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heat conduction in liquids


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1: 10.73 Physical Applications
Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging. … In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …
§10.73(v) Rayleigh Function
For applications of the Rayleigh function σ n ( ν ) 10.21(xiii)) to problems of heat conduction and diffusion in liquids see Kapitsa (1951a).
2: 14.31 Other Applications
§14.31(i) Toroidal Functions
Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). …
§14.31(ii) Conical Functions
The conical functions 𝖯 1 2 + i τ m ( x ) appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). …
3: Bibliography K
  • P. L. Kapitsa (1951a) Heat conduction and diffusion in a fluid medium with a periodic flow. I. Determination of the wave transfer coefficient in a tube, slot, and canal. Akad. Nauk SSSR. Žurnal Eksper. Teoret. Fiz. 21, pp. 964–978.
  • A. V. Kashevarov (1998) The second Painlevé equation in electric probe theory. Some numerical solutions. Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).
  • S. Kesavan and A. S. Vasudevamurthy (1985) On some boundary element methods for the heat equation. Numer. Math. 46 (1), pp. 101–120.
  • Y. A. Kravtsov (1968) Two new asymptotic methods in the theory of wave propagation in inhomogeneous media. Sov. Phys. Acoust. 14, pp. 1–17.
  • Y. A. Kravtsov (1988) Rays and caustics as physical objects. In Progress in Optics, E. Wolf (Ed.), Vol. 26, pp. 227–348.
  • 4: 9.16 Physical Applications
    Details of the Airy theory are given in van de Hulst (1957) in the chapter on the optics of a raindrop. … Extensive use is made of Airy functions in investigations in the theory of electromagnetic diffraction and radiowave propagation (Fock (1965)). … In fluid dynamics, Airy functions enter several topics. … Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics. … This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …
    5: Bibliography C
  • H. S. Carslaw and J. C. Jaeger (1959) Conduction of Heat in Solids. 2nd edition, Clarendon Press, Oxford.
  • C. J. Chapman (1999) Caustics in cylindrical ducts. Proc. Roy. Soc. London Ser. A 455, pp. 2529–2548.
  • T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.
  • H. Cohen (1993) A Course in Computational Algebraic Number Theory. Springer-Verlag, Berlin-New York.
  • J. P. Coleman (1987) Polynomial approximations in the complex plane. J. Comput. Appl. Math. 18 (2), pp. 193–211.
  • 6: 20.13 Physical Applications
    Let z , α , t . …These two apparently different solutions differ only in their normalization and boundary conditions. …Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). In the singular limit τ 0 + , the functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.
    7: 19.33 Triaxial Ellipsoids
    Application of (19.16.23) transforms the last quantity in (19.30.5) into a two-dimensional analog of (19.33.1). …
    §19.33(ii) Potential of a Charged Conducting Ellipsoid
    If a conducting ellipsoid with semiaxes a , b , c bears an electric charge Q , then the equipotential surfaces in the exterior region are confocal ellipsoids: … A conducting elliptic disk is included as the case c = 0 . … The same result holds for a homogeneous dielectric ellipsoid in an electric field. …
    8: Preface
    Executive responsibility was vested in the editors: Frank W. …Lozier directed the NIST research, technical, and support staff associated with the project, administered grants and contracts, together with Boisvert compiled the Software sections for the Web version of the chapters, conducted editorial and staff meetings, represented the project within NIST and at professional meetings in the United States and abroad, and together with Olver carried out the day-to-day development of the project. … The Web address where additional DLMF content can be found is printed in blue at appropriate places in the Handbook. … Any oversight is unintentional, and the editors apologize in advance. … Editor-in-Chief and Mathematics Editor
    9: Bibliography
  • G. B. Airy (1838) On the intensity of light in the neighbourhood of a caustic. Trans. Camb. Phil. Soc. 6, pp. 379–402.
  • N. I. Akhiezer (2021) The classical moment problem and some related questions in analysis. Classics in Applied Mathematics, Vol. 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • Z. Altaç (1996) Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118 (3), pp. 789–792.
  • G. E. Andrews (1986) q -Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS Regional Conference Series in Mathematics, Vol. 66, Amer. Math. Soc., Providence, RI.
  • T. M. Apostol (1990) Modular Functions and Dirichlet Series in Number Theory. 2nd edition, Graduate Texts in Mathematics, Vol. 41, Springer-Verlag, New York.
  • 10: Bibliography R
  • H. Rademacher (1973) Topics in Analytic Number Theory. Springer-Verlag, New York.
  • H. A. Ragheb, L. Shafai, and M. Hamid (1991) Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric. IEEE Trans. Antennas and Propagation 39 (2), pp. 218–223.
  • W. Reinhardt (1982) Complex Coordinates in the Theory of Atomic and Molecular Structure and Dynamics. Annual Review of Physical Chemistry 33, pp. 223–255.
  • D. H. Rouvray (1995) Combinatorics in Chemistry. In Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grötschel, and L. Lovász (Eds.), pp. 1955–1981.
  • J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.