J. B. Aarseth

Heat transfer in a liquid film on an unsteady stretching surface

Abstract The momentum and heat transfer in a laminar liquid film on a horizontal stretching sheet is analysed. The governing time-dependent boundary layer equations are reduced to a set of ordinary differential equations by means of an exact similarity transformation. The resulting two-parameter problem is solved numerically for some representative values of the unsteadiness parameter S for Prandtl numbers from 0.001 to 1000. The temperature is observed to increase monotonically from the elastic sheet towards the free surface except in the high diffusivity limit Pr→0 where the surface temperature approaches that of the sheet. A low stretching rate, i.e. high values of S, tends to reduce the surface temperature for all Prandtl numbers. The heat flux from the liquid to the elastic sheet decreases with S for Pr≲0.1 and increases with increased unsteadiness for Pr≳1.

Flow of a power-law fluid film on an unsteady stretching surface

Abstract Flow of a thin liquid film of a power-law fluid caused by the unsteady stretching of a surface is investigated by using a similarity transformation. This transformation reduces the unsteady boundary-layer equations to a non-linear ordinary differential equation governed by a nondimensional unsteadiness parameter S. The effect of S on the film thickness is explored numerically for different values of the power-law index n. A physical explanation for the findings is also provided.

Temperature dependence of the Casimir effect.

The temperature dependence of the Casimir force between a real metallic plate and a metallic sphere is analyzed on the basis of optical data concerning the dispersion relation of metals such as gold and copper. Realistic permittivities imply, together with basic thermodynamic considerations, that the transverse electric zero mode does not contribute. This results in observable differences from the conventional prediction, which does not take this physical requirement into account. The results are shown to be consistent with the third law of thermodynamics, as well as being not inconsistent with current experiments. However, the predicted temperature dependence should be detectable in future experiments. The inadequacies of approaches based on ad hoc assumptions, such as the plasma dispersion relation and the use of surface impedance without transverse momentum dependence, are discussed.

Casimir problem of spherical dielectrics: quantum statistical and field theoretical approaches.

The Casimir free energy for a system of two dielectric concentric nonmagnetic spherical bodies is calculated with use of a quantum statistical mechanical method, at arbitrary temperature. By means of this rather novel method, which turns out to be quite powerful (we have shown this to be true in other situations also), we consider first an explicit evaluation of the free energy for the static case, corresponding to zero Matsubara frequency (n=0). Thereafter, the time-dependent case is examined. For comparison we consider the calculation of the free energy with use of the more commonly known field theoretical method, assuming for simplicity metallic boundary surfaces.

What is the temperature dependence of the Casimir effect

The temperature dependence of the Casimir force between a real metallic plate and a metallic sphere is analyzed on the basis of optical data concerning the dispersion relation of metals such as gold and copper. Realistic permittivities imply, together with basic thermodynamic considerations, that the transverse electric zero mode does not contribute. This results in observable differences from the conventional prediction, which does not take this physical requirement into account. The results are shown to be consistent with the third law of thermodynamics, as well as being not inconsistent with current experiments. However, the predicted temperature dependence should be detectable in future experiments. The inadequacies of approaches based on ad hoc assumptions, such as the plasma dispersion relation and the use of surface impedance without transverse momentum dependence, are discussed.

Casimir problem of spherical dielectrics: Numerical evaluation for general permittivities

The Casimir mutual free energy F for a system of two dielectric concentric nonmagnetic spherical bodies is calculated, at arbitrary temperatures. The present paper is a continuation of an earlier investigation [Phys. Rev. E 63, 051101 (2001)], in which F was evaluated in full only for the case of ideal metals (refractive index n= infinity ). Here, analogous results are presented for dielectrics, for some chosen values of n. Our basic calculational method stems from quantum statistical mechanics. The Debye expansions for the Riccati-Bessel functions when carried out to a high order are found to be very useful in practice (thereby overflow/underflow problems are easily avoided), and also to give accurate results even for the lowest values of l down to l=1. Another virtue of the Debye expansions is that the limiting case of metals becomes quite amenable to an analytical treatment in spherical geometry. We first discuss the zero-frequency TE mode problem from a mathematical viewpoint and then, as a physical input, invoke the actual dispersion relations. The result of our analysis, based upon the adoption of the Drude dispersion relation at low frequencies, is that the zero-frequency TE mode does not contribute for a real metal. Accordingly, F turns out in this case to be only one-half of the conventional value at high temperatures. The applicability of the Drude model in this context has, however, been questioned recently, and we do not aim at a complete discussion of this issue here. Existing experiments are low-temperature experiments, and are so far not accurate enough to distinguish between the different predictions. We also calculate explicitly the contribution from the zero-frequency mode for a dielectric. For a dielectric, this zero-frequency problem is absent.

Temperature dependence of the Casimir effect

In view of the increasing accuracy of Casimir experiments, there is a need for performing accurate theoretical calculations. Using accurate experimental data for the permittivities we present, via the Lifshitz formula applied to the standard Casimir setup with two parallel plates, accurate theoretical results in the case of the metals Au, Cu and Al. Both similar and dissimilar cases are considered. Concentrating in particular on the finite temperature effect, we show how the Casimir pressure varies with separation for three different temperatures, T = {1, 300, 350}K. The metal surfaces are taken to be perfectly plane. The experimental data for the permittivities generally yield results that are in a good agreement with those calculated from the Drude relation with finite relaxation frequency. We give the results in a tabular form, in order to facilitate the assessment of the temperature correction which is on the 1% level. We emphasize two points: (i) the most promising route for a definite experimental verification of the finite temperature correction appears to be to concentrate on the case of large separations (optimum around 2 µm); and (ii) there is no conflict between the present kind of theory and the Nernst theorem in thermodynamics.

A new scaling property of the Casimir energy for a piecewise uniform string

An unexpected and very accurate scaling invariance of the Casimir energy of the piecewise uniform relativistic string is pointed out. The string consists of 2N pieces of equal length, of alternating type I/type II material, endowed with different tensions and mass densities but adjusted such that the velocity of transverse sound equals c. If EN(x) denotes the Casimir energy as a function of the tension ratio x=TI/TII, it turns out that the ratio fN(x)=EN(x)/EN(0), which lies between zero and one, will be practically independent of N for integers N⩾2. Physical implications of this scaling invariance are discussed. Finite temperature theory is also considered.

CASIMIR PROBLEM IN SPHERICAL DIELECTRICS: A QUANTUM STATISTICAL MECHANICAL APPROACH

The Casimir mutual free energy F for a system of two dielectric concentric nonmagnetic spherical bodies is calculated, at arbitrary temperatures. Whereas F has recently been evaluated for the special case of metals (refractive index n = ∞), here analogous results are presented for dielectrics, and shown graphically when n = 2.0. Our calculational method relies upon quantum statistical mechanics. The Debye expansions for the Riccati-Bessel functions when carried out to a high order are found to be very useful in practice (thereby overflow/underflow problems are easily avoided), and also to give accurate results even for the lowest values of l. Another virtue of the Debye expansions is that the limiting case of metals becomes quite amenable to an analytical treatment in spherical geometry. We first discuss the zero-frequency TE mode problem from a mathematical viewpoint and then, as physical input, invoke the actual dispersion relations. The result of our analysis, based upon adoption of the Drude dispersion relation as the most correct one at low frequencies, is that the zero-frequency TE mode does not contribute for a metal. Accordingly, F turns out in this case to be only one half of the conventional value.

Magnetohydrodynamic wave‐current system with constant vorticity

The basic linear theory is given for a perfectly conducting stratified wave‐current system in which there is an upper layer of incompressible fluid of constant vorticity propagating on a slightly heavier lower layer of stagnant fluid. An extraneous homogeneous horizontal magnetic field B0 is present, directed either longitudinally or transversely. Because of the magnetic field the basic governing equations for the system form a set of differential equations, in marked contrast to the single algebraic dispersion equation that would result were the magnetic field absent. The differential equations are solved, and the solutions are shown graphically for various cases of the magnetic field strength.