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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.

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]

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]

Natural convection in a vertical slot filled with a fluid-saturated porous medium modeled by the Brinkman equation is considered. A continuum of solutions are found from capped to open ends in fully-developed two-dimensional flow. This brings to flow in a porous media some aspects of the Newtonian conduction regime results of Bühler (Heat Mass Transf 39: 631, 2003) and the convection regime results of Weidman (Heat Mass Transf 43: 103, 2006). The Brinkman model, valid for sufficiently large porosity and governed by parameter α, allows one to connect Bühler’s conduction regime flow at low α to a near-Darcy convection regime boundary-layer flow at high α. [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]

The stability of flow in an annulus capped at both ends and bounded by a fixed outer cylinder and a spiraling inner rod is determined in the context of linear theory. For infinite aspect ratio and constant fluid properties the problem is governed by a Taylor number Ta, an axial Reynolds number Re, and the radius ratio η. Linear stability is tested with respect to both axisymmetric (n=0) and nonaxisymmetric (n≠0) disturbances for η=0.2, 0.4, 0.6, 0.8 over the range 0≤Re≤2000. The evolution of axial wave number, axial phase speed, and spiral inclination angle with increasing Re at each η through all mode transitions to an asymptotic state n=N is reported. It is found that N=2, 4, 7, 19 for η=0.2, 0.4, 0.6, 0.8, respectively, and that the asymptotic state is reached at Re[bar_over_tilde:_approx._equal_to]2000 for all radius ratios. The asymptotic state is characterized by critical Taylor numbers and frequencies that approach constant values while critical wave numbers fall off as Re-1. In this same limit the counter-rotating vortex pairs align themselves axially within the annulus. A conjecture is made on the influence of η on stability in the high Reynolds number limit. [ABSTRACT FROM AUTHOR]

Recent studies have elucidated the possibility of solitary wave behavior in Marangoni–Bénard instability flows. The present paper takes a look at experiments in both heat-transfer- and mass-transfer-driven Marangoni systems to discern nonlinear behavior of the type associated with solitary wave interactions. Direct observation of phase shifts incurred by two Marangoni waves having undergone a head-on collision are reported. Analysis of experiments exhibiting wave reflection at a solid wall and obliquely interacting waves in Marangoni instability flows also show remarkable similarities with nonlinear behavior in Korteweg–de Vries systems. [ABSTRACT FROM AUTHOR]

The evolution of large scale vortices as coherent structures in a free shear layer and the corresponding energetics in Fourier space are investigated numerically. The results confirm that vortex pairing as a result of selective energy transfer directly to the subharmonic is a rare event when other instability modes are present with random phases. For this naturally perturbed shear flow, vortices are observed to coalesce locally and intermittently in a manner that leaves isolated vortices between paired states. The spatial and temporal intermittency results in a gradual energy transfer from the linearly unstable mode to larger length scales. [ABSTRACT FROM AUTHOR]

The article discusses the result of a study concerning helical instability of a rotating liquid jet. It was found that initial planar disturbances, marked by vertical striations immediately downstream of the rotating tube, quickly transition into helical waves. The droplet formation process involves thin stretching ligaments which are eventually severed.

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]