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Navier–Stokes equilibrium solutions of a viscous fluid confined between two infinite parallel plates that can independently stretch or shrink in orthogonal directions are studied. It is assumed that the admissible solutions satisfy spatial self-similarity in the stretching or shrinking perpendicular coordinates. The nonlinear steady boundary-value problem is discretized using a spectral Legendre method, and equilibrium solutions are found and tracked in the two-dimensional parameter space by means of pseudo-arclength continuation Newton–Krylov schemes. Different families of solutions have been identified, some of which are two-dimensional and correspond to the classical Wang and Wu self-similar flows arising in a plane channel with one stretching–shrinking wall [Wang, C.-A. and Wu, T.-C., "Similarity solutions of steady flows in a channel with accelerating walls," Comput. Math. Appl. 30, 1–16 (1995)]. However, a large variety of three-dimensional solutions have also been found, even for low stretching or shrinking rates. When slightly increasing those rates, some of these solutions disappear at saddle-node bifurcations. By contrast, when both plates are simultaneously stretching or shrinking at higher rates, a wide variety of new families of equilibria are created and annihilated in the neighborhood of cuspidal codimension-2 bifurcation points. This behavior has similarities with the one observed in other planar and cylindrical self-similar flows. [ABSTRACT FROM AUTHOR]

The stagnation point flow over a linearly stretching or shrinking sheet is considered in the present study. The transformed ordinary differential equations are solved numerically. Dual solutions are possible for the shrinking case, while the solution is unique for the stretching case. For the shrinking case, a linear temporal stability analysis is performed to determine which one of the solution is stable and thus physically reliable. [ABSTRACT FROM AUTHOR]

The article offers information regarding the number pi, a mathematical constant. It highlights the application of computers in the search for more digits of pi as well as the development of the formulas for the total number of collisions. It also highlights information concerning the dynamic equations.

The article offers information regarding the number pi, a mathematical constant. It highlights the application of computers in the search for more digits of pi as well as the development of the formulas for the total number of collisions. It also highlights information concerning the dynamic equations.

An extensive set of experiments were performed to determine the stability boundary for the longitudinal oscillation of steel spheres settling through Hydroxy Propyl Guar (HPG) contained in long vertical tubes, a phenomenon first reported by Mollinger, et al. (1999). Results are reported in d/D-pH parameter space, where d is the sphere diameter and D is the internal diameter of a tube. Longitudinal oscillations, which owe their existence to the self-healing nature of the gel, occur over the region 0.48 < d/D < 0.86 and 8.31 < pH < 8.82. We discover that the upper pH limit of this stability domain is linked to the HPG gel point. During the course of experimentation, additional instabilities in the form of high-frequency transverse oscillations or helical motions were discovered and documented. The pure transverse oscillations found in the domain 8.035 < pH < 8.31 and 0.9 < d/D < 0.6 result from the elastic nature of the guar in this region. For 6.25 < pH < 8.035 transverse oscillations or stable descents are interspersed between bands of tight helical motion at large d/D and wide helical motion at low d/D. At the lowest value pH = 6.25 the guar is a Newtonian fluid, yet the spiral motion persists in small bands of d/D. We have verified, in the literature and through additional experiments using distilled water as the Newtonian liquid that the spiral motion of spheres falling inside vertical tubes does indeed exist. [ABSTRACT FROM AUTHOR]

A modification of Homann’s axisymmetric outer potential stagnation-point flow of strain rate $a$ is obtained by adding periodic radial and azimuthal velocities of the form $b\hspace{0.167em} r\sin 2\theta $ and $b\hspace{0.167em} r\cos 2\theta $, respectively, where $b$ is a shear rate. This leads to the discovery of a new family of asymmetric viscous stagnation-point flows depending on the shear-to-strain-rate ratio $\gamma = b/ a$ that exist over the range $\ensuremath{-} \infty \lt \gamma \lt \infty $. Numerical solutions for the wall shear stress parameters and the displacement thicknesses are given and compared with their large-$\gamma $ asymptotic behaviours. Sample similarity velocity profiles are also presented. [ABSTRACT FROM AUTHOR]

We investigate possible linear waves and nonlinear wave interactions in a bounded three-layer fluid system using both analysis and numerical simulations. For sharp interfaces, we obtain analytic solutions for the admissible linear mode-one parent/signature waves that exist in the system. For diffuse interfaces, we compute the overtaking interaction of nonlinear mode-two solitary waves. Mathematically, owing to a small loss of energy to dispersive tails during the interaction, the waves are not solitons. However, this energy loss is extremely minute, and because the dispersively coupled waves in the system exhibit the three types of Lax KdV interactions, we conclude that for all intents and purposes the solitary waves exhibit soliton behavior. [ABSTRACT FROM AUTHOR]

Over two decades ago, some numerical studies and laboratory experiments identified the phenomenon of leapfrogging internal solitary waves located on separated pycnoclines. We revisit this problem to explore the behaviour of the near resonance phenomenon. We have developed a numerical code to follow the long-time inviscid evolution of isolated mode-two disturbances on two separated pycnoclines in a three-layer stratified fluid bounded by rigid horizontal top and bottom walls. We study the dependence of the solution on input system parameters, namely the three fluid densities and the two interface thicknesses, for fixed initial conditions describing isolated mode-two disturbances on each pycnocline. For most parameter values, the initial disturbances separate immediately and evolve into solitary waves, each with a distinct speed. However, in a narrow region of parameter space, the waves pair up and oscillate for some time in leapfrog fashion with a nearly equal average speed. The motion is only quasi-periodic, as each wave loses energy into its respective dispersive tail, which causes the spatial oscillation magnitude and period to increase until the waves eventually separate. We record the separation time, oscillation period and magnitude, and the final amplitudes and celerity of the separated waves as a function of the input parameters, and give evidence that no perfect periodic solutions occur. A simple asymptotic model is developed to aid in interpretation of the numerical results. [ABSTRACT FROM AUTHOR]

Modified axisymmetric, finite-length Taylor–Couette (TC) cells with stationary outer cylinder and rotating inner cylinder are designed in an effort to produce simultaneous onset of toroidal vortices of continuously varying wavelength along the gap. For a given axial variation in the inner radius, the axial variation in the outer radius can be chosen such that at every axial position, the criterion for the onset of Taylor vortices in a corresponding classical TC cell is met. In one scenario, a conical inner cylinder is chosen and the shape of the outer cylinder is then determined by locally satisfying the onset criterion. In another scenario, the inner and outer radii are chosen such that the onset criterion is locally satisfied and the axial rate of change in the classical onset wave number is held constant. In both cases, the modified cells possess a large-scale meridional circulation wrought by the finite Ekman (Bödewadt) pumping on the inner (outer) cylinder walls. Using direct numerical simulation, it is found that for sufficiently large aspect ratio, there exists a critical rotation rate for the simultaneous transition from the base flow to counter-rotating toroidal vortices throughout the varying-radius region. The vortices propagate in the direction of decreasing gap width with a phase speed that decreases with increasing aspect ratio. [ABSTRACT FROM AUTHOR]

The design and construction of the Eiffel Tower was based, in part, on a uniform horizontal wind model giving 300 kg m-2 kinematic pressure acting on the surface of the tower. Eiffel received a patent for his method of construction that eliminates the need for diagonal trellis bars used to resist the moment of an oncoming wind. At the end of the 19th century boundary-layer theory, laminar or turbulent, was nonexistent. Now, however, models for atmospheric flow over rough landscapes are available, the simplest being a power-law distribution of velocity with height. In this paper we deduce the shape of the tower had Eiffel incorporated this information into the design and construction of his world famous tower. Moreover, we prove Eiffel’s observation that the tower profile conforms to the moment distribution wrought by the wind. [ABSTRACT FROM AUTHOR]

The stability of viscous rotating liquid columns and their application to rotating viscous liquid jets aligned under gravity is reviewed. Experiments on stable viscous fluid flow discharged from rotating vertical pipes exhibit very weak contraction. We present an elementary liquid jet analysis to understand this phenomenon. Indeed, our inviscid model of a slender rotating inviscid liquid jet shows that rotation suppresses contraction. Next we study the comparable problem for granular flow. Our model for noncohesive granular flow emanating from a vertical pipe rotating about its central axis, valid for sufficiently large rotation rate, shows that the granular profiles blossom rather than contract. The profiles of both the liquid and granular jets depend on the same dimensionless parameters—an exit Froude number Fr0 and an exit swirl parameter χ0. The limitations of both models are discussed. Experimental data for granular jet profiles compare well with the collision-free granular flow model in its range of applicability. A criterion for the rotation rate at which particles adjacent to the inner wall of the rotating pipe cease to flow is also given and compared to experiment. [ABSTRACT FROM AUTHOR]

The stability of circular Couette flow in discontinuous axisymmetric geometries is investigated using numerical simulations and physical experiments. By contouring the geometry of the inner cylinder, Taylor vortices can be made to appear in discrete sections along the length of the cylinder while adjoining sections remain stable. The disparate flows are connected by transition regions that arise from the stability of the axially nonuniform base flow state. The geometry of the inner cylinder can be tailored to produce the simultaneous onset of Taylor vortices of different wavelength in neighboring sections. In another variant, a stack of inner cylinders of common radius are made to rotate independently to produce adjacent regions of stable and unstable flow. [ABSTRACT FROM AUTHOR]

Vertical rotating viscous liquid jet experiments show a clear preference for helical instabilities that evolve from initially planar disturbances at large rotation rates for fixed fluid properties. The laboratory setup for the experiments described herein was chosen as the nearest earth-based equivalent to a uniformly rotating viscous liquid column in the absence of gravity. In the ideal situation with stress-free boundaries, the preferred modes of linear temporal instability are theoretically known over the entire physical domain spanned by the Hocking parameter L=γ/ρa3Ω2 and the rotational Reynolds number Re=a2Ω/ν, where a is the column radius, Ω is its uniform angular velocity, and ρ, ν, and γ are, respectively, the fluid density, kinematic viscosity, and surface tension. The theoretical results show that instability in L-Re parameter space is dominated by three mode types: The axisymmetric mode, the n≥2 planar modes, and the first n=1 spiral mode. Experiments reveal that, in the L-Re region for which the uniformly rotating liquid column is dominated by planar modes of instability, the rotating liquid jet spontaneously gives rise to planar disturbances of mode n≥2 that rapidly evolve into helical instabilities. However, these observed instabilities are not the spiral normal modes that exist for n≥1 as posited in linear stability theory. In spite of obvious fundamental differences between the rotating liquid jet and the uniformly rotating liquid column, some remarkable similarities associated with initial growth rates, angular frequencies, and mode transitions between the two systems are found. [ABSTRACT FROM AUTHOR]

Stability boundaries separating different states of bimodal convection in a box of saturated porous material with impermeable bounding faces are determined. Two opposing vertical end walls are always insulated, while the thermal conditions on the other set of opposing vertical side walls range from insulated to perfectly conducting, as measured by the Biot number based on box height. The temperature difference between the hot lower plate and the cooler upper plate provides the mechanism for instability. The eigenvalue problem for the critical Rayleigh number is solved numerically over a range of box sizes and side-wall heat transfer conditions. For small values of the distance H2 between conducting side walls, agreement with previous asymptotic analysis is obtained. New results for boxes with planform dimensions both comparable to the box height show the existence of isolated regions rich in bimodal structure. These islands are separated by broad regions having modal properties identical to those found for conducting side walls in the limit H2→0. The numerical results for O(1) box dimensions continue to exhibit the stabilizing effect of side-wall heat transfer as previously observed for thin slabs. [ABSTRACT FROM AUTHOR]

An experimental study of the drag on hollow cylinders and conglomerates falling in a viscous fluid under Stokes flow conditions is described. The experiments were carried out in a tank of square cross section using silicone oil as the Newtonian fluid. Settling velocities of the free falling objects were measured and corrected to conditions of zero Reynolds number flow in an unbounded fluid. The results reveal that all objects tested have Stokes settling velocities smaller than that of a sphere of equal mass and volume. Measurements are reported in terms of the settling speed ratio defined as the ratio of the Stokes settling speed to that of a sphere of equal mass and volume. For the hollow cylinders two parameters are varied: the aspect ratio (length to outside diameter) and the radius ratio (inner to outer radius). Measurements show that the settling speed ratio decreases markedly as the hollowness of the cylinder increases. Each fixed radius ratio data set exhibits a maximum settling speed ratio near an aspect ratio of 1.65. For conglomerates composed of n spheres two trends appear: one for planar configurations and the other for globular clusters. Experimental data for two spheres and three spheres in point contact are in good agreement with recent theoretical results. [ABSTRACT FROM AUTHOR]

The effect of side wall heat transfer on the stability of natural convection in a vertically oriented finite slab of saturated porous material is considered. All six bounding faces are impermeable. A temperature contrast between the top and bottom horizontal surfaces provides the mechanism for destabilization. The narrow vertical end walls are perfectly insulated. The thermal conditions on the broad vertical side walls range from perfectly insulating to fully conducting as determined by the value of B, the Biot number based on slab height. An asymptotic analysis of the general solution is made in the narrow gap limit ε<1, where ε is the cross-slab width-to-height ratio. In this case the relevant heat transfer variable is B=εB, the Biot number based on the narrow slab width. In the limit ε → 0 when B=O(1), including the case B→∞ that corresponds to a linear side wall temperature profile, tall, vertical, three-dimensional, finger-like cells are found at the critical Rayleigh number Rc=π2/ε2. In the limit B → 0 corresponding to perfect insulation, one obtains two-dimensional, O(1) aspect ratio rolls with axes normal to the side walls at the critical value Rc=4π2. These two-dimensional rolls predominate only for B=O(ε2), and transition to tall narrow three-dimensional cells occurs when O(ε2)< B < O(1). [ABSTRACT FROM AUTHOR]

An experimental study has been made of the hydrodynamic stability of viscous fluid flow contained between the differentially heated walls of a tall vertical annulus. Tests were conducted using different glycerol solutions spanning a range of Rayleigh numbers (104–106) and moderately high Prandtl numbers (15–150) for a height/gap ratio of 64 and a radius ratio equal to 0.62. Visualization studies show that the unstable flow consists of two separate progressive wave systems, one ascending the hot inner wall and the other descending the cold outer wall. The approximate Rayleigh number, phase speed, and wavelength at the onset of instability for each wave system were determined for several Prandtl numbers. The regular wave patterns observed near the stability boundary persist over only a very limited range of supercritical Rayleigh numbers above which flow dislocations set in. At higher Rayleigh numbers these random dislocations appear in increasing numbers and ultimately render the flow turbulent. [ABSTRACT FROM AUTHOR]