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Register a new userAnalysis of stationary points and their bifurcations in the ABC flow
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Analytical expressions for coordinates of stationary points and conditions for their existence in the ABC flow are received. The type of the stationary points is shown analytically to be saddle-node. Exact expressions for eigenvalues and eigenvectors of the stability matrix are given. Behavior of the stationary points along the bifurcation lines is described.
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In this paper we study resonances of the $ABC$-flow in the near integrable case ($Cll 1$). This is an interesting example of a Hamiltonian system with 3/2 degrees of freedom in which simultaneous existence of two resonances of the same order is possible. Analytical conditions of the resonance existence are received. It is shown numerically that the largest $n:1$ ($n=1,2,3$) resonances exist, and their energies are equal to theoretical energies in the near integrable case. We provide analytical and numerical evidences for existence of two branches of the two largest $n:1$ ($n=1,2$) resonances in the region of finite motion.
Viscoelastic flows through porous media become unstable and chaotic beyond critical flow conditions, impacting industrial and biological processes. Recently, Walkama textit{et al.} [Phys. Rev. Lett. textbf{124}, 164501 (2020)] have shown that geometric disorder greatly suppresses such chaotic dynamics. We demonstrate experimentally that geometric disorder textit{per se} is not the reason for this suppression, and that disorder can also promote choatic fluctuations, given a slightly modified initial condition. The results are explained by the effect of disorder on the occurrence of stagnation points exposed to the flow field, which depends on the initially ordered geometric configuration.
How anisotropic particles rotate and orient in a flow depends on the hydrodynamic torque they experience. Here we compute the torque acting on a small spheroid in a uniform flow by numerically solving the Navier-Stokes equations. Particle shape is varied from oblate (aspect ratio $lambda = 1/6$) to prolate ($lambda = 6$), and we consider low and moderate particle Reynolds numbers (${rm Re} le 50$). We demonstrate that the angular dependence of the torque, predicted theoretically for small particle Reynolds numbers remains qualitatively correct for Reynolds numbers up to ${rm Re} sim 10$. The amplitude of the torque, however, is smaller than the theoretical prediction, the more so as ${rm Re}$ increases. For Re larger than $10$, the flow past oblate spheroids acquires a more complicated structure, resulting in systematic deviations from the theoretical predictions. Overall, our numerical results provide a justification of recent theories for the orientation statistics of ice-crystals settling in a turbulent flow.
We report the results of a complete modal and nonmodal linear stability analysis of the electrohydrodynamic flow (EHD) for the problem of electroconvection in the strong injection region. Convective cells are formed by Coulomb force in an insulating liquid residing between two plane electrodes subject to unipolar injection. Besides pure electroconvection, we also consider the case where a cross-flow is present, generated by a streamwise pressure gradient, in the form of a laminar Poiseuille flow. The effect of charge diffusion, often neglected in previous linear stability analyses, is included in the present study and a transient growth analysis, rarely considered in EHD, is carried out. In the case without cross-flow, a non-zero charge diffusion leads to a lower linear stability threshold and thus to a more unstable low. The transient growth, though enhanced by increasing charge diffusion, remains small and hence cannot fully account for the discrepancy of the linear stability threshold between theoretical and experimental results. When a cross-flow is present, increasing the strength of the electric field in the high-$Re$ Poiseuille flow yields a more unstable flow in both modal and nonmodal stability analyses. Even though the energy analysis and the input-output analysis both indicate that the energy growth directly related to the electric field is small, the electric effect enhances the lift-up mechanism. The symmetry of channel flow with respect to the centerline is broken due to the additional electric field acting in the wall-normal direction. As a result, the centers of the streamwise rolls are shifted towards the injector electrode, and the optimal spanwise wavenumber achieving maximum transient energy growth increases with the strength of the electric field.
Since their development in 2001, regularised stokeslets have become a popular numerical tool for low-Reynolds number flows since the replacement of a point force by a smoothed blob overcomes many computational difficulties associated with flow singularities (Cortez, 2001, textit{SIAM J. Sci. Comput.} textbf{23}, 1204). The physical changes to the flow resulting from this process are, however, unclear. In this paper, we analyse the flow induced by general regularised stokeslets. An explicit formula for the flow from any regularised stokeslet is first derived, which is shown to simplify for spherically symmetric blobs. Far from the centre of any regularised stokeslet we show that the flow can be written in terms of an infinite number of singularity solutions provided the blob decays sufficiently rapidly. This infinite number of singularities reduces to a point force and source dipole for spherically symmetric blobs. Slowly-decaying blobs induce additional flow resulting from the non-zero body forces acting on the fluid. We also show that near the centre of spherically symmetric regularised stokeslets the flow becomes isotropic, which contrasts with the flow anisotropy fundamental to viscous systems. The concepts developed are used to { identify blobs that reduce regularisation errors. These blobs contain regions of negative force in order to counter the flows produced in the regularisation process, but still retain a form convenient for computations.
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