• Previous Article
    Effective boundary conditions of the heat equation on a body coated by functionally graded material
  • DCDS Home
  • This Issue
  • Next Article
    Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations
March  2016, 36(3): 1383-1413. doi: 10.3934/dcds.2016.36.1383

Passive scalars, moving boundaries, and Newton's law of cooling

1. 

Department of Mathematics, University of California, Riverside, Riverside, CA 92521

2. 

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States

Received  January 2015 Revised  June 2015 Published  August 2015

We study the evolution of passive scalars in both rigid and moving slab-like domains, in both horizontally periodic and infinite contexts. The scalar is required to satisfy Robin-type boundary conditions corresponding to Newton's law of cooling, which lead to nontrivial equilibrium configurations. We study the equilibration rate of the passive scalar in terms of the parameters in the boundary condition and the equilibration rates of the background velocity field and moving domain.
Citation: Juhi Jang, Ian Tice. Passive scalars, moving boundaries, and Newton's law of cooling. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1383-1413. doi: 10.3934/dcds.2016.36.1383
References:
[1]

J. Beale, Large-time regularity of viscous surface waves,, Arch. Rational Mech. Anal., 84 (): 307. doi: 10.1007/BF00250586. Google Scholar

[2]

J. Beale and T. Nishida, Large-time behavior of viscous surface waves,, Recent topics in nonlinear PDE, (1984), 1. doi: 10.1016/S0304-0208(08)72355-7. Google Scholar

[3]

P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow,, Ann. of Math. (2), 168 (2008), 643. doi: 10.4007/annals.2008.168.643. Google Scholar

[4]

A. Kiselev, R. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows,, Indiana Univ. Math. J., 57 (2008), 2137. doi: 10.1512/iumj.2008.57.3349. Google Scholar

[5]

Y. Guo and I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension,, Arch. Rational Mech. Anal., 207 (2013), 459. doi: 10.1007/s00205-012-0570-z. Google Scholar

[6]

Y. Guo and I. Tice, Decay of viscous surface waves without surface tension in horizontally infinite domains,, Anal. PDE, 6 (2013), 1429. doi: 10.2140/apde.2013.6.1429. Google Scholar

[7]

Y. Guo and I. Tice, Local well-posedness of the viscous surface wave problem without surface tension,, Anal. PDE, 6 (2013), 287. doi: 10.2140/apde.2013.6.287. Google Scholar

[8]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces,, Comm. Partial Differential Equations, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296. Google Scholar

[9]

Z. Lin, J.-L. Thiffeault and C. Doering, Optimal stirring strategies for passive scalar mixing,, J. Fluid Mech., 675 (2011), 465. doi: 10.1017/S0022112011000292. Google Scholar

[10]

A. Majda and P. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena,, Phys. Rep., 314 (1999), 237. doi: 10.1016/S0370-1573(98)00083-0. Google Scholar

[11]

T. Nishida, Y. Teramoto and H. Yoshihara, Global in time behavior of viscous surface waves: Horizontally periodic motion,, J. Math. Kyoto Univ., 44 (2004), 271. Google Scholar

[12]

J. Slattery, Advanced Transport Phenomena,, Cambridge University Press, (1999). Google Scholar

[13]

W. Strauss, Partial Differential Equations. An Introduction,, John Wiley and Sons, (1992). Google Scholar

[14]

J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport,, Nonlinearity, 25 (2012). doi: 10.1088/0951-7715/25/2/R1. Google Scholar

[15]

Z. Warhaft, Passive scalars in turbulent flows,, Annu. Rev. Fluid Mech., 32 (2000), 203. doi: 10.1146/annurev.fluid.32.1.203. Google Scholar

[16]

A. Zlatoš, Diffusion in fluid flow: Dissipation enhancement by flows in 2D,, Comm. Partial Differential Equations, 35 (2010), 496. doi: 10.1080/03605300903362546. Google Scholar

show all references

References:
[1]

J. Beale, Large-time regularity of viscous surface waves,, Arch. Rational Mech. Anal., 84 (): 307. doi: 10.1007/BF00250586. Google Scholar

[2]

J. Beale and T. Nishida, Large-time behavior of viscous surface waves,, Recent topics in nonlinear PDE, (1984), 1. doi: 10.1016/S0304-0208(08)72355-7. Google Scholar

[3]

P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow,, Ann. of Math. (2), 168 (2008), 643. doi: 10.4007/annals.2008.168.643. Google Scholar

[4]

A. Kiselev, R. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows,, Indiana Univ. Math. J., 57 (2008), 2137. doi: 10.1512/iumj.2008.57.3349. Google Scholar

[5]

Y. Guo and I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension,, Arch. Rational Mech. Anal., 207 (2013), 459. doi: 10.1007/s00205-012-0570-z. Google Scholar

[6]

Y. Guo and I. Tice, Decay of viscous surface waves without surface tension in horizontally infinite domains,, Anal. PDE, 6 (2013), 1429. doi: 10.2140/apde.2013.6.1429. Google Scholar

[7]

Y. Guo and I. Tice, Local well-posedness of the viscous surface wave problem without surface tension,, Anal. PDE, 6 (2013), 287. doi: 10.2140/apde.2013.6.287. Google Scholar

[8]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces,, Comm. Partial Differential Equations, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296. Google Scholar

[9]

Z. Lin, J.-L. Thiffeault and C. Doering, Optimal stirring strategies for passive scalar mixing,, J. Fluid Mech., 675 (2011), 465. doi: 10.1017/S0022112011000292. Google Scholar

[10]

A. Majda and P. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena,, Phys. Rep., 314 (1999), 237. doi: 10.1016/S0370-1573(98)00083-0. Google Scholar

[11]

T. Nishida, Y. Teramoto and H. Yoshihara, Global in time behavior of viscous surface waves: Horizontally periodic motion,, J. Math. Kyoto Univ., 44 (2004), 271. Google Scholar

[12]

J. Slattery, Advanced Transport Phenomena,, Cambridge University Press, (1999). Google Scholar

[13]

W. Strauss, Partial Differential Equations. An Introduction,, John Wiley and Sons, (1992). Google Scholar

[14]

J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport,, Nonlinearity, 25 (2012). doi: 10.1088/0951-7715/25/2/R1. Google Scholar

[15]

Z. Warhaft, Passive scalars in turbulent flows,, Annu. Rev. Fluid Mech., 32 (2000), 203. doi: 10.1146/annurev.fluid.32.1.203. Google Scholar

[16]

A. Zlatoš, Diffusion in fluid flow: Dissipation enhancement by flows in 2D,, Comm. Partial Differential Equations, 35 (2010), 496. doi: 10.1080/03605300903362546. Google Scholar

[1]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[2]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961

[3]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

[4]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[5]

Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799

[6]

Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685

[7]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[8]

Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473

[9]

Atte Aalto, Jarmo Malinen. Compositions of passive boundary control systems. Mathematical Control & Related Fields, 2013, 3 (1) : 1-19. doi: 10.3934/mcrf.2013.3.1

[10]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229

[11]

Robert Denk, Leonid Volevich. A new class of parabolic problems connected with Newton's polygon. Conference Publications, 2007, 2007 (Special) : 294-303. doi: 10.3934/proc.2007.2007.294

[12]

Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013

[13]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577

[14]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006

[15]

Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025

[16]

Noriaki Yamazaki. Almost periodicity of solutions to free boundary problems. Conference Publications, 2001, 2001 (Special) : 386-397. doi: 10.3934/proc.2001.2001.386

[17]

Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial & Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247

[18]

Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 77-94. doi: 10.3934/naco.2013.3.77

[19]

Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations & Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017

[20]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]