Chia-Shun Yih

Stability of Liquid Flow down an Inclined Plane

The stability of a liquid layer flowing down an inclined plane is investigated. A new perturbation method is used to furnish information regarding stability of surface waves for three cases: the case of small wavenumbers, of small Reynolds numbers, and of large wavenumbers. The results for small wavenumbers agree with Benjamins result obtained by the use of power series expansion, and the results for the two other cases are new. The results for large wavenumbers, zero surface tension, and vertical plate contradict the tentative assertion of Benjamin. The three cases are then re‐examined for shear‐wave stability, and the results compared with those for confined plane Poiseuille flow. The comparison serves to indicate the vestiges of shear waves in the free‐surface flow, and to give a sense of unity in the understanding of the stability of both flows. The case of large wavenumbers also serves as a new example of the dual role of viscosity in stability phenomena.The topological features of the ci curves for...

Instability due to viscosity stratification

The principal aim of this paper is to show that the variation of viscosity in a fluid can cause instability. Plane Couette-Poiseuille flow of two superposed layers of fluids of different viscosities between two horizontal plates is considered, and it is found that both plane Poiseuille flow and plane Couette flow can be unstable, however small the Reynolds number is. The unstable modes are in the neighbourhood of a hidden neutral mode for the case of a single fluid, which is entirely ignored in the usual theory of hydrodynamic stability, and are brought out by the viscosity stratification.

Fluid Motion Induced by Surface‐Tension Variation

Fluid motion in a long straight channel induced by longitudinally varying surface tension has been discussed by Levich. This problem is re‐examined and a different solution is given. In addition, the stability of laminar flows involving surface‐tension variation is briefly discussed, and a correction of a previous result [C.‐S. Yih, J. Fluid Mech. 28, 493 (1967)] is made.

Exact solutions for steady two-dimensional flow of a stratified fluid

Three classes of exact solutions for steady two-dimensional flows of a stratified fluid are found. The flows which correspond to these solutions have arbitrary amplitude (however defined). Two of the three classes of solutions have close bearings on the lee-wave problem in meteorology. It is also shown that the amplitudes of the lee-wave components (if there is more than one component) depend not on the details of the shape of the barrier, but only on certain simple integral properties of the function for the singularity distribution generating the barrier.

Dual Role of Viscosity in the Instability of Revolving Fluids of Variable Density

The stability of a viscous fluid between rotating cylinders and with a radial temperature gradient against the formation of axisymmetric disturbances (Taylor vortices) is considered, and it has been found that viscosity has a dual role. If the circulation increases radially outward (so that the flow would be stable in the absence of density variation) but the density decreases with the radial distance, the situation can arise that viscosity actually has a destabilizing effect. In the opposite circumstance, thermal diffusivity is always destabilizing. Detailed results for small spacing of the cylinders and sufficient conditions for stability of a revolving fluid of variable density or entropy also are given.

Instability of unsteady flows or configurations. Part 2. Convective instability

The formation of convective cells in a fluid between two horizontal rigid boundaries with time-periodic temperature distribution is studied by the use of the Floquet theory. Numerical results for the critical Rayleigh number are given for a Prandtl number of 0·73 (air) and for various values of the frequency and magnitude of the primary temperature oscillation. Some numerical results for a Prandtl number of 7·0 (water) are also given. The most striking feature of the results is that the disturbances (or convection cells) oscillate either synchronously or with half frequency.

Instability of unsteady flows or configurations Part 1. Instability of a horizontal liquid layer on an oscillating plane

A layer of viscous liquid with a free surface is set in motion by the lower boundary moving simple-harmonically in its own plane. The stability of this motion is investigated. Since the primary flow is time-dependent, the time variable cannot be separated from at least one space variable, and a new approach must be used to investigate the problem. In this paper the stability of long waves is studied by a perturbation method which has not been applied before to problems of stability of unsteady flows, and it is found that the flow under consideration can be unstable for long waves.

Gravity waves in a stratified fluid

A unified treatment of wave motion in a stratified fluid, with or without density discontinuities, is achieved by reducing the governing differential system to a Sturm-Liouville system. With the aid of Sturms comparison theorem, it is found (without detailed calculations) that, for any stratification, the phase velocity increases as the wave-number decreases and that, for the same wave-number, the phase velocity increases as the density gradient is increased everywhere and decreases as the density is increased everywhere by a constant amount. Sturms oscillation theorem provides upper and lower bounds for the phase velocity for a given stratification, a given wave-number, and a given number of zeros of the eigenfunction (or a given number of stationary surfaces in the fluid). The inequalities giving these bounds are used to explain the well-known tendency for surfaces of density discontinuities to behave as rigid boundaries when the stratification in each layer is slight. The rigid-boundary behaviour of interfaces in such cases enables one to obtain the approximate eigenvalue spectrum by superimposing the spectra of the individual layers (with the interfaces treated as rigid) on the spectrum of the interfacial (or free surface) waves, obtained by ignoring the slight continuous stratification in each layer. It is pointed out that the Ritz method can be used for calculating the eigenvalues even when the density is discontinuous, and examples are given to show the accuracy of the Ritz method. The nature of the spectrum when the depth is infinite is also clarified. In the course of the development of the theory, the effects of compressibility and of three-dimensionality are determined and given explicitly, the rate of growth of unstable stratifications is related to the phase velocity of waves in stable ones, and equipartition of energy is proved. Motion due to a wave-maker is discussed in order to bring out the connexion between the type of the governing partial differential equation and the nature (local or not local) of the disturbances. The effect of surface tension and the stability of a stratified fluid under vertical oscillation are also discussed.

Wave formation on a liquid layer for de-icing airplane wings

Wave formation on a thin liquid layer used for de-icing air-plane wings is investigated by studying the stability of air flow over a liquid-coated flat plate at zero angle of incidence. The ratio of the viscosity of the liquid to that of air is very high (over half a million), and the Reynolds number based on liquid depth and air viscosity is of the order of a few thousand in actual practice. Under these circumstances the analysis gives two formulas, in closed form, for the growth rate and phase velocity of the waves in terms of the wavenumber and other relevant parameters, including the Froude number F representing the gravity effect and a parameter S representing the surface-tension effect. In the calculation, the wavenumber is not restricted in any way. The wavenumber of the waves that one expects to observe is that for which the growth rate is the maximum. The instability is one in which the viscosity difference between the two fluids (air and liquid) plays the dominant role, and is of the kind found by Yih (1967).

Instability of a Rotating Liquid Film with a Free Surface

The centripetal acceleration of a rotating liquid film is tantamount to a centrifugal force, which tends to cause the liquid film to form rings around the circular cylinder to which it is attached. The stabilizing factors are surface tension and, presumably, viscosity. But it is shown in this paper that instability occurs even for large values of the surface-tension parameter and at small Reynolds numbers. The critical wave number is shown to depend predominantly on the surface tension. Its dependence on the Reynolds number, R, is slight if R is small, and nil if R is large. The effect of viscosity is therefore essentially to slow down the rate of amplification of the unstable disturbances. The analysis is carried out for both large and small Reynolds numbers, for various ratios of film thickness to cylinder radius, and for various surface tension parameters. (The calculation for intermediate Reynolds numbers turns out to be unnecessary for the purpose of comparison with the experiments obtained. Enough information is provided by the calculations performed for practical applications.) Numerical results are given. Comparison of results obtained from 65 experiments with pure glycerine, water+glycerine mixture, and water with the analytical results shows satisfactory agreement.