We have all noticed how the surface of a lake gets ruffled by a gentle breeze. To understand such phenomena, we have to study perturbations at an interface between two fluids. Let us consider a plane horizontal interface separating two fluids, with gravity acting downward. If this interface is disturbed, we expect that, under certain circumstances, the interface may oscillate in the form of waves and, under other circumstances, the disturbance may grow to give rise to an instability.
We shall first develop a general mathematical analysis to show how a perturbation at such an interface evolves. Then several special cases of waves and instabilities will follow from the general dispersion relation we derive. To simplify the analysis, we assume that the fluids on both sides of the interface are incompressible and ideal. Hence, if there is no vorticity inside one of these fluids at the beginning, Kelvin's vorticity theorem asserts that the velocities induced inside the fluid as a result of the perturbation remain irrotational. So velocity potentials can be introduced. We neglect the surface tension at the interface between the two fluids.
We have previously derived Bernoulli's principle for steady flows. We now obtain a similar result for a flow which is incompressible and irrotational, but varying in time. Writing and substituting in the Euler equation, we get
where is the the potential for the body force (gravity in the present case). This equation gives rise to the integral
where F(t) is constant in space, but can be a function of time. It may be noted that in deriving Bernoulli's theorem we did not restrict ourselves to irrotational flows. Hence, in order to obtain Bernoulli's theorem, we had to integrate along streamlines to make zero. In the present case, by the assumption so that it is not necessary to integrate along a streamline. Hence Eq (10.60) should ho1d between points not lying on a streamline.
Figure 10.8: Clouds over Denver, Co. The pattern is similar to that seen in the Kelvin-Helmholtz instability. This instability that accompanies wind shear occurs frequently in the atmosphere, however those currents are not normally visible. This is called clear air turbulence, and is a significant hazard to the flight of airplanes.
Figure 10.7 shows the configuration we want to study. The horizontal plane z = 0 separates the two fluids with densities (fluid below) and (fluid above). Let us also assume that the two fluids have uniform velocities U and U' in the x-direction. It is not difficult to show that this configuration satisfies steady state equations (i.e. hydrodynamic equations with ). Let the interface between the two fluids be perturbed from its undisturbed position z = 0 and let us denote the position of the interface by . The subscript 1 is to follow the convention that all perturbed quantities have subscript 1. Our aim is to find out whether this perturbation grows with time or decays or oscillates. The velocity potential in the fluid below can be written as
where the unperturbed part -Ux would give the uniform velocity U in the x direction and is the perturbed part satisfying
as a result of the incompressibility condition . Similarly the velocity potential in the fluid above can be written as
where is the perturbed part satisfying
Since the velocity perturbations are caused by the displacements of the interface, we have to connect the perturbed parts of the velocity potential with the displacements of the interface. To do this, consider a fluid element of the lower fluid lying infinitesimally close to the interface and find out its vertical velocity. In terms of the velocity potential of the lower fluid, the vertical velocity is given by . On the other hand, the vertical velocity of the fluid element is also given by the Lagrangian derivative of the displacement . Equating these two, we have
where the RHS is just the expansion of the Lagrangian derivative, in which we have kept only the terms linear in the perturbed quantities. Similarly, consider a fluid element infinitesimally above the interface and obtain
Since we are linearizing the perturbation equations, we can again write any arbitrary perturbation as a superposition of Fourier components. Because of the symmetry in the x-direction, a Fourier component of the displacement can be written as
The corresponding Fourier components of and should have similar x and t dependences. The z dependences should be such that the Laplace equations (10.62) and (10.64) are satisfied. Hence, they have to be of the form
where the signs before kz have been chosen so that the perturbations vanish as we go far away from the interface. Substituting Eq (10.67-10.69) into Eq (10.65-10.66), we get
These are two equations relating the amplitudes of perturbed quantities A, C and C'. We need a third equation to solve the problem, since we are dealing with three quantities. This is provided by the condition that the pressure has to be continuous across the interface. From Eq (10.60), the pressure inside the lower fluid at a point infinitesimally close to the interface can be written as
where we have written for the gravitational potential. Writing a similar expression for the pressure infinitesimally above the interface and equating them,
Although can in principle be a function of time, here it has to be a constant due to the boundary condition that the perturbations vanish far away from the interface at all times. We can find K by considering the unperturbed configuration for which u = U, u'= U' and , , are all zero. This gives
We note that
if we keep only the linear terms in the perturbed quantities. Substituting for from Eq (10.75) and making a similar substitution for , we find from Eq (10.73) that
after making use of Eq (10.74). Substituting from Eq (10.67-10.69) into this equation, we finally get
which is the third equation connecting the amplitudes of perturbed quantities. Combining Eq (10.70), (10.71) and (10.77), we obtain
An equation connecting the frequency and wavenumber of a Fourier component of the perturbation is called a dispersion relation. For the two-fluid interface problem, Eq (10.78) then is the dispersion relation. We note that Eq (10.78) is a quadratic equation for if k is regarded as given. The solution of this quadratic equation is
We have already seen that a positive imaginary part in leads to instability. Hence the instability occurs when the expression within the square root in Eq (10.79) is negative, i.e. when
This instability is known as the Kelvin-Helmholtz instability (Helmholtz 1868; Kelvin 1871) and causes an interface between two fluids to wrinkle if the fluids on the two sides are moving with different speeds. The most common example of the Kelvin-Helmholtz instability is provided by the observation that a wind blowing over a water surface causes the water surface to undulate.
Even when |U - U'| is arbitrarily small, we notice from Eq (10.82) that the Fourier components satisfying
become unstable. Since large k means small wavelengths, we conclude that only perturbations with small wavelengths are unstable if | U - U'| is sufficiently small. This conclusion, however, gets completely modified when we include surface tension, because surface tension tends to suppress very small wavelengths. On inclusion of surface tension, it is found that the Kelvin-Helmholtz instability takes place only when |U - U'| is larger than a critical value.
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