Effect of Wave Instabilities in Thermal and Solutal Mixing of Double-Diffusive Opposed Jets Impinging in a Passive-Mixer

2-D numerical experiments are performed to investigate the flow instabilities and mixing of different non-isothermal counterflowing jets in a Passive-Mixer. The fluid is modelled as a binary mixture with thermal and solutal buoyancy effects considered through the Boussinesq approximation. The streams are arranged in a thermal and solutal buoyancy aiding configuration. Computations are carried out for three different ratios of the upper jet bulk velocity to the lower jet bulk velocity (VR), namely, VR = 0.5, 1.0 and 2. Within the parametric domain of RiT and RiC defined by region (RiT + RiC) = 3, the instability causing transition from steady to unsteady flow regime is observed for VR = 1 and 2 while no transition is found to occur at VR = 0.5. Using Landau theory, it is established that the transition from steady to unsteady flow regime is a supercritical Hopf bifurcation. A complete regime map identifying the steady and unsteady flow regimes, within the parametric space of the present study, is obtained by plotting the neutral curves of RiC and RiT (obtained using Landau theory) for different values of VR. POD analysis of the unsteady flows at VR = 1, establishes the presence of standing waves. However, for VR = 2, the presence of degenerate pairs in the POD eigenspectrum ascertains the presence of travelling waves in the unsteady flows. The standing wave unsteady flow mode is found to yield the highest rate of mixing.

The effects of Coriolis force and radial stretch on the convective instability characteristics of the Bödewadt flow

The Bödewadt boundary-layer flow is induced by the rotation of a viscous fluid rotating with a constant angular velocity over a stationary disk. In this paper, the Bödewadt boundary-layer flow has been studied in the presence of the Coriolis force to observe the effect of radial stretch of the lower disk on the flow. For the first time in the literature, a numerical investigation of the effects of both stretching mechanism and the Coriolis force on the flow behaviour and on the convective instability characteristics of the above flow has been carried out. In this paper, the Kármán similarity transformations have been considered in order to convert the system of PDEs representing the momentum equations of the flow into a system of highly non-linear coupled ODEs and solved numerically to obtain the velocity profiles of the Bödewadt flow. Then, a convective instability analysis has been performed by using the Chebyshev collocation method in order to obtain the neutral curves. From the neutral curves it is observed that radial stretch has a globally stabilising effect on both the inviscid Type-I and the viscous Type-II instability modes. This underlying physical phenomena has been verified by performing an energy analysis of the flow. The results obtained excellently supports the previous works and will be prominently treated as a benchmark for our future studies.

Instability thresholds for thermal convection in a Kelvin–Voigt fluid of variable order

We present numerical techniques for calculating instability thresholds in a model for thermal convection in a complex viscoelastic fluid of Kelvin–Voigt type. The theory presented is valid for various orders of an exponential fading memory term, and the strategy for obtaining the neutral curves and instability thresholds is discussed in the general case. Specific numerical results are presented for a fluid of order zero, also known as a Navier–Stokes–Voigt fluid, and fluids of order 1 and 2. For the latter cases it is shown that oscillatory convection may occur, and the nature of the stationary and oscillatory convection branches is investigated in detail, including where the transition from one to the other takes place.

About force of interaction of a point charge with a dielectric ball

The decision of a problem on force of interaction of a point charge with a dielectric ball which is represented in the form of the infinite sum is resulted. It is shown that in a case when dielectric permeability of a ball aspires in infinity, force of interaction aspires to force of interaction of a point charge with the conducting isolated ball which contains only one composed. n the basis of the received results the simple approached formula for calculation of force of interaction of a point charge and the dielectric ball, containing too only one composed is offered. Using the received formula, the problem about force of interaction of a point charge and the same charged dielectric ball is solved. For various values of parameters the neutral curves dividing scopes of forces of pushing away and an attraction are constructed.

Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow

The thermal convection of a Jeffreys fluid subjected to a plane Poiseuille flow in a fluid-porous system composed of a fluid layer and a porous layer is studied in the paper. A linear stability analysis and a Chebyshev τ-QZ algorithm are employed to solve the thermal mixed convection. Unlike the case in a single layer, the neutral curves of the two-layer system may be bi-modal in the proper depth ratio of the two layers. We find that the longitudinal rolls (LRs) only depend on the depth ratio. With the existence of the shear flow, the effects of the depth ratio, the Reynolds number, the Prandtl number, the stress relaxation, and strain retardation times on the transverse rolls (TRs) are also studied. Additionally, the thermal instability of the viscoelastic fluid is found to be more unstable than that of the Newtonian fluid in a two-layer system. In contrast to the case for Newtonian fluids, the TRs rather than the LRs may be the preferred mode for the viscoelastic fluids in some cases.

On Approximate Aesthetic Curves

Symmetry plays an essential role for generating aesthetic forms. The curve is the basic element used by designers to obtain aesthetic forms. A curve with a linear logarithmic curvature graph gradient is called aesthetic curve. The aesthetic value of a curve increases when its gradient is close to a straight line. We introduce the notions of approximate aesthetic curves and approximate neutral curves and obtain estimations between the curvature of an approximate aesthetic/neutral curve and an aesthetic curve.

On Langmuir circulation in 1 : 2 and 1 : 3 resonance

This paper is concerned with the nonlinear dynamics of spanwise periodic longitudinal vortex modes (Langmuir circulation (LC)) that arise through the instability of two-dimensional periodic flows (waves) in a non-stratified uniformly sheared layer of finite depth. Of particular interest is the excitation of the vortex modes either in the absence of interaction or in resonance, as described by nonlinear amplitude equations built upon the mean field Craik–Leibovich (CL) equations. Since Y-junctions in the surface footprints of Langmuir circulation indicate sporadic increases (doubling) in spacing as they evolve to the scale of sports stadiums, interest is focused on bifurcations that instigate such changes. To that end, surface patterns arising from the linear and nonlinear excitation of the vortex modes are explored, subject to two parameters: a Rayleigh number ${\mathcal{R}}$ present in the CL equations and a symmetry breaking parameter $\unicode[STIX]{x1D6FE}$ in the mixed free surface boundary conditions that relax to those at the layer bottom where $\unicode[STIX]{x1D6FE}=0$. Looking first to linear instability, it is found as $\unicode[STIX]{x1D6FE}$ increases from zero to unity, that the neutral curves evolve from asymmetric near onset to almost symmetric. The nonlinear dynamics of single modes is then studied via an amplitude equation of Ginzburg–Landau type. While typically of cubic order when the bifurcation is supercritical (as it is here) and the neutral curves are parabolic, the Ginzburg–Landau equation must instead here be of quartic order to recover the asymmetry in the neutral curves. This equation is then subjected to an Eckhaus instability analysis, which indicates that linearly unstable subharmonics mostly reside outside the Eckhaus boundary, thereby excluding them as candidates for excitation. The surface pattern is then largely unchanged from its linear counterpart, although the character of the pattern does change when $\unicode[STIX]{x1D6FE}\ll 1$ as a result of symmetry breaking. Attention is then turned to strong resonance between the least stable linear mode and a sub-harmonic of it, as described by coupled nonlinear amplitude equations of Stuart-Landau type. Both 1 : 2 and 1 : 3 resonant interactions are considered. Phase plots and bifurcation diagrams are employed to reveal classes of solution that can occur. Dominant over much of the ${\mathcal{R}}$-$\unicode[STIX]{x1D6FE}$ range considered are non-travelling pure- and mixed-mode equilibrium solutions that act singly or together. To wit, pure modes solutions alone act to realise windrows with spacings in accord with linear theory, while bistability can realise Y-junctions and, depending upon initial conditions, double or even triple the dominant spacing of LC.

Fractal sets of neutral curves for stably stratified plane Couette flow

The linear stability of plane Couette flow is investigated when the plates are horizontal, and the fluid is stably stratified with a cubic basic density profile. The disturbances are treated as inviscid and diffusion of the density field is neglected. Previous studies have shown that this density profile can develop multiple neutral curves, despite the stable stratification, and the fact that plane Couette flow of homogeneous fluid is stable. It is shown that when the neutral curves are plotted with wave angle on one axis, and location of the density inflexion point on the other axis, they produce a self-similar fractal pattern. The repetition on smaller and smaller scales occurs in the limit when the waves are highly oblique, i.e. longitudinal vortices almost aligned with the flow; the corresponding limit for two-dimensional waves is that of strong buoyancy/weak flow. The fractal set of neutral curves also represents a fractal of bifurcation points at which nonlinear solutions can be continued from the trivial state, and these may be helpful for understanding turbulent states. This may be the first example of a fractal generated by a linear ordinary differential equation.

Neutral stability curves of low-frequency Görtler flow generated by free-stream vortical disturbances

The neutral curves of the boundary layer Görtler-vortex flow generated by free-stream disturbances, i.e., curves that distinguish the perturbation flow conditions of growth and decay, are computed through a receptivity study for different Görtler numbers, wavelengths, and low frequencies of the free-stream disturbance. The perturbations are defined as Klebanoff modes or strong and weak Görtler vortices, depending on their growth rate. The critical Görtler number below which the inviscid instability due to the curvature never occurs is obtained and the conditions for which only Klebanoff modes exist are thus revealed. A streamwise-dependent receptivity coefficient is defined and we discuss the impact of the receptivity on the $N$-factor approach for transition prediction.