Viscous Case Research Articles

Behaviour of an isolated rimmed elliptical inclusion in 2D slow incompressible viscous flow

For 2D linear viscous flow, it is shown that the rates of rotation and stretch of an isolated elliptical inclusion with a coaxial elliptical rim are fully determined by two corresponding scalar values. For power-law viscosity, effective viscosity ratios of the inclusion and rim to the matrix depend on orientation and the system is more complex but, in practice, the simplification with two scalar values still provides a good approximation. Finite-element modelling (FEM) is used to determine the two characteristic values across a wide parameter space for the linear viscous case, with a viscosity ratio (relative to the matrix) of the inclusion from 106 to 1, of the rim from 10−6 to 1, axial ratios from 1.00025 to 20, and rim thicknesses relative to the inclusion axes of 5 to 20%. Results are presented in a multi-dimensional data table, allowing continuous interpolation over the investigated parameter range. Based on these instantaneous rates, the shape fabric of a population of inclusions is forward modelled using an initial value Ordinary Differential Equation (ODE) approach, with the simplifying but unrealistic assumption that the rim remains elliptical in shape and coaxial with respect to the inclusion. However, comparison with accurate large strain numerical experiments demonstrates that this simplified model gives qualitatively robust predictions and, for a range of investigated examples, also remarkably good quantitative estimates for shear strains up to at least γ = 5. A statistical approach, allowing random variation in the initial orientation, axial ratio and rim viscosity, can reproduce the characteristic shape preferred orientation (SPO) of natural porphyroclast populations. However, vorticity analysis based on the SPO or the interpreted stable orientation of inclusions is not practical. Varying parameters, such as inclusion and rim viscosity, rim thickness, and power law-exponents for non-linear viscosity, can reproduce the range of naturally observed behaviour (e.g., back-rotation, effectively stable orientations at back-rotated angles, a cut-off axial ratio separating rotating from stable inclusions) even for constant simple shear and these features are not uniquely characteristic of the vorticity of the background flow.

Crawling motility through the analysis of model locomotors: Two case studies

We study model locomotors on a substrate, which derive their propulsive capabilities from the tangential (viscous or frictional) resistance offered by the substrate. Our aim is to develop new tools and insight for future studies of cellular motility by crawling and of collective bacterial motion. The purely viscous case (worm) is relevant for cellular motility by crawling of individual cells. We re-examine some recent results on snail locomotion in order to assess the role of finely regulated adhesion mechanisms in crawling motility. Our main conclusion is that such regulation, although well documented in several biological systems, is not indispensable to accomplish locomotion driven by internal deformations, provided that the crawler may execute sufficiently large body deformations. Thus, there is no snail theorem. Namely, the crawling analog of the scallop theorem of low Reynolds number hydrodynamics does not hold for snail-like crawlers. The frictional case is obtained by assuming that the viscous coefficient governing tangential resistance forces, which act parallel and in the direction opposite to the velocity of the point to which they are applied, depends on the normal force acting at that point. We combine these surface interactions with inertial effects in order to investigate the mechanisms governing the motility of a bristle-robot. This model locomotor is easily manufactured and has been proposed as an effective tool to replicate and study collective bacterial motility.

Rayleigh–Taylor instabilities in axi-symmetric fluid outflow from a point source

This paper studies outflow of a light fluid from a point source, starting from an initially spherical bubble. This region of light fluid is embedded in a heavy fluid, from which it is separated by a thin interface. A gravitational force directed radially inward toward the mass source is permitted. Because the light inner fluid is pushing the heavy outer fluid, the interface between them may be unstable to small perturbations, in the Rayleigh– Taylor sense. An inviscid model of this two-layer flow is presented, and a linearized solution is developed for early times. It is argued that the inviscid solution develops a point of infinite curvature at the interface within finite time, after which the solution fails to exist. A Boussinesq viscous model is then presented as a means of quantifying the precise effects of viscosity. The interface is represented as a narrow region of large density gradient. The viscous results agree well with the inviscid theory at early times, but the curvature singularity of the inviscid theory is instead replaced by jet formation in the viscous case. This may be of relevance to underwater explosions and stellar evolution. doi:10.1017/S1446181112000090

A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity

We consider a one dimensional transport model with nonlocal velocity given by the Hilbert transform and develop a global well-posedness theory of probability measure solutions. Both the viscous and non-viscous cases are analyzed. Both in original and in self-similar variables, we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction potential. Existence, uniqueness, self-similar asymptotic behavior and inviscid limit of solutions are obtained in the space P2(R) of probability measures with finite second moments, without any smallness condition. Our results are based on the abstract gradient flow theory developed by Ambrosio et al. (2005) [2]. An important byproduct of our results is that there is a unique, up to invariance and translations, global in time self-similar solution with initial data in P2(R), which was already obtained by Deslippe et al. (2004) [17] and Biler et al. (2010) [6] by different methods. Moreover, this self-similar solution attracts all the dynamics in self-similar variables. The crucial monotonicity property of the transport between measures in one dimension allows to show that the singular logarithmic potential energy is displacement convex. We also extend the results to gradient flow equations with negative power-law locally integrable interaction potentials.

Extended scaling invariance of one‐dimensional models of liquid dynamics in a horizontal capillary

In this paper, we consider the adaptive numerical solution of one‐dimensional models of liquid dynamics in a horizontal capillary. The bulk liquid is assumed to be initially at rest and is put into motion by capillarity: the smaller is the capillary radius, the steeper becomes the initial transitory of the meniscus location derivative, and as a consequence, the numerical solution to a prescribed accuracy becomes harder to achieve. Therefore, in order to solve a capillary problem effectively, it would be advisable to apply an adaptive numerical method.Here, we show how an extended scaling invariance that can be used to define a family of solutions from a computed one. In the viscous case, the similarity transformation involves solutions of liquids with different density, surface tension, viscosity, and capillary radii, whereas in the inviscid case, we can generate a family of solutions for the same liquid and capillaries with different radii. With our study, we are able to prove that the monitor function, used in the adaptive algorithm, is invariant with respect to the considered scaling group. It follows, from this particular results, that all the solutions within the generated family verify the adaptive criteria used for the computed one. Moreover, all the solutions have the same order of accuracy even if the maximum value of the step size varies under the action of the scaling group. Copyright © 2012 John Wiley & Sons, Ltd.

Invariant Gibbs measures of the energy for shell models of turbulence: the inviscid and viscous cases

Gaussian measures of Gibbsian type are associated with some shell model of turbulence; they are constructed by means of the energy, a conserved quantity for the 3D inviscid and unforced shell model. We prove the existence of a unique global flow for a stochastic viscous shell model with the property that these Gibbs measures are invariant for this flow. Moreover, we prove that the deterministic inviscid shell model has a stationary solution with respect to these measures.

On the efficient application of weighted essentially nonoscillatory scheme

SUMMARYIn this paper, the efficient application of high‐order weighted essentially nonoscillatory (WENO) reconstruction to the subsonic and transonic engineering problems is studied. On the basis of the physical considerations, two techniques are proposed to enhance the accuracy and efficiency of the WENO reconstruction. First, it is observed that the WENO scheme using characteristic variable has better accuracy and convergence speed than the scheme using primitive variable. For engineering problems with shock of moderate amplitude, on the basis of the Rankine–Hugoniot conditions, a simplified characteristic‐variable‐based WENO is developed. The simplified version significantly reduces the cost overhead without sacrificing the shock‐capturing capability. Second, in this work, it is found for viscous case that it is better to include the viscous effect. On the basis of a simple analysis, the viscous correction to the parameter ε in the WENO reconstruction is proposed. Numerical results indicate, with the proposed simplified characteristic‐variable‐based reconstruction and the viscous correction, that the nonlinear WENO interpolation is sharply activated in the region of shock jump, whereas in the shockless area, the WENO interpolation weights are tuned towards the designed optimal value for better accuracy. Compared with the original characteristic‐variable‐based WENO, the current implementation has similar accuracy and reduced cost. At the same time, compared with the primitive variable‐based WENO, better accuracy and convergence speed are obtained at marginal cost overhead. Several practical cases are calculated to demonstrate the accuracy and efficiency of the current methodology. Copyright © 2012 John Wiley & Sons, Ltd.

Pore scale analysis of the impact of mixing-induced reaction dependent viscosity variations

Expanding interest in enhanced subsurface natural resource recovery and carbon sequestration motivates study of reacting flows in porous media. In this work, we examine the case of reaction products that increase or decrease the viscosity of the fluid. Parallel reactant streams flow through porous media and react transversely along the centerline. We utilize a pore scale, finite element numerical method that couples the reaction with fluid flow through two arrangements of porous media at three Damkohler (Da) numbers and two viscosity conditions. When the product increases the fluid viscosity, the flow velocity is reduced and higher amounts of product are formed due to increased diffusion time. Conversely, reduced fluid viscosity leads to greater fluid velocity and lower amount of product formation. An exception is the viscous thinning case of high Peclet (Pe) number and high Da where an instability develops (in low Reynolds (Re) number flow) that enhances mixing between the reactants, resulting in increasing product formation.

Agglomeration based discontinuous Galerkin discretization of the Euler and Navier–Stokes equations

In this work we exploit the flexibility associated to discontinuous Galerkin methods to perform high-order discretizations of the Euler and Navier–Stokes equations on very general meshes obtained by means of agglomeration techniques. Agglomeration is here considered as an effective mean to decouple the geometry representation from the solution approximation being alternative to standard isoparametric or breakthrough isogeometric discretizations. The mesh elements can be composed of a set of standard sub-elements belonging to a fine low-order mesh whose cardinality can be freely chosen according to the domain discretization capabilities. The number of mesh sub-elements still have an impact on the cost of numerical integration. Since the agglomerated elements are arbitrarily shaped, physical space orthonormal basis functions are here considered as a key ingredient to build a suitable discrete dG space. As a result we are allowed to perform high-order discretizations on top of the set of (possibly) subparametric sub-cells composing an agglomerated element. Our approach is validated on challenging viscous and inviscid test cases. We demonstrate the use of low-order meshes as a starting point to obtain high-order accurate solutions on coarse meshes.

The second act of hydro: small perturbations

Hydrodynamical description of the ‘Little Bang’ in heavy ion collisions is surprisingly successful: here we systematically study propagation of small perturbations treated hydrodynamically. Using analytic description of the expanding fireball known as the ‘Gubser flow’, we proceed to linearized equations for perturbations. As all variables are separated and all equations solved (semi)analytically, we can collect all the harmonics and reconstruct the complete Green function of the problem, even in the viscous case. Applying it to the power spectrum, we found acoustic minimum at m = 7 and maximum at m = 9, which remarkably have some evidence for both in the data. We estimate effective viscosity and the size of the perturbation from the fit to power spectrum. The shape of the two-point correlator is also reproduced remarkably well. At the end, we argue that independent perturbations are local, and thus harmonics phases are correlated.

A Laboratory Study of Nonlinear Western Boundary Currents, with Application to the Gulf Stream Separation due to Inertial Overshooting*

Abstract Various dynamical aspects of nonlinear western boundary currents (WBCs) have been investigated experimentally through physical modeling in a 5-m-diameter rotating basin. The motion of a piston with a velocity up that can be as low as up = 0.5 mm s−1 induces a horizontally unsheared current of homogeneous water that, flowing over a topographic beta slope, experiences westward intensification. First, the character of WBCs for various degrees of nonlinearity is investigated. By varying up, flows ranging from the highly nonlinear inertial Charney regime down to a weakly nonlinear regime can be simulated. In the first case, the dependence of zonal length scales on up is found to be in agreement with Charney’s theory; for weaker flows, a markedly different functional dependence emerges describing the initial transition toward the linear, viscous case. This provides an unprecedented coverage of nonlinear WBC dependence on an amplitude parameter in terms of experimental data. WBC separation from a wedge-shaped continent past a cape (simulating Cape Hatteras) due to inertial overshooting is then analyzed. By increasing current speed, a critical behavior is identified according to which a very small change of up marks the transition from a WBC that follows the coast past the cape to a WBC (nearly dynamically similar to a full-scale Gulf Stream) that separates from the cape without any substantial deflection, as with the Gulf Stream Extension. The important effect of the deflection angle of the continent is analyzed as well. Finally, the qualitative effect of a sloping sidewall along a straight coast is considered: the deflection of the flow away from the western wall due to the tendency to preserve potential vorticity clearly emerges.

Fate of the initial state perturbations in heavy ion collisions. III. The second act of hydrodynamics

The hydrodynamical description of the ``Little Bang'' in heavy-ion collisions is surprisingly successful, mostly owing to the very small viscosity of the quark-gluon plasma. In this paper we systematically study the propagation of small perturbations, also treated hydrodynamically. We start with a number of known techniques allowing for the analytic calculation of the propagation of small perturbations on top of the expanding fireball. The simplest approximation is the ``geometric acoustics,'' which substitutes the wave equation by mechanical equations for the propagating ``phonons.'' Next we turn to the case in which variables can be separated, where one can obtain not only the eikonal phases but also the amplitudes of the perturbation. Finally, we focus on the so-called Gubser flow, a particular conformal analytic solution for the fireball expansion, on top of which one can derive closed equations for small perturbations. Perfect hydrodynamics allows all variables to be separated and all equations to be solved in terms of known special functions. We can thus collect the analytical expression for all the harmonics and reconstruct the complete Green's function of the problem. In the viscous case the equations still allow for variable separation, but one of the equations has to be solved numerically. Summing all the harmonics we show real-time perturbation evolution, observing the viscosity-induced changes in the spectra and the correlation functions. The calculated angular shape of the correlation function is remarkably similar to the shape emerging from the experimental data, for sufficiently large viscosity. We predict a minimum at $m\ensuremath{\sim}7$ and maximum at $m\ensuremath{\sim}9$ harmonics, which also have some experimental evidence for it. We conclude that local ``hot spots'' in the initial state are the only visible origin of the observed correlations.

RAYLEIGH–TAYLOR INSTABILITIES IN AXI-SYMMETRIC OUTFLOW FROM A POINT SOURCE

AbstractThis paper studies outflow of a light fluid from a point source, starting from an initially spherical bubble. This region of light fluid is embedded in a heavy fluid, from which it is separated by a thin interface. A gravitational force directed radially inward toward the mass source is permitted. Because the light inner fluid is pushing the heavy outer fluid, the interface between them may be unstable to small perturbations, in the Rayleigh–Taylor sense. An inviscid model of this two-layer flow is presented, and a linearized solution is developed for early times. It is argued that the inviscid solution develops a point of infinite curvature at the interface within finite time, after which the solution fails to exist. A Boussinesq viscous model is then presented as a means of quantifying the precise effects of viscosity. The interface is represented as a narrow region of large density gradient. The viscous results agree well with the inviscid theory at early times, but the curvature singularity of the inviscid theory is instead replaced by jet formation in the viscous case. This may be of relevance to underwater explosions and stellar evolution.

Optimal decay rate of solutions for Cahn–Hilliard equation with inertial term in multi-dimensions

The Cauchy problem of the Cahn–Hilliard equation with inertial term is considered. Based on Greenʼs function method together with energy estimates, we get the global-in-time existence and optimal decay rate of solutions. Furthermore, the viscous case is investigated in the last section.

Capillary instability of elliptic liquid jets

Instability of a liquid jet issuing from an elliptic nozzle in Rayleigh mode is investigated and its behavior is compared with a circular jet. Mathematical solution of viscous free-surface flow for asymmetric geometry is complicated if 3-D analytical solutions are to be obtained. Hence, one-dimensional Cosserat (directed curve) equations are used which can be assumed as a low order form of Navier-Stokes equations for slender jets. Linear solution is performed using perturbation method. Temporal dispersion equation is derived to find the most unstable wavelength responsible for the jet breakup. The obtained results for a circular jet (i.e., an ellipse with an aspect ratio of one) are compared with the classical results of Rayleigh and Weber for inviscid and viscous cases, respectively. It is shown that in the Rayleigh regime, which is the subject of this research, symmetric perturbations are unstable while asymmetric perturbations are stable. Consequently, spatial analysis is performed and the variation of growth rate under the effect of perturbation frequencies for various jet velocities is demonstrated. Results reveal that in comparison with a circular jet, the elliptic jet is more unstable. Furthermore, among liquid jets with elliptical cross sections, those with larger ellipticities have a larger instability growth rate.

Equations of stochastic quasi-gas dynamics: Viscous gas case

The article reports some results of test calculations of one model from the hierarchical set of models of gas dynamics obtained (a brief scheme of the deduction is presented) from a system of stochastic differential equations describing gas at moderate and small Knudsen numbers.

Peristaltic motion of Phan–Thien–Tanner fluid in the presence of slip condition

This investigation considers the peristaltic flow of a Phan–Thien–Tanner fluid in the presence of slip condition and induced magnetic field. By use of the long wavelength and low Reynolds number approximations, closed form series solutions for stream function, pressure gradient, magnetic force function, axial induced magnetic field, and current density were obtained. The pressure gradient and frictional forces per wavelength were computed by numerical integration. The velocity slip condition in terms of shear stress is taken into account. Graphical results show the comparison between no-slip and viscous fluid cases. Pumping and trapping phenomena are discussed.

Shock pattern solutions for viscous-collisional plasma ion acoustic waves in view of the linear theory of the non-equilibrium thermodynamics

In the framework of irreversible thermodynamics, we study nonlinear ion-acoustic waves (IAWs) in viscous and collisional plasmas. Electrons, which form the background, are assumed to be nonthermal. On account of ion viscosity and ion-electron collisions, we investigate using ion fluid equations. We study the effects of the nonthermally distributed electrons β and the temperature ratio σ (= Ti/Te) on the stability, where the stability for Burger’s equation is analyzed by two methods: the phase portrait method and irreversible thermodynamics relations at different values of σ and β. We usa a reductive perturbation technique, where the nonlinear evolution of an IAW is governed by the driven Burger equation. This equation is solved exactly by using two methods: the tanh-function method and the Cole–Hopf transformation. Both methods produce shock wave solutions, their results compared, and good agreement exists in most predictions. The analytical calculations show that an IAW propagates as a shock wave with subsonic speed. The flow velocity, pressure, number density, electrostatic potential, and thermodynamic characteristics are estimated and illustrated as functions of time t and the distance x. It is found via the tanh-function method that the amplitudes of the sought-for functions of the system are suppressed and move towards an equilibrium state at the highest value of β. The tanh-function method reveals an advantage over the Cole–Hopf method in the viscous and collisional cases of IAWs, where it satisfies the stability conditions at the highest value of β with the chosen σ values when applied to evaluate the Onsager relation.

Deformation of an elliptical inclusion in two-dimensional incompressible power-law viscous flow

An iterative, semi-analytical solution is derived for deformation of an elliptical (in cross-section) power-law viscous inclusion within an infinite linear viscous matrix undergoing a general 2D incompressible flow. Finite-element numerical models are used to extend the analysis to that of a power-law viscous matrix. The general behaviour of a deformable elliptical inclusion is not dramatically changed by power-law viscous rheology, but the effective viscosity is now a function of the orientation and axial ratio of the inclusion. Overall, the effect is similar to a markedly increased viscosity ratio for a stronger inclusion, or a decreased ratio for a weaker inclusion, when compared to the linear viscous case. As a result, rather low reference state viscosity ratios between inclusion and matrix (e.g., 2 to 3, determined at the same effective strain rate for both materials) can produce marked differences in behaviour for the range of power-law stress exponents established experimentally for many minerals and rocks (typically 3–6). Even for very high strain within a shear zone ( γ > 100), initially nearly circular inclusions ( R < 2) can maintain low axial ratios ( R < 2–3) and widely variable orientations. These inclusions deform internally and are not rigid, but continue to rotate or oscillate without strong elongation.

On the Self-Similar Solutions of the 3D Euler and the Related Equations

We generalize and localize the previous results by the author on the study of self-similar singularities for the 3D Euler equations. More specifically we extend the restriction theorem for the representation for the vorticity of the Euler equations in a bounded domain, and localize the results on asymptotically self-similar singularities. We also present progress towards relaxation of the decay condition near infinity for the vorticity of the blow-up profile to exclude self-similar blow-ups. The case of the generalized Navier-Stokes equations having the laplacian with fractional powers is also studied. We apply the similar arguments to the other incompressible flows, e.g. the surface quasi-geostrophic equations and the 2D Boussinesq system both in the inviscid and supercritical viscous cases.