Lighthill

Lighthill, J. 1978. Waves in fluids. Cambridge Cambridge University F ress. 

Iight86] Lighthill, J., An Informal Introduction to Theoretical Fluid Mechanics, Oxford University Press (1986). 

Eqs. (40H41) are obtained from the analytical solution using the first two terms in the 0-series expansion of the concentration profile. As a result, they are accurate only for small values of meridional angle, 8. To correct for large values of 6, Newman [45] used Lighthill s transformation and Eq. (15) for the meridional velocity gradient to calculate the local mass transfer rate as Sc - oo. His numerical result is plotted in Fig. 5 in the form of Shloc/Re1/2 Sc1/3 vs. 8 as the thin solid line. The dashed line is... 

In early type stars, the bottom boundary penetrates into convective core (Osaki 1975). Accordingly, convective motion of eddies excites sound waves, as in the case of acoustic noise emmision from incompressible turbulence, shown by Lighthill (1978). Since the frequency of excited waves is higher than the Brunt-Vaisala frequency at the photosphere, the waves are not trapped, but running outward (cf. Unno et al 1979). 

Chien (C7) has proposed an energy mechanism for explaining the initial decrease in film thickness and subsequent formation of waves. Mayer (M7) has dealt with the manner in which the waves, which are initially fairly regular near the inception line, later develop into roll waves, etc., downstream, and Lighthill (L10) has also considered the manner in which waveforms may change by amplitude and frequency dispersion. 

L10. Lighthill, M. J., in First Symposium on Naval Hydrodynamics, Sept. 24-28, 1956, Washington, D.C., pp. 17-40, Publication No. 515. National Academy of Sciences—National Research Council, Washington, D.C. 1957. 

Golay, M. J. E. 1958 Contribution to the Gas Chromatography Symposium, Amsterdam. Lighthill, M. J. 1958 Fourier analysis and generalized functions. Cambridge University Press. Taylor, Sir Geoffrey 1953 Proc. Roy. Soc. A, 219,186. 

The Mathematical Institute, University of Edinburgh (Communicated by M. J. Lighthill, F.R.S.—Received 31 December 1957)... 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

A kinematic wave may be called linear if the relationship between the flow and the concentration can be expressed by one or more linear equations, algebraic or differential. The term linear may also be applied when a diffusion term is included in the continuity equation as is done in 3 of Lighthill ... 

The inversion of this transform gives a somewhat cumbersome integral, of which the physical meaning is far from obvious, and Lighthill Whitham naturally prefer to elucidate this form the asymptotic behaviour of the transform, by the method of steepest descents. The method presented here also uses the transform without the need for inversion and obtains a description of the wave in terms of its moments. 

That this is essentially a kinematic wave is seen by dropping the conductivity term in (24) and writing k = Hf + Hs, the concentration of heat and q = vhfT, the flow of heat. We then recover the kinematic wave equation given by Lighthill Whitham. If thermal equilibrium were instantaneously attained so that... 

Lighthill, M. J. 1956 Viscosity effects in sound waves of finite amplitude. Surveys in mechanics. Cambridge University Press. 

Cylindrical Boundary Layer Laminar boundary layers on cylindrical surfaces, with flow parallel to the cylinder axis, are described by Glauert and Lighthill (Proc. R. Soc. [London], 230A, 188—203 [1955]), Jaffe and Okamura (Z. Angew. Math. Phys., 19,564-574 [1968]), and Stewartson (Q. Appl. Math., 13, 113—122 [1955]). For a turbulent boundary layer, the total drag may be estimated as... 

M. J. Lighthill, An Introduction to Fourier Analysis and Generalised Functions (Cambridge University Press, Cambridge, 1958). 

Figure 3 summarizes the results obtained when the contribution of London forces In the transport rate is negligible. When the boundary-layer analysis of Levich and Lighthill is valid (Case la) the Sherwood number is given by... 

Dashed lines (—) represent the Levich-Lighthill equation (19). For Peclet numbers less than unity, the Shor-wood numbers correspond to those computed for a stagnant fluid (Case lc or Case 3) using Equation (26). At very large Peclet numbers the solid curves become tangent to the dotted lines (.) which represent the Sherwood... 

Fig. 6. Sherwood numbers computed for the transport of finite particles to a spherical collector under the combined action of convective-diffusion and London forces. Values of the aspect ratio are (a) ft - 104, (b) ft — 105, (c) ft =- 10s, and (d) ft = 10. For each aspect ratio, the value of A/kT was taken (upper curves to lower curves) as ID2, 1, 10 2 and 10 4. Dashed lines represent the Levich-Lighthill equation (19), while the dotted curves represent Sherwood numbers deduced from Figure 4 which ignores the transport from diffusion.

Brownian fluctuations, inertia, nonhydrodynamic interactions, etc.) to lead to exponential divergence of particle trajectories, and (2) a lack of predictability after a dimensionless time increment (called the predictability horizon by Lighthill) that is of the order of the natural logarithm of the ratio of the characteristic displacement of the deterministic mean flow relative to the RMS displacement associated with the disturbance to the system. This weak, logarithmic dependence of the predictability horizon on the magnitude of the disturbance effects means that extremely small disturbances will lead to irreversibility after a very modest period of time. 

Lighthill (LI) suggested a general form which gives Om = —0.16 but does not correctly represent the diagonal terms,... 

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