An approximate theory is given for the generation of internal gravity waves in a viscous Boussinesq fluid by the rectilinear vibrations of an elliptic cylinder. A parameter λ which is proportional to the square of the ratio of the thickness of the oscillatory boundary layer that surrounds the cylinder to a typical dimension of its cross-section is introduced. When λ 1 (or equivalently when the Reynolds number R 1), the viscous boundary condition at the surface of the cylinder may to first order in λ be replaced by the inviscid one. A viscous solution is proposed for the case λ 1 in which the Fourier representation of the stream function found in Part 1 (Hurley 1997) is modified by including in the integrands a factor to account for viscous dissipation. In the limit λ 4 0 the proposed solution becomes the inviscid one at each point in the flow field.For ease of presentation the case of a circular cylinder of radius a is considered first and we take a to be the typical dimension of its cross-section in the definition of λ above. The accuracy of the proposed approximate solution is investigated both analytically and numerically and it is concluded that it is accurate throughout the flow field if λ is sufficiently small, except in a small region near where the characteristics touch the cylinder where viscous effects dominate.Computations indicate that the velocity on the centreline on a typical beam of waves, at a distance s along the beam from the centre of the cylinder, agrees, within about 1 %, with the (constant) inviscid values provided λs\a is less than about 10 −$ . This result is interpreted as indicating that those viscous effects which originate from the characteristics that touch the cylinder (places where the inviscid velocity is singular) reach the centreline of the beam when λs\a is about 10 −$ . For larger values of s, viscous effects are significant throughout the beam and the velocity profile of the beam changes until it attains, within about 1 % when λs\a is about 2, the value given by the similarity solution obtained by Thomas & Stevenson (1972). For larger values of λs\a, their similarity solution applies.In an important paper Makarov et al. (1990) give an approximate solution for the circular cylinder that is very similar to ours. However, it does not reduce to the inviscid one when the viscosity is taken to be zero.Finally it is shown that our results for a circular cylinder apply, after small modifications, to all elliptical cylinders.

This paper concerns steady plane periodic waves on the surface of an ideal liquid flowing above a horizontal bottom. The flow is irrotational. The volume flow rate is denoted by Q, the velocity potential by ø, the period in ø of the waves by 2L, and the maximum angle of inclination between the tangent to the surface and the horizontal by θm.Krasovskii (12) established that, at each fixed Q and L, there exist wave solutions for each value of θm strictly between zero and ⅙π. We establish that, at each fixed Q and L, there exist wave solutions for each value of qc strictly between c and zero. Here qc is the flow speed at the crest, andwhere g is the acceleration due to gravity. Krasovskii's set of solutions is included in the set that we obtain.

Flow resistance laws, as used for example in water-supply pipe networks, are formulae relating the volume flow rate q along a pipe to the pressure-head difference t between its ends, q = ψ(t). ψ is monotonic. The simple Hazen-Williams power "law" is often used: in appropriate circumstances the more complicated Colebrook-White law (CW) may better represent aspects of the experimental data. Result 1, the first and easiest-to-state result in the paper, is that φ CW , the inverse to ψ CW , can be expressed in terms of the Lambert W -function (Corless et al. 1993). One use of this and of related results, in connection with convex optimization problems describing equilibrium flows in pipe networks, is summarised in Result 2.

Steady plane periodic gravity waves on the surface of an ideal liquid flowing over a horizontal bottom are considered. The flow is rotational with a vorticity distribution ω(ψ) and has flux Q. Let R/g denote the total head and S the flow force of the wavetrain. The diagrams (Fig. 1) show combinations of R and S which are possible in the general case. (We normalise so that Q = 1 throughout. The axes are R/R* and S/S*, where the suffix * refers to the critical flow.) It is proved that no waves are possible below γ+ or to the right of γ here γ+ corresponds to unidirectional supercritical streams, and thus is the best possible barrier, while γ is a barrier to the right of the line of waves of greatest height. Bounds on wave properties are found in the process of establishing the above results. When ω ≡ 0 these bounds were conjectured by Benjamin and Lighthi U (1954) and established in Keady and Norbury (1975). The generalisation to flows with vorticity is accomplished under the condition that ω′(ψ)<π2/hc  2, where hc is the height of the crest of the wave.

In these notes we give proofs of some properties of surface solitary waves. Assuming the existence of solitary-wave solutions to the nonlinear boundary-value problem (P) defined below, it is shown (i) that the wave is a wave of elevation alone, and (ii) that at large distances it is asymptotic to a uniform supercritical stream (i.e. the Froude number , where c is the speed of the stream, h is its depth, and g is the gravity constant).We also deduce a number of inequalities relating F2 to a/h, where a is the maximum displacement of the free surface from its value at infinity. In particular, it is shown for the wave of greatest height that 1·480 < F2 < 2.