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We assert that the solutions to the Cauchy problem of the inviscid vorticity equation remain regular and unique for any smooth initial data of finite energy. However, the primitive formulation of the Euler equations is not well-posed, due to the passive pressure. One of the implications is that the anomalous energy dissipation, anticipated by Onsager (1949), cannot occur in inviscid flows. In the complete absence of viscous effects, the ultimate accumulation of enstrophy in sealed domains is bound to become arbitrarily excessive, if there is a sustained supply of shears and strains., Comment: 7 pages

The Navier-Stokes equations in the primitive formulation for incompressible flow describe the evolution of velocity and pressure, without recourse to vorticity. We show that, beyond the finite Leray-Hopf regularity interval, every postulated strong solution is accompanied by infinitely many diffusion-dominated percolations of arbitrary size, while the momentum deficit caused by the non-linearity is compensated by the pressure gradient. In the upper half space, we demonstrate how sequences of these collective companions can be re-scaled into an absurd singularity. Owning to the passive nature of the pressure, there exist no essential a priori bounds for establishing the uniqueness of primitive solutions. With the illustration of well-exploited examples of closed-form basic flows, we elucidate the reason why perturbations, infinitesimal or finite, instigate indeterminate states that render the concept of flow instability inadmissible. An effort has also been made to reappraise a number of important issues in fluid dynamics. Unfortunately, the primitive theory cannot serve as a reliable tool for prediction. Nevertheless, a dedicated effort has been made to elaborate a priori bounds for vorticity dynamics for ideal as well as real fluids. As a result, we are able to establish long-time regularity of the Cauchy problem for incompressible flows. A main conclusion is that no events of finite-time blow-up can ever occur in the Euler or Navier-Stokes equations for initial data of finite energy., Comment: 49 pages; 7 references; 2 new sections (7 and 8); updated section 1; some minor modifications

In the present note, we show that, as a priori bounds, the vorticity dynamics derived from Leray's backward self-similarity hypothesis admits only trivial solution in viscous as well as inviscid flows. By analogy, there is no non-zero solution in the forward self-similar equation. Since the Navier-Stokes or Euler equations are invariant under space translation in the whole space, our analysis establishes that technically flawed arguments have been exploited in a number of past papers, notably in Necas, Ruzicka & Sverak (1996); Tsai (1998); and Pomeau (2016), where the presumed decays or bounds at infinity are ill-defined and non-existent. Furthermore, an effort has been made to exemplify an inappropriate application of the familiar extremum principles in the theory of linear elliptic equation. In the appendix, we give a counterexample to the Sobolev inequality and, hence illustrate the nature of self contradiction. In the totality comparison of Lp norms, its scope of application is not significant., Comment: 14 pages; 5 more references; add one appendix

We examine the blow-up claims of the incompressible Euler equations for several specific flow-fields, (1) the columnar eddies in the vicinity of stagnation; (2) a quasi-three-dimensional structure for illustrating oscillations and concentrations in shears; (3) a 2-and-1/2D flow. We assert that these claimed finite-time singularities are not genuine. Over the whole space, the potential or the velocity coincides with singularities of harmonic functions. We show that the existence of the potential for unique velocity is necessary but insufficient, as the velocity fields are merely specified up to multiples of the curl of a vector potential, which satisfies a system of degenerated Laplace's equations. We have derived several closed-form formulas in simply-connected domains, as well as orthogonal curvilinear co-ordinates. It follows that steady solutions of Euler's equations are indeterminate. Hence, use of singular potential flow in expounding fluid motions is inept. The formulation of the primitive variables (velocity and pressure) gives rise to non-unique flow fields. In essence, the notion of a finite-time singularity is an ill-defined proposition. In Appendix A, we show that the well-posedness claim (Yudovich, 1995) of planar vorticity in Lebesgue's class has been made on the basis of flawed technicalities. Certain inequalities of functional analysis play key roles in the regularity study. In Appendix B, we assert that their derivations have not been rigorously formulated. Concretely, applications of Sobolev's inequality (as well as Ladyzhenskaya's inequalities) to non-trivial solenoidal velocity fields are unattainable. There have been claims of singular flow solutions in the technical literature. However, a large majority of the deductions rely on these imprecise functional inequalities and hence suffer from logical inconsistency., Comment: 36 pages, 1 figure, 2 appendices

In the present technical note, we establish that the setting of the primitive variables of the unsteady incompressible fluid dynamics is ill-formulated in spatially periodic domains as the specification of the boundary velocity is too broad to sidestep time-dependency and approximation errors. As an illustration, we show that the Taylor-Green solution in planes suffers from the Hadamard-divergence, and the ABC flow in cubes is non-unique. In direct numerical simulations of homogeneous turbulence with no corrective precautions on the boundary values, our assertion helps us understand the well-experienced nuisances, such as slow rates of convergence in energy dissipation, fluctuations in the statistics moments, or spontaneous surges in the time-averaged flow quantities. In particular, vorticity dynamics is not described by singular integral equations., Comment: Add 2 more references, updated space-time spurts, anomalous dissipation, Fourier analysis. Total 11 pages

The initial-boundary value problem of the vorticity equation has been solved numerically by an iterative method. A variety of initial vorticity distributions is specified. All of them can be described by simple mathematical functions: there are a vorticity source-sink pair, circular shears out of a localised monopole, a cat's-eye topology and a few flows originating from single or multiple vortices. Our computational results show diverse flow phenomena, such as roll-ups of shear layers, vortex merging or impingement, as well as birth of spiral structures., Comment: 13 pages, 21 plots

In the present note, we solved numerically the viscous vorticity equation of the initial-boundary value problem describing the classic Helmholtz phenomena of vortex interaction. In the leapfrogging of vortex pairs, we demonstrate the fact that there exists a variety of initial vortex configurations, such as the initial vortex core structures, the starting speeds, the lateral and longitudinal separations as well as the fluid viscosity. To simulate leapfrogging appears to be a straightforward task, based on the vorticity calculations for a number of initial vortices with different core structures. Indeed, the evaluation of the unsteady vortex interactions requires accurate numerical simulations, and the solutions are diverse and intriguing. In particular, the impact of two asymmetric approaching vortices can produce peeled-off fission vortices or cross-bred eddies of mushroom topology. Over the course of flow development, we are unable to define any unique consistent Reynolds number which may be used to classify the individual flows, because of the multiplicity of characteristic lengths and velocity scales. Our key observation is that the initial-boundary value problems of fluid motion do not necessarily imply the realization of dynamically similar flows., Comment: 17 pages, 26 plots

In this technical note, we demonstrate the robustness of our numerical scheme of vorticity iteration in dealing with the dipole-wall interaction at small viscosity, with emphasis on mesh convergence, boundary vorticity as well as wall viscous dissipation. In particular, it is found that, among the four different dipole configurations, the processes of vortex-wall collision at no-slip surfaces are exceedingly complex and are functions of the initial conditions. The critical issue direct numerical simulations is to establish mesh convergence which appears to be case-dependent. Roughly speaking, converged $2D$ meshes are found to be inversely proportional to viscosity at uniform spacings. Essentially, we have ruled out the possibility of anomalous energy dissipation in the limit of small viscosity. Our computational results show that the rate of the energy degradation follows the predictive trend of the well-known Prandtl scaling., Comment: 17 pages. 20 figures. A reference added and quoted. Several updates with rewording

In the present paper, we examine the viscous flow evolution in a square cavity. Coupled with the stream function, the initial-boundary value problem of the vorticity is numerically solved by a method of iteration. The only boundary condition is the wall velocity which in turn defines the Dirichlet value for the stream function. We assert that the corner singularity in the cavity flow is in fact a theoretical artefact. By adopting suitably regulated lid velocity, excellent comparison with experiments is found because of the converged palinstrophy field. Our calculations demonstrate that asymmetrical shears initiate the formation of the vortices with counter-rotating strained cores, while the consecutive re-birth and the subsequent disintegration of the developed eddies proliferate the shears into smaller scales. This production process is particularly intense in close proximity to the solid surfaces., Comment: 20 pages, 13 plots

In the theory of hydrodynamic stability, the procedure to decompose an incompressible flow field into its basic motion and disturbances is imprecise and problematic because the disturbances, infinitesimal or finite, are ill-defined quantities in analysis. The linearised equations contravene the first principles of classical mechanics while the disturbance-driven non-linear formulation can hardly be considered as exact science. The notion that unstable and amplified disturbances precipitating the early stages of laminar-turbulent transition is vague, speculative or fundamentally flawed. Similarly, turbulence is unjustifiably assumed to involve a statistical mean, and zero-on-average fluctuations. This simplistic postulation is unpromising, and inevitably renders turbulent flows to a recondite dynamics of self-contradiction. By consequence, the closure problem of turbulence modelling the Reynolds stresses deals with issues of no physical relevance., Comment: 11 pages. Correction of figure 1. On p.9, line 5, 'the full Navier-Stokes ...', instead of `the Navier-Stokes ...'

We demonstrate that numerical solutions of Burgers' equation can be obtained by a scale-totality algorithm for fluids of small viscosity (down to one billionth). Two sets of initial data, modelling simple shears and wall boundary layers, are chosen for our computations. Most of the solutions are carried out well into the fully turbulent regime over finely-resolved scales in space and in time. It is found that an abrupt spatio-temporal concentration in shear constitutes an essential part during the flow evolution. The vorticity surge has been instigated by the non-linearity complying with instantaneous enstrophy production, while ad hoc disturbances play no role in the process. In particular, the present method predicts the precipitous vorticity re-distribution and accumulation, predominantly over localised regions of minute dimension. The growth rate depends on viscosity and is a strong function of initial data. Nevertheless, the long-time energy decay is history-independent and is inversely proportional to time. Our results provide direct evidence of the vorticity proliferation embedded in the equations of motion. The non-linear intensification is a robust feature, and is ultimately responsible for the drastic succession in boundary layer profiles over the intrinsic laminar-turbulent transition (Schubauer and Klebanoff 1955). The dynamical inception of turbulence can be decrypted by solving the full time-dependent Navier-Stokes equations which ascribe no instability stages., Comment: 32 pages and 17 figures (17 ps files). Corrections of typos with minor re-wording

We explain why the theory of Escauriaza, Seregin, and Sverak (Russian Math. Surveys, 2003) on potential finite time singularity in Navier-Stokes solutions must be largely misapprehended. It is found that the proofs of the backward uniqueness theorem for parabolic equations contain technical errors. The stated validity of a theorem for vorticity is established on ill-informed analyses as the solenoidal constraint is not taken into account. There are many cases where parabolic scalings are erroneously applied. We briefly discuss a number of related issues., Comment: 7 pages, no figures, add one more reference, correction of typos, and minor re-wording

In this paper, we show that the spatio-temporal evolution of incompressible flows in a long circular pipe can be described by vorticity dynamics. The principal techniques to obtain solutions are similar to those used for flows in the whole space. As the consideration of the Navier-Stokes equations is given in a cylindrical co-ordinates system, two aspects of complication arise. One is the interaction of the velocity components in the radial and azimuthal directions, due to the fictitious centrifugal force in the equations of motion. The rate of the vorticity production at the pipe wall depends on the initial data at entry, and hence is unknown a priori; it must be determined as part of the solution. The vorticity solution obtained defines an intricate flow-field of multitudinous degrees of freedom. As the Reynolds number increases, the analytical solution predicts vorticity-scale proliferations in succession. For sufficiently large initial data, pipe flows are of a turbulent nature. The solution of the governing equations is globally regular and does not bifurcate in space or in time. It is asserted that laminar-turbulent transition is a dynamic process inbred in the non-linearity. The presence of exogenous disturbances, due to imperfect test environments or purpose-made artificial forcing, distorts the course of the intrinsic transition. The flow structures observed by Reynolds (1883) and others can be synthesised and elucidated in light of the current theory., Comment: 41 pages and 7 figures (7 ps files). Correction of typos, and some rewording

In the present note, we show that the uni-directional flows in a rectangular channel and in a circular pipe are exact spatio-temporal solutions of the Navier-Stokes equations over a short time interval. We assert that the classical plane Poiseuille-Couette flow and Hagen-Poiseuille flow are time-independent approximations of the exact solutions if an appropriate initial velocity distribution at starting location is specified. Conceptually, there do not exist absolute steady flows starting from unspecified initial data. The classic experimental measurements by Poiseuille can be explained in terms of the evolutional solutions. In particular, the pipe flow does not have a time-independent characteristic velocity. The orthodox notion that the parabolic profile exists for arbitrary Reynolds numbers is unwarranted., Comment: 16 pages; 5 figures (postscript colour eps) Version 3: Included examples of numerical computations showing the effect of entry velocity profiles, updated reference list and typo corrections

In the present paper we prove that the pipe Poiseuille flow of parabolic velocity profile attenuates exponentially in time with respect to three dimensional infinitesimal disturbances at all finite wave numbers and Reynolds numbers for given azimuthal periodicity if the equations of motion are linearized. The spectra of the eigenvalue are shown to consist of infinitely many discrete eigen-modes. Results of asymptotic analysis, expressed in simple algebraic formulas and functional relations, are given. Good comparison has been found in the approximations and numerical computations. The present results are best interpreted as a description of the pipe flow regime, where the linear diffusion due to viscosity dominates., Comment: 27 pages, 2 B/W eps plots

It is shown that the Cauchy problem of the equations in magnetohydrodynamics in the whole space is globally well-posed for any initial smooth and localized data. In general, the mathematical structure of solution shows that the coupled magnetic-vortical field has the characters of turbulence, if the initial data exceed certain size. In particular, if a strong magnetic force dominates the flow evolution, the current density possesses a cubic non-linearity. The solution of the non-linear problem has been constructed and has been expressed as an infinite series., Comment: 23 pages, no plots. Revised version with corrections

An effort has been made to solve the Cauchy problem of the Navier-Stokes equations in the whole space by two methods. It is proved that the sum of the three vorticity components is a time-invariant in fluid motion. It has been proved that, given smooth, localized initial data with finite energy and enstrophy, the vorticity equation admits a global, unique and smooth solution. Second, the vorticity equation has been converted into a non-linear integral equation by means of similarity reduction. The solution of the integral equation has been constructed in a series expansion. The series is shown to converge for initial data of finite size. The complete vorticity field is characterized, as an instantaneous description, by a multitude of vorticity constituents. The flow field is composed of vortical elements of broad spatio-temporal scales. Inference of the solutions leads itself to a satisfactory account for the observed dynamic characteristics of transition process, and of turbulent motion. In the limit of vanishing viscosity, the equations of motion cannot develop flow-field singularities in finite time. In the Maxwell-Boltzmann kinetic theory, the density function of the Maxwellian molecules possesses a phase-space distribution resembling the continuum turbulence. Qualitatively, the apparent macroscopic randomness of turbulence can be attributed to a ramification of molecular fluctuations., Comment: 139 pages. 6 BW figures (7 ps files). 13 sections. 4 appendices. In S3, the Jacobian of velocity (and pressure) is shown to be essentially bounded in space (bounded by the initial vorticity), updated application of derived apriori bounds. Typos in the pressure kernel cleared up in S5. Typos and minor re-wording in S7. Missing minus sign in inverting Poisson's eqns

The normal-mode analysis of the Reynolds-Orr energy equation governing the stability of viscous motion for general three-dimensional disturbances has been revisited. The energy equation has been solved as an unconstrained minimization problem for the Couette-Poiseuille flow. The minimum Reynolds number for every Couette-Poiseuille velocity profile has been computed and compared with those available in the literature. For fully three-dimensional disturbances, it is shown that the minimum Reynolds number is in general smaller than the corresponding two-dimensional counterpart for all the Couette-Poiseuille profiles except plane Couette flow., Comment: 11 pages, 6 Postscript figures

The English compositions of two groups of Hong Kong secondary school students learning English as a Second Language were compared. Results found that students who used computer word processors, on the whole, wrote better compositions than students who used pen and paper. (46 references) (MDM)