(311a) Wave Formation On Nonparallel Axisymmetric Flows or the Unstable Career of William B. Krantz

When one thinks of the career of Bill Krantz, “unstable” does not come immediately to mind.  Perhaps “cosmopolitan” as he has worked at universities in Turkey, Germany, England, India, the Netherlands, Australia, and Singapore in addition to Colorado, Cincinnati and Notre Dame in the US.  “Multitalented” would be appropriate as he has published seminal papers in the areas of fluid mechanics, in-situ combustion of coal, the formation of patterned ground in the Arctic, and membrane separations.  But at the heart of almost all of his research, since his days as a graduate student, is stability analysis.  Whether this is applied to a free surface flow, a combustion front during an in-situ gasification process, or the formation of an asymmetric membrane Bill Krantz has been able to see the similarities and make significant contributions in these seemingly disparate areas.  And so, “unstable” is an apt description.

My work with Bill involved a stability analysis of a free surface flow over a cone.  The reason for studying such a seemingly esoteric phenomena lies in the effect that waves have on mass transfer in such devices as wetted-wall columns and gas-liquid absorption towers.  While Bill had studied wave formation on a planar flow, the flow over packings in an absorption tower are obviously more complex.  Flow over a conical surface offered the advantage of a known geometry but with a more complex underlying flow pattern (2-D rather than 1-D) than previously studied. 

To study the stability of such a system, the two dimensional Navier-Stokes equations for an axisymmetric flow were non-dimensionalized using an unspecified characteristic length.  The two equations were cross differentiated and subtracted to eliminate the pressure term.  When a stream function was used to replace the velocity terms, a single equation for this system was derived.  This equation is a partial differential equation, containing fourth-order derivatives in both the streamwise (x) and cross-stream (y) directions, with non-constant coefficients.  In nondimensionalizing the Navier-Stokes equations, a parameter (δ) representing the ratio of the characteristic length for the diffusion of vorticity in the cross-stream direction to the characteristic length for the diffusion of vorticity in the streamwise direction arises.  For low Reynolds number flows, this parameter will be small, thus suggesting the use of a regular perturbation analysis. 

The resulting equation then can be solved sequentially for each increasing order of magnitude of the perturbation parameter (δ).  The zeroth-order solution turns out to be the solution to Problem 3.W₄ in Bird, Stewart and Lightfoot (Transport Phenomena, John Wiley & Sons, Inc., 1960).  This solution assumes that the flow profile at any point along the conical surface is semi-parabolic, i.e., it neglects the influence of the cross-stream velocity on the streamwise velocity profile.  When the equation representing the next higher order of the perturbation parameter is solved, the effect of the cross-stream velocity is introduced.  The effect of this cross-stream velocity is to make the basic flow thinner thus increasing the average streamwise velocity.

With the stream function of the basic flow now solved, the stream function for the oscillating flow is expressed as a sum of a non-time dependent function (basic flow) and a time-varying function (dependent on both the streamwise and cross-stream directions as well as time).  Since a solution corresponding to small amplitude oscillations is sought, the resulting equation can be linearized to yield the 2-D analog of the Orr-Sommerfeld equation.  Again, the presence of the small parameter (δ) suggests the use of a perturbation analysis to solve this equation. 

Because of the thinning nature of the basic flow in the streamwise direction, the stability of the flow can be expressed in either absolute or relative terms.  A flow which is unstable in an absolute sense is one in which the amplitude of the oscillations increase.  A flow which is unstable in a relative sense is one in which the amplitude of the oscillations is not increasing in size but where the ratio of the amplitude of the oscillation to the depth of the unperturbed flow is increasing.  Thus a flow that is unstable in the absolute sense is also unstable in the relative sense but a flow can be unstable in the relative sense while simultaneously being stable in the absolute sense.   

From this analysis it is determined that there exist conditions under which the flow of a Newtonian fluid down a conical surface will be unstable in an absolute sense.  However, as these instabilities move down the cone they will eventually become stable (in the absolute sense but still unstable in the relative sense) regardless of the magnitude of the instability near the apex of the cone.  Under other conditions, no unstable oscillations (in the absolute sense) are observed but oscillations in the relative sense do exist.  As these oscillations (unstable in the relative sense) move further down the cone, they will eventually become stable in both absolute and relative terms.  Increasing the flow rate or decreasing the apex angle of the cone both lead to more unstable conditions, in both an absolute and relative sense.  This solution also predicts that as oscillations move down the cone their wave velocity will decrease, thus leading to shorter wave lengths as oscillations move down the conical surface.  While no experiments were performed to verify these results, certain geologic features do provide qualitative support for these predictions.

This same approach can be used to also predict the stability of flows over other axisymmetric flows, such as flows on cylindrical surfaces.  Unlike the flows over conical surfaces, where no temporally growing oscillations are predicted, flows on cylindrical surfaces do allow temporally growing solutions.  In the case of flows over cylindrical surfaces the effect of the surface tension of the liquid is to make the flow more unstable in all situations. 

These solutions, for both the basic flow and the perturbed flow on either cylinders or cones, were the first of their kind to be reported in the literature.  But more than that, they represent the faith that Bill Krantz had in a young graduate student.  As is typical for Bill the mentoring does not stop on graduation.  Through Bill’s inspiration and guidance I have followed his lead into academia and into government service at the National Science Foundation.  This is not the only time that Bill Krantz has worked to bring out the best in his students.  In fact, this is common for Bill Krantz.  Hopefully many more students in the future will benefit from his guidance and friendship.