The Casimir effect, reflecting quantum vacuum fluctuations in the electromagnetic field in a region with material boundaries, has been studied both theoretically and experimentally since 1948. The forces between dielectric and metallic surfaces both plane and curved have been measured at the 10 to 1 percent level in a variety of room-temperature experiments, and remarkable agreement with the zero-temperature theory has been achieved. In fitting the data various corrections due to surface roughness, patch potentials, curvature, and temperature have been incorporated. It is the latter that is the subject of the present article. We point out that, in fact, no temperature dependence has yet been detected, and that the experimental situation is still too fluid to permit conclusions about thermal corrections to the Casimir effect. Theoretically, there are subtle issues concerning thermodynamics and electrodynamics which have resulted in disparate predictions concerning the nature of these corrections. However, a general consensus has seemed to emerge that suggests that the temperature correction to the Casimir effect is relatively large, and should be observable in future experiments involving surfaces separated at the few micrometer scale.

In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersion model gives consistent results for the Casimir free energy at low temperatures. Specifically, for the free energy near T = 0 we find the leading term proportional to T 2 and the next-to-leading term proportional to T 5/2 . These terms give rise to zero Casimir entropy as T → 0, and is thus in accordance with Nernst's theorem.

Lord Kelvin's result that waves behind a ship lie within a half-angle 19 deg 28' is perhaps the most famous and striking result in the field of surface waves. We solve the linear ship wave problem in the presence of a shear current of constant vorticity S, and show that the Kelvin angles (one each side of wake) as well as other aspects of the wake depend closely on the "shear Froude number" Frs=VS/g (based on length g/S^2 and the ship's speed V), and on the angle between current and the ship's line of motion. In all directions except exactly along the shear flow there exists a critical value of Frs beyond which no transverse waves are produced, and where the full wake angle reaches 180 deg. Such critical behaviour is previously known from waves at finite depth. For side-on shear, one Kelvin angle can exceed 90 deg. On the other hand, the angle of maximum wave amplitude scales as 1/Fr (Fr based on size of ship) when Fr >> 1, a scaling virtually unaffected by the shear flow.Comment: 10 pages, 6 figures, accepted for J. Fluid Mech. Rapid

An approximate dispersion relation is derived and presented for linear surface waves atop a shear current whose magnitude and direction can vary arbitrarily with depth. The approximation, derived to first order of deviation from potential flow, is shown to produce good approximations at all wavelengths for a wide range of naturally occuring shear flows as well as widely used model flows. The relation reduces in many cases to a 3‐D generalization of the much used approximation by Skop (1987), developed further by Kirby and Chen (1989), but is shown to be more robust, succeeding in situations where the Kirby and Chen model fails. The two approximations incur the same numerical cost and difficulty. While the Kirby and Chen approximation is excellent for a wide range of currents, the exact criteria for its applicability have not been known. We explain the apparently serendipitous success of the latter and derive proper conditions of applicability for both approximate dispersion relations. Our new model has a greater range of applicability. A second order approximation is also derived. It greatly improves accuracy, which is shown to be important in difficult cases. It has an advantage over the corresponding second‐order expression proposed by Kirby and Chen that its criterion of accuracy is explicitly known, which is not currently the case for the latter to our knowledge. Our second‐order term is also arguably significantly simpler to implement, and more physically transparent, than its sibling due to Kirby and Chen.

We present a comprehensive theory for linear gravity-driven ship waves in the presence of a shear current with uniform vorticity, including the effects of finite water depth. The wave resistance in the presence of shear current is calculated for the first time, containing in general a non-zero lateral component. While formally apparently a straightforward extension of existing deep water theory, the introduction of finite water depth is physically non-trivial, since the surface waves are now affected by a subtle interplay of the effects of the current and the sea bed. This becomes particularly pronounced when considering the phenomenon of critical velocity, the velocity at which transversely propagating waves become unable to keep up with the moving source. The phenomenon is well known for shallow water, and was recently shown to exist also in deep water in the presence of a shear current [Ellingsen, J. Fluid Mech. 742 R2 (2014)]. We derive the exact criterion for criticality as a function of an intrinsic shear Froude number S b/g (S is uniform vorticity, b size of source), the water depth, and the angle between the shear current and the ship's motion.Formulae for both the normal and lateral wave resistance force are derived, and we analyse its dependence on the source velocity (or Froude number Fr) for different amounts of shear and different directions of motion. The effect of the shear current is to increase wave resistance for upstream ship motion and decrease it for downstream motion. Also the value of Fr at which R is maximal is lowered for upstream and increased for downstream directions of ship motion. For oblique angles between ship motion and current there is a lateral wave resistance component which can amount to 10-20% of the normal wave resistance for side-on shear and S b/g of order unity.The theory is fully laid out and far-field contributions are carefully separated off by means of Cauchy's integral theorem, exposing potential pitfalls associated with a slightly different method (Sokhotsky-Plemelj) used in several previous works.