Taylor Problem Research Articles

The Saffman–Taylor problem and several sets of remarkable integral identities

The Saffman–Taylor problem and several sets of remarkable integral identities

Unified framework of forced magnetic reconnection and Alfvén resonance

A unified linear theory that includes forced reconnection as a particular case of Alfvén resonance is presented. We consider a generalized Taylor problem in which a sheared magnetic field is subject to a time-dependent boundary perturbation oscillating at frequency ω0. By analyzing the asymptotic time response of the system, the theory demonstrates that the Alfvén resonance is due to the residues at the resonant poles, in the complex frequency plane, introduced by the boundary perturbation. Alfvén resonance transitions towards forced reconnection, described by the constant-psi regime for (normalized) times t≫S1/3, when the forcing frequency of the boundary perturbation is ω0≪S−1/3, allowing the coupling of the Alfvén resonances across the neutral line with the reconnecting mode, as originally suggested in Uberoi and Zweibel, (1999). Additionally, it is shown that even if forced reconnection develops for finite, albeit small, frequencies, the reconnection rate and reconnected flux are strongly reduced for frequencies ω0≫S−3/5.

On the dynamic instability of non‐Newtonian fluids driven by gravity

In this paper, we investigate the instability for Rayleigh–Taylor problem of three‐dimensional incompressible non‐Newtonian fluids with Eills‐type. If the density profile in equilibrium‐state is heavier with increasing height, we prove that the equilibrium‐state solution is unstable under ‐norm by exploiting a bootstrap method.

On Rayleigh–Taylor instability in Navier–Stokes–Korteweg equations

This paper focuses on the Rayleigh–Taylor instability in the two-dimensional system of equations of nonhomogeneous incompressible viscous fluids with capillarity effects in a horizontal periodic domain with infinite height. First, we use the modified variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh–Taylor problem. Then, motivated by the Grenier’s idea in (Grenier in Commun. Pure Appl. Math. 53(9):1067–1091, 2000), we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh–Taylor problem and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in (Hwang and Guo in Arch. Ration. Mech. Anal. 167(3):235–253, 2003), and thus obtain the nonlinear Rayleigh–Taylor instability result. Our instability result presents that the Rayleigh–Taylor instability can occur in the fluids with capillarity effects for any capillary coefficient kappa >0 if the critical capillary coefficient is infinite. In particular, it improves the previous Zhang’s result in (Zhang in J. Math. Fluid Mech. 24(3):70–23, 2022) with the assumption of smallness of the capillary coefficient.

Stabilizing Effect of Capillarity in the Rayleigh–Taylor Problem to the Viscous Incompressible Capillary Fluids

Stabilizing Effect of Capillarity in the Rayleigh–Taylor Problem to the Viscous Incompressible Capillary Fluids

Structure of pressure-gradient-driven current singularity in ideal magnetohydrodynamic equilibrium

Singular currents typically appear on rational surfaces in non-axisymmetric ideal magnetohydrodynamic (MHD) equilibria with a continuum of nested flux surfaces and a continuous rotational transition. These currents have two components: a surface current (Dirac δ-function in flux surface labeling) that prevents the formation of magnetic islands, and an algebraically divergent Pfirsch–Schlüter current density when a pressure gradient is present across the rational surface. On flux surfaces adjacent to the rational surface, the traditional treatment gives the Pfirsch–Schlüter current density scaling as , where is the difference of the rotational transform relative to the rational surface. If the distance s between flux surfaces is proportional to , the scaling relation will lead to a paradox that the Pfirsch–Schlüter current is not integrable. In this work, we investigate this issue by considering the pressure-gradient-driven singular current in the Hahm–Kulsrud–Taylor problem, which is a prototype for singular currents arising from resonant magnetic perturbations. We show that not only the Pfirsch–Schlüter current density but also the diamagnetic current density are divergent as . However, due to the formation of a Dirac δ-function current sheet at the rational surface, the neighboring flux surfaces are strongly packed with . Consequently, the singular current density , making the total current finite, thus resolving the paradox. Furthermore, the strong packing of flux surfaces causes a steepening of the pressure gradient near the rational surface, with . In general non-axisymmetric MHD equilibrium, contrary to Grad’s conjecture that the pressure profile is flat around densely distributed rational surfaces, our result suggests a pressure profile that densely steepens around them.

On the dynamic Rayleigh–Taylor instability in the Euler–Korteweg model

This paper focuses on the Rayleigh–Taylor instability in the system of equations of the two-dimensional nonhomogeneous incompressible Euler–Korteweg equations in a horizontal periodic domain with infinite height. First, we use variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh–Taylor problem. Then, motivated by the Grenier's idea in [21], we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh–Taylor problem due to the absence of viscosity in the system, and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in [28], and thus obtain the nonlinear Rayleigh–Taylor instability result, which presents that the Rayleigh–Taylor instability can occur in the capillary fluids for any capillary coefficient κ>0 if the critical capillary number is infinite.

On Rayleigh–Taylor instability in nonhomogeneous incompressible elasticity fluids

This paper focuses on the Rayleigh–Taylor instability in the system of equations of the two-dimensional nonhomogeneous incompressible elasticity fluid in a horizontal periodic domain with infinite height. First, we use variational method to construct (linear) unstable solutions for the linearized elastic Rayleigh–Taylor problem. Then, motivated by the Grenier's idea in [10], we further construct approximate solutions with higher-order growing modes to the elastic Rayleigh–Taylor problem due to the absence of viscosity in the system, and derive the error estimates between both the approximate solutions and nonlinear solutions of the elastic Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in [25], and thus obtain the nonlinear Rayleigh–Taylor instability result, which presents that the Rayleigh–Taylor instability can occur in elasticity fluids with small elasticity coefficient.

Linear instability of viscoelastic interfacial Hele-Shaw flows: A Newtonian fluid displacing an UCM fluid

We theoretically study the linear stability of the Saffman–Taylor problem where a viscous Newtonian fluid (with viscosity η l ) displaces an Upper Convected Maxwell (UCM) fluid (with viscosity η r ) in a rectilinear Hele-Shaw cell. The dispersion relation is given by the roots of a cubic polynomial with coefficients depending on wavenumber along with several dimensionless groups as parameters. Using Routh–Hurwitz stability criterion, we show that the viscosity ratio η r / η l still plays a decisive role in determining stability (stable if η r / η l ≤ 1 ). If η r / η l > 1 , the flow is more unstable than an identical Newtonian–Newtonian setup and the most unstable wavenumber is larger. Increasing Deborah number, capillary number or flow speed worsens the instability. Elasticity has a variety of effects and can give rise up to three types of singular behaviors: (i) there exists infinitely many distinct wavenumbers at which the velocity becomes singular, (ii) stress becomes singular when the wavenumber exceeds a certain value; and (iii) a resonance phenomenon occurs when η r / η l is large, where the growth rate increases very rapidly near certain wavenumber and eventually becomes singular when the displacing fluid is inviscid. • Explicit formulas for the dispersion relation. • Singular behaviors in the velocity, stress, and growth rate. • Viscosity contrast determines long wave stability. • Destabilizing effect if increasing flow speed, Deborah number or Capillary number. • More unstable than Newtonian case with shorter most unstable wave.

Numerical study of δ-function current sheets arising from resonant magnetic perturbations

General three-dimensional toroidal ideal magnetohydrodynamic equilibria with a continuum of nested flux surfaces are susceptible to forming singular current sheets when resonant perturbations are applied. The presence of singular current sheets indicates that, in the presence of non-zero resistivity, magnetic reconnection will ensue, leading to the formation of magnetic islands and potentially regions of stochastic field lines when islands overlap. Numerically resolving singular current sheets in the ideal magnetohydrodynamics (MHD) limit has been a significant challenge. This work presents numerical solutions of the Hahm–Kulsrud–Taylor (HKT) problem, which is a prototype for resonant singular current sheet formation. The HKT problem is solved by two codes: a Grad–Shafranov (GS) solver and the Stepped Pressure Equilibrium Code (SPEC) code. The GS solver has built-in nested flux surfaces with prescribed magnetic fluxes. The SPEC code implements multi-region relaxed magnetohydrodynamics (MRxMHD), whereby the solution relaxes to a Taylor state in each region while maintaining force balance across the interfaces between regions. As the number of regions increases, the MRxMHD solution appears to approach the ideal MHD solution assuming a continuum of nested flux surfaces. We demonstrate agreement between the numerical solutions obtained from the two codes through a convergence study.

Quasi-linear theory of forced magnetic reconnection for the transition from the linear to the Rutherford regime

Using the in-viscid two-field reduced magneto-hydrodynamic (MHD) model, a new analytical theory is developed to unify the Hahm–Kulsrud–Taylor (HKT) linear solution and the Rutherford quasi-linear regime. Adopting a quasi-linear approach, we obtain a closed system of equations for plasma response in the Taylor problem. An integral form of an analytical solution is obtained for the forced magnetic reconnection, uniformly valid throughout the entire regimes from the HKT linear solution to the Rutherford quasi-linear solution. In particular, the quasi-linear effect can be described by a single coefficient , where and ψ c are the Lundquist number and amplitude of external magnetic perturbation, respectively. The HKT linear solution for the response can be recovered when the index K s → 0. On the other hand, the quasi-linear effect plays a key role in the island growth when K s ∼ 1. Our new analytical solution has also been compared with reduced MHD simulations with agreement.

Instability of the abstract Rayleigh–Taylor problem and applications

Based on a bootstrap instability method, we prove the existence of unstable strong solutions in the sense of [Formula: see text]-norm to an abstract Rayleigh–Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates. In the proof we develop a method to modify the initial data of the linearized abstract RT problem by exploiting the existence theory of a unique solution to the stratified (steady) Stokes problem and an iterative technique, such that the obtained modified initial data satisfy the necessary compatibility conditions on boundary of the original (nonlinear) abstract RT problem. Applying an inverse transform of Lagrangian coordinates to the obtained unstable solutions and taking then proper values of the parameters, we can further obtain unstable solutions of the RT problem in viscoelastic, magnetohydrodynamics (MHD) flows with zero resistivity and pure viscous flows (with/without interface intension) in Eulerian coordinates.

Stability of the viscoelastic Rayleigh–Taylor problem with internal surface tension

We investigate the stability of the Rayleigh–Taylor (RT) problem for stratified viscoelastic fluids with internal surface tension. More precisely, under the stability condition Dis(ϑ,κ)<1, we prove the existence of unique strong solution with exponential decay in time for the (stratified) viscoelastic RT (VRT) problem with proper initial data in Lagrangian coordinates. This shows that a sufficiently large elasticity coefficient or a sufficiently large surface tension coefficient has a stabilizing effect so that it can inhibit the development of (stratified) RT instability.

Rigidity of MHD equilibria to smooth incompressible ideal motion near resonant surfaces

In ideal MHD, the magnetic flux is advected by the plasma motion, freezing flux-surfaces into the flow. An MHD equilibrium is reached when the flow relaxes and force balance is achieved. We ask what classes of MHD equilibria can be accessed from a given initial state via smooth incompressible ideal motion. It is found that certain boundary displacements are formally not supported. This follows from yet another investigation of the Hahm–Kulsrud–Taylor (HKT) problem, which highlights the resonant behaviour near a rational layer formed by a set of degenerate critical points in the flux-function. When trying to retain the mirror symmetry of the flux-function with respect to the resonant layer, the vector field that generates the volume-preserving diffeomorphism vanishes at the identity to all order in the time-like path parameter.

Instability solutions for the Rayleigh–Taylor problem of non-homogeneous viscoelastic fluids in bounded domains

It is well-known that the Rayleigh–Taylor (RT) problem of an inhomogeneous viscoelastic fluid defined on a bounded domain is unstable, if the elasticity coefficient κ is less than some threshold κC. In this paper, we rigorously prove the existence of a unique unstable strong solution in the sense of L1-norm for the RT problem in Lagrangian coordinates based on a bootstrap instability method, when κ<κC. Applying an inverse transformation of Lagrangian coordinates to the obtained unstable solution, we can further get a unique unstable solution for the RT problem of inhomogeneous viscoelastic fluids in Eulerian coordinates.

Simulation of dynamic channel angular pressing of copper samples using experimental data of loading

The paper represents the numerical simulation of severe plastic deformation of a copper sample under dynamic channel-angular pressing. The initial sample velocity and the value of pressure acting on the back surface of the sample during the process were determined based on the analysis of experimental data. Numerical computations were carried out using the author’s numerical code created in the integrated development environment Delphi. A modified finite element method (without constructing a global stiffness matrix) was used as a numerical method. The behavior of the material was described by an elastic-plastic model using the active type fracture model. Testing of the numerical code was carried out on the Taylor problem. The velocity and pressure values required for the successful dynamic channel-angular pressing were determined. A uniform distribution of the ultrafine-grained structure in almost the entire volume of the copper sample was revealed.

Magnetohydrodynamical equilibria with current singularities and continuous rotational transform

We revisit the Hahm–Kulsrud–Taylor (HKT) problem, a classic prototype problem for studying resonant magnetic perturbations and 3D magnetohydrodynamical equilibria. We employ the boundary-layer techniques developed by Rosenbluth, Dagazian, and Rutherford (RDR) for the internal m = 1 kink instability, while addressing the subtle difference in the matching procedure for the HKT problem. Pedagogically, the essence of RDR's approach becomes more transparent in the reduced slab geometry of the HKT problem. We then compare the boundary-layer solution, which yields a current singularity at the resonant surface, to the numerical solution obtained using a flux-preserving Grad–Shafranov solver. The remarkable agreement between the solutions demonstrates the validity and universality of RDR's approach. In addition, we show that RDR's approach consistently preserves the rotational transform, which hence stays continuous, contrary to a recent claim that RDR's solution contains a discontinuity in the rotational transform.

Nonlinear stability and instability in the Rayleigh–Taylor problem of stratified compressible MHD fluids

We establish the stability/instability criteria for the stratified compressible magnetic Rayleigh–Taylor (RT) problem in Lagrangian coordinates. More precisely, under the stability condition $$\varXi <1$$ (under such case, the third component of impressed magnetic field is not zero), we show the existence of a unique solution with an algebraic decay in time for the (compressible) magnetic RT problem with proper initial data. The stability result in particular shows that a sufficiently large (impressed) vertical magnetic field can inhibit the growth of the RT instability. On the other hand, if $$\varXi >1$$ , there exists an unstable solution to the magnetic RT problem in the Hadamard sense. This in particular shows that the RT instability still occurs when the strength of an magnetic field is small or the magnetic field is horizontal. Moreover, by analyzing the stability condition in the magnetic RT problem for vertical magnetic fields, we can observe that the compressibility destroys the stabilizing effect of magnetic fields in the vertical direction. Fortunately, the instability in the vertical direction can be inhibited by the stabilizing effect of the pressure, which also plays an important role in the proof of the stability of the magnetic RT problem. In addition, we extend the results for the magnetic RT problem to the compressible viscoelastic RT problem, and find that the stabilizing effect of elasticity is stronger than that of the magnetic fields.

Modelling of turbulence energy decay based on hybrid methods

PurposeThe purpose of this study is to present an exact and fast-calculated algorithm for the modelling of turbulent energy decay based on two different methods: finite-difference and spectral methods.Design/methodology/approachThe filtered three-dimensional non-stationary Navier–Stokes equation is used for simulating the turbulent process. The problem is solved using hybrid methods, where the equation of motion is solved using finite difference methods in combination with cyclic penta-diagonal matrix, which allowed to reach high order of accuracy, and Poisson equation is solved using the spectral methods, which is proposed to speed up the solution procedure. For validation of the given algorithm, the turbulent characteristics were compared with the exact solution of the classical Taylor and Green vortex problem, showing good agreement.FindingsThe proposed method shows high computational efficiency and good estimation quality. A numerical algorithm for solving non-stationary three-dimensional Navier–Stokes equations for modelling of isotropic turbulence decay using hybrid methods was developed. The simulation’s turbulence characteristics show good agreements with analytical solution. The developed numerical algorithm can be used for simulation of turbulence decay with different values of viscosity.Originality/valueAn efficient algorithm for simulation of turbulence processes depending on the properties of the viscosity was developed.

The initial development of a jet caused by fluid, body and free surface interaction with a uniformly accelerated advancing or retreating plate. Part 2. Well-posedness and stability of the principal flow

We consider the problem of a rigid plate, inclined at an angle $\unicode[STIX]{x1D6FC}\in (0,\unicode[STIX]{x03C0}/2)$ to the horizontal, accelerating uniformly from rest into, or away from, a semi-infinite strip of inviscid, incompressible fluid under gravity. Following on from Gallagher et al. (J. Fluid Mech., vol. 841, 2018, pp. 109–145) (henceforth referred to as GNB), it is of interest to analyse the well-posedness and stability of the principal flow with respect to perturbations in the initially horizontal free surface close to the plate contact point. In particular we find that the solution to the principal unperturbed problem, denoted by [IBVP] in GNB, is well-posed and stable with respect to perturbations in initial data in the region of interest, localised close to the contact point of the free surface and the plate, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}\geqslant -\cot \,\unicode[STIX]{x1D6FC}$, while the solution to [IBVP] is ill-posed with respect to such perturbations in the initial data, when the plate is accelerated with dimensionless acceleration $\unicode[STIX]{x1D70E}&lt;-\cot \,\unicode[STIX]{x1D6FC}$. The physical source of the ill-posedness of the principal problem [IBVP], when $\unicode[STIX]{x1D70E}&lt;-\cot \,\unicode[STIX]{x1D6FC}$, is revealed to be due to the leading-order problem in the innermost region localised close to the initial contact point being in the form of a local Rayleigh–Taylor problem. As a consequence of this mechanistic interpretation we anticipate that, when the plate is accelerated with $\unicode[STIX]{x1D70E}&lt;-\cot \,\unicode[STIX]{x1D6FC}$, the inclusion of weak surface tension effects will restore well-posedness of the problem [IBVP] which will, however, remain temporally unstable.