, we develop an implicit generalization of a constitutive relation introduced by Korteweg (1901) that can describe the phenomenon of capillarity. Second, using a sub-class of the constitutive relations (implicit Euler equations), we show that even in that simple situation more than one of the members of the sub-class may be able to describe one or a set of experiments one is interested in describing, and we must determine which amongst these constitutive relations is the best by culling the class by systematically comparing against an increasing set of observations. (The implicit generalization developed in this paper is not a sub-class of the implicit generalization of the Navier-Stokes fluid developed by Rajagopal (2003), (2006) or the generalization due to Průša and Rajagopal (2012), as spatial gradients of the density appear in the constitutive relation developed by Korteweg (1901).) Third, we introduce a challenging set of partial differential equations that would lead to new techniques in both analysis and numerical analysis to study such equations.

How to cite

top

Rajagopal, Kumbakonam R.. "A generalization of the classical Euler and Korteweg fluids." Applications of Mathematics 68.4 (2023): 485-497. <http://eudml.org/doc/299507>.

@article{Rajagopal2023,
abstract = {The aim of this short paper is threefold. First, we develop an implicit generalization of a constitutive relation introduced by Korteweg (1901) that can describe the phenomenon of capillarity. Second, using a sub-class of the constitutive relations (implicit Euler equations), we show that even in that simple situation more than one of the members of the sub-class may be able to describe one or a set of experiments one is interested in describing, and we must determine which amongst these constitutive relations is the best by culling the class by systematically comparing against an increasing set of observations. (The implicit generalization developed in this paper is not a sub-class of the implicit generalization of the Navier-Stokes fluid developed by Rajagopal (2003), (2006) or the generalization due to Průša and Rajagopal (2012), as spatial gradients of the density appear in the constitutive relation developed by Korteweg (1901).) Third, we introduce a challenging set of partial differential equations that would lead to new techniques in both analysis and numerical analysis to study such equations.},
author = {Rajagopal, Kumbakonam R.},
journal = {Applications of Mathematics},
keywords = {compressible fluid; Euler fluid; Korteweg fluid; implicit constitutive equation},
language = {eng},
number = {4},
pages = {485-497},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A generalization of the classical Euler and Korteweg fluids},
url = {http://eudml.org/doc/299507},
volume = {68},
year = {2023},
}

TY - JOUR
AU - Rajagopal, Kumbakonam R.
TI - A generalization of the classical Euler and Korteweg fluids
JO - Applications of Mathematics
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 485
EP - 497
AB - The aim of this short paper is threefold. First, we develop an implicit generalization of a constitutive relation introduced by Korteweg (1901) that can describe the phenomenon of capillarity. Second, using a sub-class of the constitutive relations (implicit Euler equations), we show that even in that simple situation more than one of the members of the sub-class may be able to describe one or a set of experiments one is interested in describing, and we must determine which amongst these constitutive relations is the best by culling the class by systematically comparing against an increasing set of observations. (The implicit generalization developed in this paper is not a sub-class of the implicit generalization of the Navier-Stokes fluid developed by Rajagopal (2003), (2006) or the generalization due to Průša and Rajagopal (2012), as spatial gradients of the density appear in the constitutive relation developed by Korteweg (1901).) Third, we introduce a challenging set of partial differential equations that would lead to new techniques in both analysis and numerical analysis to study such equations.
LA - eng
KW - compressible fluid; Euler fluid; Korteweg fluid; implicit constitutive equation
UR - http://eudml.org/doc/299507
ER -

References

top
  1. Blechta, J., Málek, J., Rajagopal, K. R., 10.1137/19M1244895, SIAM J. Math. Anal. 52 (2020), 1232-1289. (2020) Zbl1432.76075MR4076814DOI10.1137/19M1244895
  2. Coleman, B. D., Noll, W., 10.1007/BF00276168, Arch. Ration. Mech. Anal. 6 (1960), 355-370. (1960) Zbl0097.16403MR0119598DOI10.1007/BF00276168
  3. Euler, L., Sur le mouvement de l'eau par des tuyaux de conduite, Mémoires de l’académie des sciences de Berlin 8 (1754), 111-148 French Available at https://scholarlycommons.pacific.edu/euler-works/206. (1754) 
  4. Euler, L., Principes généraux du mouvement des fluides, Mémoires de l’académie des sciences de Berlin 11 (1757), 274-315 French Available at https://scholarlycommons.pacific.edu/euler-works/226/. (1757) 
  5. Euler, L., Principia motus fluidorum, Novi Commentarii academiae scientiarum Petro-politanae 6 (1761), 271-311 Latin Available at https://scholarlycommons.pacific.edu/euler-works/258/. (1761) 
  6. Fosdick, R. L., Rajagopal, K. R., 10.1007/BF00250858, Arch. Ration. Mech. Anal. 81 (1983), 317-332. (1983) MR683193DOI10.1007/BF00250858
  7. Goodman, M. A., Cowin, S. C., 10.1007/BF00284326, Arch. Ration. Mech. Anal. 44 (1972), 249-266. (1972) Zbl0243.76005MR1553563DOI10.1007/BF00284326
  8. Green, G., 10.1017/CBO9781107325074.009, Cambr. Phil. Soc. 7 (1837), 1-24 9999JFM99999 03.0565.01. (1837) DOI10.1017/CBO9781107325074.009
  9. Hutter, K., Rajagopal, K. R., 10.1007/BF01140894, Contin. Mech. Thermodyn. 6 (1994), 81-139. (1994) Zbl0804.73003MR1277702DOI10.1007/BF01140894
  10. Korteweg, D. J., Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Arch. Néerl. (2) 6 (1901), 1-24 French 9999JFM99999 32.0756.02. (1901) 
  11. Roux, C. Le, Rajagopal, K. R., 10.1007/s10492-013-0008-4, Appl. Math., Praha 58 (2013), 153-177. (2013) Zbl1274.76039MR3034820DOI10.1007/s10492-013-0008-4
  12. Málek, J., Průša, V., Rajagopal, K. R., 10.1016/j.ijengsci.2010.06.013, Int. J. Eng. Sci. 48 (2010), 1907-1924. (2010) Zbl1231.76073MR2778752DOI10.1016/j.ijengsci.2010.06.013
  13. Málek, J., Rajagopal, K. R., 10.1016/j.mechmat.2005.05.020, Mech. Mater. 38 (2006), 233-242. (2006) DOI10.1016/j.mechmat.2005.05.020
  14. Málek, J., Rajagopal, K. R., 10.1007/s00033-010-0061-8, Z. Angew. Math. Phys. 61 (2010), 1097-1110. (2010) Zbl1273.76025MR2738306DOI10.1007/s00033-010-0061-8
  15. Maxwell, J. C., 10.1017/CBO9781139057943, Green and Co., London (1871),9999JFM99999 24.1095.01. (1871) DOI10.1017/CBO9781139057943
  16. McLeod, J. B., Rajagopal, K. R., Wineman, A. S., On the existence of a class of deformations for incompressible isotropic elastic materials, Proc. R. Ir. Acad., Sect. A 88 (1988), 91-101. (1988) Zbl0676.73015MR986216
  17. Noll, W., 10.1007/BF00277929, Arch. Ration. Mech. Anal. 2 (1958), 198-226. (1958) Zbl0083.39303MR0105862DOI10.1007/BF00277929
  18. Perlácová, T., Průša, V., 10.1016/j.jnnfm.2014.12.006, J. Non-Newton. Fluid Mech. 216 (2015), 13-21. (2015) MR3441833DOI10.1016/j.jnnfm.2014.12.006
  19. Průša, V., Rajagopal, K. R., 10.1016/j.jnnfm.2012.06.004, J. Non-Newton. Fluid Mech. 181-182 (2012), 22-29. (2012) MR2263984DOI10.1016/j.jnnfm.2012.06.004
  20. Rajagopal, K. R., 10.1023/A:1026062615145, Appl. Math., Praha 48 (2003), 279-319. (2003) Zbl1099.74009MR1994378DOI10.1023/A:1026062615145
  21. Rajagopal, K. R., 10.1017/S0022112005008025, J. Fluid Mech. 550 (2006), 243-249. (2006) Zbl1097.76009MR2263984DOI10.1017/S0022112005008025
  22. Rajagopal, K. R., 10.1007/s00033-006-6084-5, Z. Angew. Math. Phys. 58 (2007), 309-317. (2007) Zbl1113.74006MR2305717DOI10.1007/s00033-006-6084-5
  23. Rajagopal, K. R., 10.1016/j.mechrescom.2014.11.005, Mech. Res. Commun. 64 (2015), 38-41. (2015) DOI10.1016/j.mechrescom.2014.11.005
  24. Rajagopal, K. R., 10.1016/j.ijnonlinmec.2014.11.031, Int. J. Non-Linear Mech. 71 (2015), 165-172. (2015) DOI10.1016/j.ijnonlinmec.2014.11.031
  25. Rajagopal, K. R., Wineman, A., 10.1016/0020-7462(80)90002-5, Int. J. Non-Linear Mech. 15 (1980), 83-91. (1980) Zbl0442.73002MR0580724DOI10.1016/0020-7462(80)90002-5
  26. Souček, O., Heida, M., Málek, J., 10.1016/j.ijengsci.2020.103316, Int. J. Eng. Sci. 154 (2020), Article ID 103316, 27 pages. (2020) Zbl07228661MR4114183DOI10.1016/j.ijengsci.2020.103316
  27. Spencer, A. J. M., 10.1016/B978-0-12-240801-4.50008-X, Continuum Physics. Vol. 1 Academic Press, New York (1971), 239-353. (1971) MR0597343DOI10.1016/B978-0-12-240801-4.50008-X
  28. Truesdell, C. A., A First Course in Rational Continuum Mechanics. Vol. 1. General Concepts, Pure and Applied Mathematics 71. Academic Press, New York (1977). (1977) Zbl0357.73011MR0559731
  29. Truesdell, C., Noll, W., 10.1007/978-3-662-10388-3, Springer, Berlin (2004). (2004) Zbl1068.74002MR2056350DOI10.1007/978-3-662-10388-3
  30. Truesdell, C., Rajagopal, K. R., 10.1007/978-0-8176-4846-6, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (2000). (2000) Zbl0942.76001MR1731441DOI10.1007/978-0-8176-4846-6

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.