Bruce J Bayly

Interests

Research

Fluid mechanics. Magnetohydrodynamics, Magnetic field generation in stars and planets. Math education, informal education. Math and science outreach.

Teaching

Differential equations for science and engineering majors. Applied linear algebra, matrix analysis. Advanced mathematical methods for science and engineering majors. Mathematical modeling.

Courses

Scholarly Contributions

Journals/Publications

• Bazant, M. Z., Chu, K. T., & Bayly, B. J. (2005). Current-voltage relations for electrochemical thin films. SIAM Journal on Applied Mathematics, 65(5), 1463-1484.
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  Abstract: The DC response of an electrochemical thin film, such as the separator in a micro-battery, is analyzed by solving the Poisson-Nernst-Planck equations, subject to boundary conditions appropriate for an electrolytic/galvanic cell. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer, which mediates Faradaic reactions with nonlinear Butler-Volmer kinetics. Analytical results are obtained by matched asymptotic expansions in the limit of thin double layers and compared with full numerical solutions. The analysis shows that (i) decreasing the system size relative to the Debye screening length decreases the voltage of the cell and allows currents higher than the classical diffusion-limited current; (ii) finite reaction rates lead to the important possibility of a reaction-limited current; (iii) the Stern-layer capacitance is critical for allowing the cell to achieve currents above the reaction-limited current; and (iv) all polarographic (current-voltage) curves tend to the same limit as reaction kinetics become fast. Dimensional analysis, however, shows that "fast" reactions tend to become "slow" with decreasing system size, so the nonlinear effects of surface polarization may dominate the DC response of thin films. © 2005 Society for Industrial and Applied Mathematics.
• Bayly, B. J. (2004). Asymptotic structure of fast dynamo eigenfunctions. Fluid Mechanics and its Applications, 71, 157-168.
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  Abstract: The eigenfunctions of the kinematic dynamo problem exhibit complicated spatial structure when the magnetic diffusivity is small. When the base flow is spatially periodic, we may study this structure by examining the Fourier components of the eigenfunction at large wavevectors. In this regime we may seek a WKB form in terms of slowly-varying functions of wavevector. The resulting hierarchy of equations may be systematically analysed for both zero and small nonzero diffusivities. © 2002 Kluwer Academic Publishers.
• Bayly, B. J., Holm, D. D., & Lifschitz, A. (1996). Three-dimensional stability of elliptical vortex columns in external strain flows. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 354(1709), 895-926.
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  Abstract: The Kirchhoff-Kida family of elliptical vortex columns in flows with uniform strain and rotation displays a rich variety of dynamical behaviours, even in a purely two-dimensional setting. In this paper, we address the stability of these columns with respect to three-dimensional perturbations via the geometrical optics method. In the case when the external strain is equal to zero, the analysis reduces to the stability of a steady elliptical vortex in a rotating frame. When the external strain is non-zero, the stability analysis reduces to the theory of a Schrödinger equation with quasi-periodic potential. We present stability results for a variety of different Kirchhoff-Kida flows. The vortex columns are typically unstable except when the interior vorticity is approximately the negative of the background vorticity, so that the flow in the inertial frame is nearly a potential flow.
• Bayly, B. J. (1993). Scalar dynamo models. Geophysical and Astrophysical Fluid Dynamics, 73(1-4), 61-74.
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  Abstract: The equation (δt + u•∇)C = R(x, t)C + κ∇2C, is a scalar analogue of the magnetic induction equation. If the velocity field u(x, t) and the 'stretching' function R(x, t) are explicitly given, then we have the analogue of the dynamo problem. The scalar problem displays many of the same features as the vector kinematic dynamo problem. The fastest growing modes have growth rates that approach a finite limit as K→0 while the eigenfunctions develop more and more complex structure at smaller and smaller length scales. Some insight is provided by an analysis which finds a lower bound on the growth rate that is asymptotically independent of the diffusivity © 1993 Gordon and Breach Science Publishers S.A.
• Bayly, B. J., Levermore, C. D., & Passot, T. (1992). Density variations in weakly compressible flows. Physics of Fluids A, 4(5), 945-954.
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  Abstract: Density variations in real fluids are related to both pressure and entropy variations, even in the incompressible limit. The behavior of density variations depends crucially on the relative sizes of the pressure, temperature, and entropy fluctuations. It is shown how this arises from a formal asymptotic expansion of the compressible Navier-Stokes equations about a uniform state. Direct numerical simulations of the full compressible equations verify the consistency of different asymptotic regimes. In the case of turbulent flow, the Kolmogorov-Obukhov theory allows one to predict inertial range scalings for the density fluctuation power spectra in various situations. The results may be relevant to observations of the interstellar medium. © 1992 American Institute of Physics.
• Cohen, J., Amitay, M., & Bayly, B. J. (1992). Laminar-turbulent transition of wall-jet flows subjected to blowing and suction. Physics of Fluids A, 4(2), 283-289.
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  Abstract: A new family of solutions describing the mean flow of an incompressible two-dimensional wall jet subjected to boundary suction or blowing was found, in which the vertical velocity at the wall varies in a power law with downstream distance. Linear stability calculations indicate that two unstable modes, an inviscid inflectional point instability and a viscous instability, may coexist in the flow. It is demonstrated theoretically that the relative importance of each mode can be controlled by subjecting the wall jet to small amounts of blowing or suction. Experimental results, in the absence of suction and blowing, support theoretical findings about the coexistence of the two instability modes. © 1992 American Institute of Physics.
• Bayly, B. J., & Herbert, T. (1988). Instability mechanisms in shear-flow transition.. ANN. REV. FLUID MECH., 20 , 1988, 359-391.
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  Abstract: Results from classical inviscid and viscous stability theory are reviewed in relation to turbulent shear flow at high Reynolds numbers. The nonlinear growth and equilibration of two dimensional waves in wall bounded and free shear flows is next considered. Three dimensional secondary instability is then described and interpreted in terms of the theory of instability of elliptical vortices.
• Bayly, B. J. (1987). THREE-DIMENSIONAL INSTABILITIES IN QUASI-TWO DIMENSIONAL INVISCID FLOWS.. American Society of Mechanical Engineers, Applied Mechanics Division, AMD, 87, 71-77.
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  Abstract: The three-dimensional inviscid Navier-Stokes equations have a large family of exact steady 'quasi-two-dimensional' solutions, in which the velocity in the x,y plane is determined by a stream function psi (x,y), with the z-velocity and z-vorticity functions of psi (x,y) alone. If the projected streamlines in the x,y plane are closed curves, the flow may be subject to a broad band three-dimensional instability in the form of a wave packet centered on a particular surface of constant psi . The structure of the wave is determined by a Floquet system of ordinary differential equations around the corresponding psi contour in the x,y plane, and the Floquet exponent gives the growth rate. This family of instabilities includes the Rayleigh centrifugal instability, the Leibovich-Stewartson columnar vortex instability, and the secondary instability of finite-amplitude waves in plane shear flows.
• Bayly, B. J., & Childress, S. (1987). Fast dynamo action in unsteady flows and maps in three dimensions. Physical Review Letters, 59(14), 1573-1576.
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  Abstract: Unsteady fast dynamo action is obtained in a family of stretch-fold-shear maps applied to a spatially periodic magnetic field in three dimensions. Exponential growth of a mean field in the limit of vanishing diffusivity is demonstrated by a numerical method which alternates instantaneous deformations with molecular diffusion over a finite time interval. Analysis indicates that the dynamo is a coherent feature of the large scales, essentially independent of the cascade of structure to small scales. © 1987 The American Physical Society.
• Bayly, B. J., Goldhirsch, I., & Orszag, S. A. (1987). Independent degrees of freedom of dynamical systems. Journal of Scientific Computing, 2(2), 111-121.
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  Abstract: Numerical evidence is presented demonstrating the possibility of obtaining reduced equations of motion for dynamical systems in terms of a small number of variables. Some implications of these results are discussed. © 1987 Plenum Publishing Corporation.
• Bayly, B. J. (1986). Fast magnetic dynamos in chaotic flows. Physical Review Letters, 57(22), 2800-2803.
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  Abstract: A generic structure is proposed for rapidly growing magnetic fields in a class of steady, incompressible flows which are everywhere chaotic with positive Liapunov exponent. In such a flow at high magnetic Reynolds number Rm, the magnetic field is approximately aligned everywhere with the dilating direction of the flow, and may have extremely complicated spatial structure. In the limit of infinite Rm, the leading-order growth rate is bounded below by the positive Liapunov exponent of the flow; thus the field is a fast dynamo. © 1986 The American Physical Society.
• Bayly, B. J. (1986). Three-dimensional instability of elliptical flow. Physical Review Letters, 57(17), 2160-2163.
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  Abstract: A theory is presented for Pierrehumbert's three-dimensional short-wave inviscid instability of the simple two-dimensional elliptical flow with velocity field u(x,y,z)=(-Ey,E-1x,0). The fundamental modes, which are also exact solutions of the nonlinear equations, are plane waves whose wave vector rotates elliptically around the z axis with period 2. The growth rates are the exponents of a matrix Floquet problem, and agree with those calculated by Pierrehumbert. © 1986 The American Physical Society.
• Bayly, B. J., & Yakhot, V. (1986). Positive- and negative-effective-viscosity phenomena in isotropic and anisotropic Beltrami flows. Physical Review A, 34(1), 381-391.
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  Abstract: A field-theoretic approach, analogous to Kraichnans direct-interaction approximation, to the stability theory of complex three-dimensional flows is developed. The long-wavelength stability of a class of Beltrami flows in an unbounded, viscous fluid is considered. We examine two flows in detail, to illustrate the effects of strong isotropy versus strong anisotropy in the basic flow. The effect of the small-scale flow on the long-wavelength perturbations may be interpreted as an effective viscosity. Using diagrammatic techniques, we construct the first-order smoothing and direct-interaction approximations for the perturbation dynamics. It is argued that the effective viscosity for the isotropic flow is always positive, and approaches a value independent of the molecular viscosity in the high-Reynolds-number limit; this flow is thus stable to long-wavelength disturbances. The anisotropic flow has negative effective viscosity for some orientations of the disturbance, and is therefore unstable, when its Reynolds number exceeds 2. © 1986 The American Physical Society.
• Bayly, B. J. (1985). Heat transfer from a cylinder in a time-dependent cross flow at low Peclet number. Physics of Fluids, 28(12), 3451-3456.
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  Abstract: The heat transfer from a constant-temperature cylinder in a uniform, time-dependent cross flow at low Peclet number is considered. The time dependence is allowed to be strong, so that the velocity fluctuations may be comparable to, or larger than, the mean flow. The first nontrivial term in the Oseen approximation is calculated using matched-expansion theory, and its physical significance is discussed. As an illustration, the time-dependent heat transfer is calculated for a steady cross flow with large sinusoidal perturbations. © 1985 American Institute of Physics.

Presentations

• Pepper, E., Greivenkamp, J., Bayly, B. J., & Cuillier, C. A. (2017, June). Break Out of Your Library: Set Physics Loose with e-Books, Outreach, and Open Textbooks!. Special Libraries Association Annual Conference. Mesa, AZ: Special Libraries Association - Physics, Astronomy and Math Division.
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  The 2017 Physics Roundtable will offer innovative ways to reach a wider library audience in the sciences. Discover how to utilize e-books to make learning more portable, outreach to make science accessible and unforgettable, and open textbooks to lower the cost of higher education.