S. I. Rubinow

Spatial instability of a jet

The instability of a circular cylindrical jet of liquid in air is studied on the assumption that the wavenumber k of the disturbance is complex while its frequency σ is real. This implies that the disturbance grows with distance along the jet, but that it does not grow with time. The occurence of such disturbances is called spatial instability, in contrast to the temporal instability studied by Rayleigh and others, in which k is real and σ is complex. It is found that there are infinitely many unstable modes for the axially symmetric case and also for each of the asymmetric cases. In the case of high velocity jets, one of these modes for the symmetric case corresponds to the mode Rayleigh found. However, it is not the most rapidly growing mode. Both analytical and numerical solutions of the dispersion equation are given for k as a function of σ and of the dimensionless jet velocity.

Slender-body theory for slow viscous flow

Slow flow of a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions. The main result is an integral equation for the force per unit length exerted on the body by the fluid. The novelty is that the body is permitted to twist and dilate in addition to undergoing the translating, bending and stretching, which have been considered by others. The method of derivation is relatively simple, and the resulting integral equation does not involve the limiting processes which occur in the previous work.

Post Buckling Behavior of Elastic Tubes and Rings with Opposite Sides in Contact

An elastic tube of circular cross section or an elastic ring can buckle if p, the outside pressure minus the inside pressure, exceeds the buckling pressure

Compartment analysis: An inverse problem

p_{b2}

Swimming of flagellated microorganisms.

. As p increases above

Recurrent precipitation and Liesegang rings

p_{b2}

Wave Propagation in a Fluid‐Filled Tube

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A mathematical framework for the study of morphogenetic development in the slime mold

Abstract A review is presented of the central mathematical problem of compartment analysis: What is the set of coefficient matrices appearing in the compartment equations that are compatible with the part of the solution that is observed? The matrix elements are shown to depend on a generally underdetermined set of nonlinear algebraic equations. Some general properties of the coefficient matrix for an n -compartment system are deduced. The matrix may be made “minimal” by the imposition of null constraints on the matrix elements. The compatibility of such arbitrary constraints with the algebraic equations is investigated. The algebraic simplicity of two-compartment and three-compartment systems permits a complete characterization of the set of minimal matrices in parametric form. Properties of the graphs representing some minimal matrices of a four-compartment system are obtained.

On closed or almost closed compartment systems

The swimming motion of a microorganism with a single flagellum is investigated for both helical and planar flagellar motion. First the force and torque exerted on the organism by the surrounding fluid are calculated in terms of the specified flagellar motion and the unknown linear and angular velocity of the whole organism. Then these unknown velocities are determined by the condition that the net force and torque on the organism are zero. Using these velocities, the trajectory of the organism is found. In the case of helical flagellar motion, the path of the entire organism is found to be a helix of small radius. The axis of the flagellum is not parallel to the axis of the helical path, but makes a small angle with it and precesses around it. If the flagellar motion is planar and sinusoidal, then the trajectory of the organism is found to be a straight line with small oscillations about it. Each point of the flagellum also oscillates longitudinally with double the frequency of the transverse oscillation, producing a figure eight motion. However if the flagellar motion is planar and asymmetric, then the trajectory is found to be a circle with small superposed oscillations. These conclusions account for the observed helical and circular trajectories of sperm, and for the figure eight motion of the tip of the flagellum in the planar case.

Force on a Rigid Sphere in an Incompressible Inviscid Fluid

A mathematical formulation is presented of Wilhelm Ostwald’s supersaturation theory of Liesegang ring formation. The theory involves diffusion of two reactants toward one another, their chemical reaction to form a product, the reverse reaction, and diffusion of the product. When the product concentration reaches a certain supersaturation value, it begins to precipitate. It is shown that this theory can lead to recurrent precipitation, resulting in rings or bands of precipitate. Conditions under which this occurs are determined. In addition, the locations and times of formation of the bands are calculated and shown to agree with experimental results.