The effect of viscosity is masked by the algebraic technique in solving the problem. THE RAYLEIGH-TAYLOR PROBLEM A. The theoretical solution to the Rayleigh-Taylor problem was first presented by Rayleigh [ 4. Rayleigh solved the problem of two unconfined, inviscid, incompressible fluids with no surface tension at the interface. This situation is as shown in Figure 1. The liquid interface, T'J, between two liquids with p 1 > p.
Considering the problem in three dimensions adds nothing to the nature of the solution and only serves to complicate the algebra of the solution. At the interface you will also find it. If a nontrivial solution exists, the determinant of the coefficients must vanish. The exact dependence on the growth rate of these quantities is given by (2. 50) and is quite complicated.
Because of this, the previous solution is probably not very good at modeling this phenomenon, so a solution in the case of a finite top layer thickness should be considered.
APPROXIMATIONS TO THE THEORY A. INTRODUCTION
SMALL DENSITY DIFFERENCE APPROXIMATION
It should be noted that this approximation is based on the assumption that. and that the assumption of a short wavelength is not valid at the maximum growth rate. This approximation predicts the preferred wavelength to be about ten percent longer than the short wavelength approximation. This is the same as the short wavelength approximation for the small density difference case, and as before, for T = 0,.
Compared to the full exact theory, we find that the short-wavelength approximation is superior. A dispersion relation is obtained that does not require a long or short wavelength limit, but only the conditions for density and viscosity described earlier.
EQUAL KINEMATIC VISCOSITY APPROXIMATION When both fluids have the same kinematic viscosity µ
SUMMARY OF THEORETICAL RESULTS FOR THE LIMITING CASES CONSIDERED
APPROXIMATIONS BASED UPON PHYSICAL ARGUMENTS A. INTRODUCTION
Simple approximations of the behavior of superimposed fluids can be made using basic physics arguments, coupled with the well-known results for gravitational waves, as described for example by Lamb. The purpose of this method is to produce approximate descriptions of the behavior without the lengthy algebraic manipulations of the previous chapter. It is hoped that this approach will provide greater insight into the nature of the movement.
Since only the solutions of the equations of motion are of interest, the complete derivations are in the appendix.
GRAVITATIONAL EFFECTS
SURFACE TENSION EFFECTS
The surface tension is given by the product of the surface tension constant, T, and the curvature. There is no mathematical or physical reason that limits the applicability of the discussion to the case where p. The description of the instability is of course only valid as long as the amplitude remains small, but we must expect that the wavelengths for which a is greatest for, as given by the small-amplitude theory, will. continue to lead in growth beyond the amplitude range for which the description of the small amplitude is valid.
If µ is the dynamic viscosity of this fluid, and v = µIp is its kinematic viscosity, then the damping of the oscillations from dimensional considerations should depend only on VK 2 From the well-known expression for a damped. simple harmonic oscillator, the damped surface wave will satisfy a form equation in an approximate sense. The second reason for the short wavelength approximation is. considered better than the long wavelength approximation is that the values of the growth rate calculated from the short wavelength approximation are closer to those of the full theory than the values calculated from the long wavelength approximation. The great advantage of the simple model which leads to Eq. 4. 21) is that it provides a direct physical insight into an expectation of a maximum in n(K ), a maximum which must occur in the unstable physical situation.
The density and dynamic viscosity of the air are set equal to zero, but the interfacial tension is included. However, this close agreement cannot be taken as justification of either approximate theory, since surface tension, not viscosity, gives the significant modification of the gravitational effect. A comparison of the approximate solution and the exact solution shows that the approximate formulation is quite accurate (see Fig. 6).
Through these two solutions, an improvement of the short-wavelength approximation can be made for the situations where the depth of the upper fluid cannot be considered infinite. As the depth of the upper layer decreases, the approximations significantly improve the short wavelength approximations for an unbound liquid. This discrepancy is not surprising since some of the important boundary conditions at the upper surface are ignored.
BIOCONVECTION
Finally, an explanation was proposed (15] based on disturbances in the uniform density of the upper layer. To show that Rayleigh-Taylor instability is the process by which the bioconvective patterns are formed, the distance between clusters in the descending patterns must be correlated. with the wavelength predicted by the theoretical model.Observations by Winet[l 3] indicate that when they reach the surface, 17% of the cells remain in a clearly defined layer on the surface, a layer between 0.
This last statement indicates that the models must depend on a cooperative nature of the cells since individual cells could not produce such a high speed. Cell self-motion balances both viscous drag and gravitational force. This gravitational force is partially balanced by a buoyant force VTp g, where p is the density of the medium.
This is the only significant effect the cell has on the fluid, and results in a net increase in the fluid's density. To get an estimate of the ratio of the acceleration force to that of gravity, we consider the worst case, namely that of one cell accelerating from rest. This is the worst case, because in fact all cells would never accelerate in the same direction and their effects on the fluid would disappear (except when the cells reach the surface and stop).
If we consider one square centimeter of an upper layer of thickness h above an infinite underlying fluid with cell concentration CL, we find that the net migration of cells upward in the underlying fluid is at a rate aU. The only significant effect of the cells on the fluid is gravitational, and the consequence of this is simply an increase in the density of the upper layer. It should be emphasized that this distance is only about twice the previously calculated nearest neighbor distance, but in most experimental cases the concentration of the top layer is more than one order of magnitude higher than the concentration used to calculate this neighbor distance, hence the measurements lower to 0.
VISCOUS FLUID CONTAINING GLASS SPHERES
FURTHER COMMENTS ON BIOCONVECTION A. INTRODUCTION
A case of steady state where the cells swim back to the surface and a circulation exists was not taken into account. The previous section was a model of a transient phenomenon; here a model for a steady state phenomenon is presented and the use of the transient model is justified. In this situation, a clearly defined upper layer of increased concentration is not always observed; instead, a rather gentle, continuous concentration gradient occurs.
In deeper cultures, gentle concentration gradients also occur below the initial concentration jump in the upper layer. A transient model of this situation is presented in this chapter, with an exponential density gradient taken as a continuous density change.
STEADY ST ATE CIRCULATION
These lines transport the cells to the jets, which occur at the intersections of the lines. The second event is the growth of the jets into long, narrow fingers (see Figure 11). The final phenomenon is the disintegration of the jets and the return of the cells to the surface.
The first possibility is that the jet may reach the bottom of the container and be stopped. As the jets lose cells, the speed of the jet decreases because the weight of the cells provides the driving force. Through an area A of interface between the upper and lower layers, the flux is from the upper layer.
The ratio of the flux into the top layer and that out of the top layer is. If the top layer has thickness h, the area of density C has thickness h' and the total depth of the fluid d, we can calculate. The first effect is the transfer of mass and momentum due to the movement of the cells.
From continuity considerations, this means that the velocity of the fluid medium (excluding the jets) is quite small; and in the present case it is 0. In the present case the flow of the medium is very small, and we are mainly concerned with the circulation of cells. For the microorganism example, the cells swim upwards at a speed more than twenty times that of the.
EXPONENTIAL DENSITY GRADIENTS
SUMMARY AND CONCLUSIONS
It has been shown that the Rayleigh-Taylor instability can give rise to Benard cell-like patterns, but the patterns are quite viscosity dependent. The theory that was developed showed excellent correlation with the experimental results. Even the approximate solutions turned out to be quite good and much more convenient to use than the exact solution.
The specific cases of microbial cycling in steady state and exponential density gradient culture were also discussed, and calculations were performed to show how these situations affect the previously developed cycling model. Although some modification may be required to describe these cases, the approximations could prove quite useful for qualitative descriptions of motion. Subtracting the first column from the second, adding the third column to the fourth, then subtracting the first column from the third is the result.
This corresponds to the three by three determinant obtained by. omitting the first row and column. Multiplying the second column by -n/K, and dividing the second row by n gives Eq. If we temporarily ignore the fact that the m. n, and collect terms as coefficients of n, n or n, those terms that are coefficients of n2.
The approximations of Chapter III are similar and clearer than the method used by Lamb, and will therefore be used.
When the upper surface of the fluid is free, we assume a solution of the form Chandrasekhar, S., "Equilibrium Character of an Incompressible Heavy Viscous Fluid of Variable Density", Proc. Rayleigh, Lord, "Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density, 11.
WAVELENGTH
MATE 35
WAVELENGTH, cm
WAVE NUMBER
