Onset Of First Instability Research Articles

Onset of oscillatory double-diffusive buoyancy instability in an inclined rectangular cavity

Double-diffusive buoyancy convection in an inclined rectangular closed cavity with imposed temperatures and concentrations along two opposite sidewalls is considered. Attention is restricted to the case where the opposing thermal and solutal buoyancy effects are of equal magnitude (buoyancy ratio Rρ=−1). In this case a quiescent equilibrium solution exists and can remain stable up to a critical thermal Grashof number Grc. For both infinite and finite layers, linear stability analysis shows that, when the cavity inclination α with respect to gravity decreases from 0° to −90°, Grc for the onset of stationary instability increases exponentially while that for the onset of oscillatory instability decreases exponentially. Below a critical αc, the first onset of instability is oscillatory, rather than stationary. For the infinite layer, the influences of α on the critical wave number and frequency of the oscillatory mode are shown and the corresponding flow structure of the eigenfunction consists of counterrotating vortices travelling from one end to the other. For a bounded layer, the neutral stability curves of the first two oscillatory modes, centrosymmetric and anticentrosymmetric, cross each other successively at a series of double Hopf bifurcation points as the aspect ratio increases. These two curves are not smooth, but each contains several abrupt changes, after every one of which a pair of counterrotating vortices is added to the flow field and thus the parity of the mode remains unchanged. The neutral curves showing the influences of Pr and Le are also obtained. The present work expands the work of Bergeon et al. (1999) [8] in which the same physical problem was studied and yet no oscillatory onset of instability was considered.

Second ballooning stability in high-β, compact stellarators

Second ballooning stability is examined in quasipoloidally symmetric, compact stellarator configurations. These high-β (volume-average β>4%) free-boundary equilibria are calculated using a reference Quasi-Poloidal Stellarator (QPS) configuration. QPS plasmas have low-shear, stellarator-like rotational transform profile with |B| that is approximately poloidally symmetric. The high-β QPS equilibria are similar in their magnetic configuration to previously studied tokamak-stellarator hybrids which have a high-shear, tokamak-like rotational transform profile. Both types of configurations have strong magnetic wells and consequently high interchange stability β limits. Free-boundary QPS equilibria have regions of second stability at high β. For infinite-n ballooning modes in QPS plasmas, the boundary for first instability is 〈β〉∼2% and the boundary for second stability is 〈β〉∼6%. Finite-n ballooning mode calculations show higher β limits, 〈β〉>5%. Increasing plasma current (for fixed plasma pressure) can lower the finite-n ballooning mode β limit to 〈β〉=3% by reducing magnetic shear. QPS plasmas with Ohmic current profiles (peaked on-axis) have both a lower infinite-n ballooning β-limit for the onset of first instability and a higher β-limit for the onset of second stability relative to QPS plasma with bootstrap current profiles (peaked off-axis). QPS plasmas are stable to low-n ideal magnetohydrodynamic kink modes and vertical modes for values of β in this range (〈β〉∼6%) due to the low level of plasma current in QPS relative to an equivalent tokamak.

Stability of fcc crystals under hydrostatic loading

Elastic moduli of three perfect fcc crystals (Al, Cu, Ag) are calculated via ab initio study of the energy changes under an hydrostatic strain imposed to the crystal system. Stability conditions for the system are expressed in terms of the computed elastic moduli (the bulk modulus, tetragonal and trigonal shear moduli) and system stability is analyzed along whole deformation path from the minimum up to the inflection point of the dependence of the total energy on the volume. Theoretical tensile strength under hydrostatic loading is evaluated as a stress at the first onset of instability. The calculated values of the elastic moduli and the equilibrium lattice parameters agree well with available experimental data.

Time-dependent pattern formation for convection in two layers of immiscible liquids

A linear and weakly nonlinear stability analysis is performed for specific two-layer systems which have been examined experimentally in the past for purely buoyancy driven convection. Time-dependent oscillations arise at the first onset of instability. For the water/silicone oil system, oscillations are predicted for a range of wavelengths and depth fractions. For the Fluorinert/silicone oil system, oscillations are theoretically predicted in a very narrow parameter range. A 3D Hopf bifurcation on a hexagonal lattice is investigated for time-periodic patterns that arise at onset in extended domains. In both fluid systems, travelling rolls, wavy rolls of type 1, and oscillating triangles are stable for most regimes.

Experiments on the stability of viscous flow between two concentric rotating spheres

Experimental results concerning the onset of instability in a spherical gap are presented, for the case in which the gap width is not too large with respect to the radius. The two spheres may be rotated in the same direction or in opposite directions. The resulting flow field and some special observations are described. The first onset of instability is recorded and the results are summarized in stability diagrams. It is shown that Taylor's calculations for rotating cylinders are also valid for rotating spheres. Finally, further experimental results by several authors concerning the onset of instabilitty between rotating cylinders are compared with those for rotating spheres.

The equilibrium and stability of two dimensional pendent drops

The paper gives a mathematical account of the equilibrium and the stability of equilibrium for two dimensional pendent drops hanging under the action of gravity and surface tension forces. The drops may be formed under conditions of constant volume, as when they hang from the horizontal lower edge of a vertical plate. Alternatively, equilibrium may be maintained under conditions of constant pressure, as, for example, when a drop is formed in a narrow gap between two plane parallel vertical plates when the fluid between the plates is connected to a reservoir of large free surface area. Equilibrium profiles, symmetric about a vertical centre plane, are calcu­lated for drops hanging from horizontal apertures of varying widths. It is shown that the volume of the suspended drop achieves a maximum as a function of, say, its depth, for each given aperture. For apertures of width less than a critical value π in capillary units of length, it is found that the pressure, as measured by the internal pressure at the aperture in excess of the external pressure, also achieves a positive maximum. If the volume and depth of the drop are gradually raised from zero the pressure maximum occurs before the volume maximum is reached. For an aperture of width π the pressure maximum is reduced to zero and the maximum point occurs at the initial position where the volume and depth are zero. For apertures greater than π the pressure decreases monotonically from zero as the depth is raised from zero. In the discussion of the stability we distinguish between two and three dimensional disturbances, and between disturbances which are symmetric or asymmetric about the centre plane of symmetry of the equilibrium profile. Drops maintained at constant volume are shown to have a limit point instability to two dimensional weak disturbances symmetric about the centre plane, which occurs at the point of maximum volume. Likewise, drops maintained at constant pressure in an aperture of width less than π, have a limit point instability to disturbances of the same form, which occurs at the point of maximum pressure. The analysis shows that for apertures greater than π drops cannot be formed under conditions of constant pressure, since they are unstable from the initial position of zero depth and volume. A brief discussion is given for the instability to plane asymmetric disturbances, which we find to be unimportant in the sense that these instabilities cannot occur before the equilibrium has already become unstable to disturbances of symmetric type. The most important source of instability of the two dimensional drop is the three dimensional disturbance, that is, one having a sinuous variation along the length of the drop. The stability to such displacements is studied in the last main section of the paper. It is shown there that instability of this form, with disturbances symmetric about the vertical centre plane, will begin to arise for apertures less than π when the equilibrium profile reaches the pressure maximum, for drops maintained both at constant volume and at constant pressure. For drops maintained at constant pressure this does not make any change in the position of the first onset of instability, since, then, two and three dimensional instabilities arise at the same point. However, for drops maintained at constant volume there is a marked change in behaviour. Since the pressure maximum occurs at a lower volume than the volume maximum, and at a lower depth, the drop maintained at constant volume will always become unstable to three dimensional instability first. The nature of the equilibrium for apertures greater than π is such that the three dimensional instability sets in at the outset, in this case for drops of both kinds. This is in agreement with a result of Plateau (1873) and Maxwell (1875), who studied the stability of a plane horizontal interface in a slit, and showed that for a slit of width greater than π the plane interface is not stable to small displacements which conserve the volume. Finally, we have considered three dimensional disturbances which are asymmetric about the centre plane, and we have shown that instabilities of this form cannot arise.

Stability of time-dependent rotational Couette flow. Part 1. Experimental investigation

The stability of viscous time-dependent rotational Couette flow, induced by an impulsively started inner cylinder was experimentally investigated. The ratio of the radius of the inner cylinder to that of the outer cylinder was . The space between the cylinders was filled with distilled water. The time-dependent instabilities were observed by means of dye injection. They appeared as a series of disks more or less evenly spaced along the inner cylinder. The spacing and growth of these instabilities were recorded using a motion picture camera. From the motion pictures the critical time (which is the time from the impulsive start to the first onset of instability) and spacing of the instabilities were experimentally determined, and a marginal stability curve (Reynolds number versus critical time) was constructed.