Arbitrary Reynolds Number Research Articles

Equatorial blowup and polar caps in drop electrohydrodynamics

We illuminate effects of surface-charge convection intrinsic to leaky-dielectric electrohydrodynamics by analyzing the symmetric steady state of a circular drop in an external field at arbitrary electric Reynolds number ReE. In formulating the problem, we identify an exact factorization that reduces the number of dimensionless parameters from four—ReE and the conductivity, permittivity and viscosity ratios—to two: a modified electric Reynolds number Rẽ and a charging parameter ϖ. In the case ϖ<0, where charge relaxation in the drop phase is slower than in the suspending phase, and, as a consequence, the interface polarizes antiparallel to the external field, we find that above a critical Rẽ value the solution exhibits a blowup singularity such that the surface-charge density diverges antisymmetrically with the −1/3 power of distance from the equator. We use local analysis to uncover the structure of that blowup singularity, wherein surface charges are convected by a locally induced flow towards the equator where they annihilate. To study the blowup regime, we devise a numerical scheme encoding that local structure where the blowup prefactor is determined by a global charging-annihilation balance. We also employ asymptotic analysis to construct a universal problem governing the blowup solutions in the regime Rẽ≫1, far beyond the blowup threshold. In the case ϖ>0, where charge relaxation is faster in the drop phase and the interface polarizes parallel to the external field, we numerically observe and asymptotically characterize the formation at large Rẽ of stagnant, perfectly conducting surface-charge caps about the drop poles. The cap size grows and the cap voltage decreases monotonically with increasing conductivity or decreasing permittivity of the drop phase relative to the suspending phase. The flow in this scenario is nonlinearly driven by electrical shear stresses at the complement of the caps. In both polarization scenarios, the flow at large Rẽ scales linearly with the magnitude of the external field, contrasting the familiar quadratic scaling under weak fields. Published by the American Physical Society 2024

Stability of plane Couette flow under anisotropic superhydrophobic effects

We study the linear stability of plane Couette flow subject to an anisotropic slip boundary condition that models the slip effect of parallel microgrooves with a misalignment about the direction of the wall motion. This boundary condition has been reported to be able to destabilize channel flow far below the critical Reynolds number of the no-slip case. Unlike channel flow, the no-slip plane Couette flow is known to be linearly stable at arbitrary Reynolds numbers. Nevertheless, the results show that the slip can cause linear instability at finite Reynolds numbers also. The misalignment angle of the microgrooves that maximizes the destabilizing effect is nearly π/4, and the unstable modes are of small streamwise wavenumbers and relatively large spanwise wavenumbers. The flow is always more destabilized by two slippery walls compared to a single slippery wall. These observations are in qualitative agreement with the slippery channel flow with the same boundary condition, indicating that such an anisotropic superhydrophobic effect has a rather general destabilizing effect in shear flows regardless of the profile of the base flow. The absence of the Tollmien–Schlichting instability allows us to reveal the inverse relationship between the critical Reynolds number and the slip length as well as the misalignment in the small-parameter regime. The results suggest that arbitrary nonvanishing slip length and misalignment, with arbitrarily weak anisotropy, may suffice to destabilize plane Couette flow.

Stability of plane Couette flow with constant wall transpiration

Plane Couette flow with constant wall transpiration, i.e., constant blowing from below and constant suction from above, is a solution of the Navier-Stokes equations in terms of exponentials. It is characterized by two Reynolds numbers Re and ReV, where Re parametrizes the moving wall and ReV describes the influence of the transpiration. For this flow the modified Orr-Sommerfeld equation admits one of the very few exact solutions in terms of hypergeometric functions. Based on this, we solve the stability problem, though for the main part focus on temporal stability. For small wall-transpiration rates up to ReV≃6.71, the flow remains unconditionally stable, analogous to the classical Couette flow for arbitrary Reynolds numbers Re. By further increasing ReV an instability sets in and a minimum critical Reynolds number of Re=668350.491 is reached at ReV=9.799. Hence, around this point, the destabilizing effect of blowing outweighs the stabilizing effect of suction. By further increasing the transpiration rate beyond this point, the corresponding critical Reynolds number Re increases again and continues to grow. This limiting case is accompanied by the development of a strong boundary-layer-like velocity profile near the upper wall and the flow transitions to the asymptotic suction boundary layer (ASBL). Thus, the present analysis comprises the whole range from classical Couette flows extended by transpiration to the ASBL, which is known to have a strongly stabilizing effect. Published by the American Physical Society 2024

Acoustic streaming: insights across Reynolds numbers

When a fluid system is subjected to an acoustic wave (or another periodic actuation), the response of the fluid is not purely periodic, but is rather characterized by the combination of a periodic flow and a steady Stokes drift component, where the former is, in many cases, an acoustic wave and the latter is commonly referred to as acoustic streaming. Classical theories of acoustic streaming have focused on slow acoustic streaming, where the periodic flow is the leading-order flow, and is insensitive to the steady flow component which appears as a small correction and is characterized by a small hydrodynamic Reynolds number. In contrast, Dubrovski et al. (J. Fluid Mech. vol. 975, 2023, A4) tackle the fast acoustic streaming regime – conceived by Zarembo (Acoustic streaming. In High-Intensity Ultrasonic Fields, 1971, pp. 135–199. Springer) approximately fifty years ago – where both the periodic and steady flow components are of a similar order of magnitude such that the periodic flow both supports and is simultaneously impacted by the steady flow. They present a novel theoretical framework that accounts for the convection of momentum both within and between the periodic and steady flow to extend slow-streaming equations to the case of steady flow with arbitrary hydrodynamic Reynolds number. They leverage a scaling analysis of the resulting system of equations and a case study to demonstrate the compatibility of their equations with slow streaming theories and highlight the distinctive features of fast streaming.

Theory of acoustic streaming for arbitrary Reynolds number flow

We study a fifty-year-old problem of fast acoustic streaming, that is, the generation of moderate or large hydrodynamic Reynolds number ( $\textit {Re}$ ) acoustic streaming (or steady flow) by the convection of momentum in an acoustic wave (or another periodic flow), while the latter is simultaneously altered by the former. The intrinsic disparity of length and time scales makes a brute-force solution of the full Navier–Stokes and continuity equations a formidable problem. Circumventing this difficulty, we split the problem into a time-averaged system of equations for the steady flow component and a dynamic system of equations for its quasi-periodic flow counterpart. The latter system of equations is obtained by subtracting the time-averaged Navier–Stokes equation from its original dynamic form, and is rendered a nonlinear wave equation using the continuity equation and an adiabatic connection between density and pressure. The resulting equations are compatible with the theory by Eckart for small $\textit {Re}$ flow, and capture large- $\textit {Re}$ effects. Scaling analysis and a case study show that acoustic streaming is weak and does not contribute to the acoustic wave close to the wave source, relevant to many microfluidic systems. At small $\textit {Re}$ , the streaming magnitude is proportional to an inverse Strouhal number, a small quantity in experiments. Moderate and large $\textit {Re}$ render the streaming magnitude comparable to the pre-attenuating periodic flow (or particle velocity of the wave) at approximately a wave attenuation length away from the wave source or further; the wave is altered by the streaming that it generates, and the streaming dominates the flow far from the wave source.

Irreversibility of heat transfer with the curvature effect on peristaltic thrusting of micropolar fluid in a resilient channel in the presence of heat generation: nonlinear analysis

In this work, we perform an investigation of the peristaltic thrusting of an incompressible viscous micropolar fluid in a symmetric resilient channel with consideration of streamline curvature effects and inertia forces. We obtained the partial differential equations for fluid flow in two dimensions for micropolar fluids in addition to the heat equation. These equations have been transformed to another domain by using a transformation between the fixed and wave frames. Asymptotic solutions of have been obtained by using the compatible boundary conditions using the perturbation technique of arbitrary Reynolds number and long-drawn wavelength. The solutions for longitudinal, normal and spin velocities, stream function, heat transfer and rate of working walls have been obtained. The several of biophysical parameters influence has been demonstrated through the graphs. Both time flow rate and pressure gradient relations are obtained. Different characteristics for fluid flow, temperature profile, normal force, microrotation (spin) velocity and Nusslet number are greatly influenced at the channel walls by the presence of micropolar fluid. Trapping and pumping phenomena are further discussed. It is found that the maximum value for the Bejan number occurs near the centre of the channel, i.e. the heat transfer irreversibility dominates around the centre of the channel.

Direct numerical simulation of moderate‐Reynolds‐number flow past arrays of ellipsoids

AbstractThrough particle‐resolved direct numerical simulations of flow past arrays of ellipsoids, the hydrodynamic force on ellipsoids depends on the particle orientation, aspect ratio, particle Reynolds number, and solid volume fraction is revealed at moderate Reynolds numbers. The results show that the mean drag force on arrays of prolate/oblate ellipsoids decreases/increases as the Hermans orientation factor increases when flows are in the reference direction defined by the average symmetric axis of particles. The individual drag force on a prolate/oblate ellipsoid increases/decreases with the increase of incidence angle, and it is also affected by the orientation of surrounding particles. The individual lift force is also significant when the aspect ratio is away from unity at large particle Reynolds numbers. Based on simulation results, correlations for the hydrodynamic force on ellipsoids at arbitrary particle Reynolds numbers, solid volume fractions, Hermans orientation factors, incidence angles, and aspect ratios are formulated.

Stabilization of the flat Poiseuille-type flow for viscoelastic polymeric liquid

This paper presents a numerical study for the problem of the one-dimensional flow of viscoelastic liquid polymers between two parallel plates. The equations of a rheologically modified Vinogradov–Pokrovskii (mVP) model is used for the formulation of the problem. It is shown that the problem could have multiple steady-state solutions. The evaluation of non-steady solutions was performed to see if the time-dependent solutions got eventually attracted by the steady ones. Also for the case of multiple steady solutions, it was checked which one attracts the non-steady solution if any. The evaluation of time-dependent solutions was used to estimate the stability of equilibrium states. It is revealed that stable steady-state regimes of the problem exist under certain conditions, and also there could be no more than one stable regime for any given set of parameters. The calculations were performed to estimate the values of Reynolds and Weissenberg numbers corresponding to either stable or unstable steady regimes. The result indicates that instability of the steady flow could possibly occur for arbitrary low Reynolds numbers under certain balance of viscous and elastic forces.

Linear stability of a ferrofluid centred around a current-carrying wire

Investigated first is the linear stability of a Newtonian ferrofluid centred on a rigid wire, surrounded by another ferrofluid with a different magnetic susceptibility. An electric current runs through the wire, generating an azimuthal magnetic field that produces a magnetic stress at the interface of the fluids. Three-dimensional disturbances to the system are considered, and the linearised Navier–Stokes equations are solved analytically in terms of an implicit expression for the growth rate of the disturbance. The growth rate is found numerically for arbitrary Reynolds number, and given explicitly in the inviscid and Stokes regimes. Investigated next is a ferrofluid whose magnetic susceptibility varies radially, centred on a rigid wire, subject to a non-uniform azimuthal field. It is proven that if the gradient of the susceptibility is positive anywhere in the fluid, then the system is linearly unstable. Moreover, it is proven that applying an axial field can stabilise disturbances for both continuous and discontinuous susceptibilities.

Drag and lift forces on a rigid sphere immersed in a wall-bounded linear shear flow

Drag and lift forces experienced by a rigid spherical particle translating close to a flat wall in a linear shear flow are determined through fully resolved simulations. Particles are assumed to obey either a non-rotating or a torque-free condition and may either lead the fluid or lag behind it. The simulations cover a wide range of flow regimes, from low-but-finite Reynolds numbers to conditions close to the onset of wake unsteadiness. Results are used to build accurate semiempirical expressions for the two force components valid at arbitrary Reynolds number, relative shear rate, and separation distance from the wall.

Mixed Convection Heat Transfer and Entropy Generation in a Water-filled Square Cavity Partially Heated from Below: The Effects of Richardson and Prandtl Numbers

In the present study, fluid flow, heat transfer, and entropy generation for mixed convection inside a water-filled square cavity were investigated numerically. The sidewalls of the cavity, which move upwards, are kept at low-temperature Tc while only a part in the center of the bottom wall is kept at high-temperature Th and the remaining parts are kept adiabatic. The governing equations, in stream function–vorticity form, are discretized and solved using the finite difference method. Particular attention was paid to the influence of the Prandtl numbers of 5.534, 3.045 and 2, corresponding respectively to the water temperatures of 303.15 K, 333.15 K and 363.15 K. The numerical results are presented in the form of streamlines, isotherms, and entropy generation contours for different values of the Richardson numbers at an arbitrary Reynolds number Re=102. Besides this, the evolution of the average Nusselt number and the average entropy generation is also reported. The obtained results show interesting behaviors of the flow and thermal fields, which mainly involve stable symmetric and non-symmetric steady-state solutions, as well as unsteady regimes, depending on specific values of the Richardson and Prandtl numbers. It is additionally observed that the average Nusselt number increases and the average entropy generation decreases when both the Richardson and Prandtl numbers increase.

Stability of plane Couette flow past an initially stressed solid

In this paper, we carry out linear stability analysis of a coupled solid–fluid system in a plane Couette flow past an initially stressed neo-Hookean solid at creeping flow limit (Re=0) and arbitrary Reynolds number (Re), as well as for different solid thicknesses. In biological and engineering systems, we often encounter flows past initially stressed deformable solids. For these cases, initial strain or initial stress appears as an additional symmetry representing tensor in the constitutive relation. Consequently, initial stress induces anisotropy (transverse isotropy in the present case) and affects material properties. We employ a hyperelastic model for the initially stressed deformable solid to study the effects of both uni-axial and equi-biaxial initial stresses on flow instability. This study also presents a case where anisotropy in the solid interferes with the stability of the solid–fluid coupled system. In contrast with the previous formulations with two or three state Eulerian–Lagrangian formulations, the present formulation requires four configurations for the initially stressed solid, viz. stress-free, initially stressed, deformed, and the perturbed state. This formulation applies to any fluid–solid coupled problems where the solid contains initial/residual stress. Our results show that initial tensile stress stabilizes the flow, while compressive initial stress destabilizes the same. Fortuitously, this observation of coupled fluid–solid stability is in agreement with the stability of solid column-like structures, where compressive axial stress leads to buckling. We report similar results for uni-axial and equi-biaxial initial stress, for different wave modes at creeping flow limit and their extensions at higher Re, and various solid thicknesses. Two modes of instability, viz., short-wave and finite-wave modes present in the creeping-flow limit are significantly affected by tensile/compressive initial stress for both uni-axial and equi-biaxial cases. For uni-axial tensile initial stress, the finite-wave mode does not extend to high Re. Instead of them, we observe multiple upstream modes to become unstable for higher Re, which show the wall mode scaling as O(Re−1/3). These upstream modes are not observed for stress-free solid. This behavior is triggered by the additional coupling terms between initial stress and perturbed variable in the governing equations of initially stressed solids. However, for the equi-biaxial case, the finite-wave mode does extend to higher Re to show wall mode scaling. This mode further stabilizes/destabilizes depending on the tensile/compressive nature of the initial stress present in the solid.

The Batchelor Spectrum of Passive Scalar Turbulence in Stochastic Fluid Mechanics at Fixed Reynolds Number

In 1959 Batchelor predicted that the stationary statistics of passive scalars advected in fluids with small diffusivity k should display a power spectrum along an inertial range contained in the viscous‐convective range of the fluid model. This prediction has been extensively tested, both experimentally and numerically, and is a core prediction of passive scalar turbulence.In this article we provide a rigorous proof of a version of Batchelor's prediction in the limit when the scalar is subjected to a spatially smooth, white‐in‐time stochastic source and is advected by the 2D Navier‐Stokes equations or 3D hyperviscous Navier‐Stokes equations in forced by sufficiently regular, nondegenerate stochastic forcing. Although our results hold for fluids at arbitrary Reynolds number, this value is fixed throughout. Our results rely on the quantitative understanding of Lagrangian chaos and passive scalar mixing established in our recent works. Additionally, in the limit, we obtain statistically stationary, weak solutions in to the stochastically forced advection problem without diffusivity. These solutions are almost‐surely not locally integrable distributions with nonvanishing average anomalous flux and satisfy the Batchelor spectrum at all sufficiently small scales. We also prove an Onsager‐type criticality result that shows that no such dissipative, weak solutions with a little more regularity can exist. © 2021 Wiley Periodicals LLC.

Smoothed profile method for direct numerical simulations of hydrodynamically interacting particles.

A general method is presented for computing the motions of hydrodynamically interacting particles in various kinds of host fluids for arbitrary Reynolds numbers. The method follows the standard procedure for performing direct numerical simulations (DNS) of particulate systems, where the Navier-Stokes equation must be solved consistently with the motion of the rigid particles, which defines the temporal boundary conditions to be satisfied by the Navier-Stokes equation. The smoothed profile (SP) method provides an efficient numerical scheme for coupling the continuum fluid mechanics with the dispersed moving particles, which are allowed to have arbitrary shapes. In this method, the sharp boundaries between solid particles and the host fluid are replaced with a smeared out thin shell (interfacial) region, which can be accurately resolved on a fixed Cartesian grid utilizing a SP function with a finite thickness. The accuracy of the SP method is illustrated by comparison with known exact results. In the present paper, the high degree of versatility of the SP method is demonstrated by considering several types of active and passive particle suspensions.

Hydrodynamic forces on a clean spherical bubble translating in a wall-bounded linear shear flow

The three-dimensional flow around a spherical clean bubble translating steadily in a wall-bounded linear shear flow is studied numerically. The present work is concerned with the drag and lift forces experienced by the bubble over a wide range of Reynolds number (0.1 ≤ Re ≤ 103, Re being based on the bubble diameter and relative velocity with respect to the ambient fluid), wall distance (1.5 ≤ LR ≤ 8, LR being the distance from the bubble center to the wall normalized by the bubble radius), and relative shear rate (−0.5 ≤ Sr ≤ 0.5, Sr being the ratio between the velocity difference across the bubble and the relative velocity). Based on the above range of parameters, situations where the bubble is repelled from or attracted to the wall are both covered. The flow structure and vorticity field are analyzed to obtain qualitative insight into the interaction mechanisms at work. The drag and lift forces are computed as well. Their variations agree well with theoretical predictions available in the limit of low-but-finite Reynolds number and, when the fluid is at rest, in the potential flow limit. Numerical results and analytical expressions are combined to provide accurate semi-empirical expressions for the drag and lift forces at arbitrary Reynolds number and separation distance.

Forced Convection Heat Transfer From a Particle at Small and Large Peclet Numbers

Abstract We theoretically study forced convection heat transfer from a single particle in uniform laminar flows. Asymptotic limits of small and large Peclet numbers Pe are considered. For Pe≪1 (diffusion-dominated regime) and a constant heat flux boundary condition on the surface of the particle, we derive a closed-form expression for the heat transfer coefficient that is valid for arbitrary particle shapes and Reynolds numbers, as long as the flow is incompressible. Remarkably, our formula for the average Nusselt number Nu has an identical form to the one obtained by Brenner for a uniform temperature boundary condition (Chem. Eng. Sci., vol. 18, 1963, pp. 109–122). We also present a framework for calculating the average Nu of axisymmetric and two-dimensional (2D) objects with a constant heat flux surface condition in the limits of Pe≫1 and small or moderate Reynolds numbers. Specific results are presented for the heat transfer from spheroidal particles in Stokes flow.

On large-scale quasi-geostrophic dynamical processes in a rotating layer of conducting liquid with arbitrary magnetic Reynolds number

The paper constructs the matheamtical model of dynamics of spatial large-scale quasi-geostrophic motions in a rotating layer of ideal incompressible conducting liquid of variable depth with arbitrary values of the magnetic Reynolds number. Our finding can have certain value in astrophysics and geophysics, for example, in the study of processes occurring in stellar interiors and the liquid Earth core; besides, they can be used in evaluating magnetohydrodynamic characteristics of marine environments, in assessing parameters of wave perturbations sources from the electromagnetic field generated by the corresponding hydrodynamic excitement, which contains important information about geophysical characteristics of the shelf zone. This information is relevant for marine geological surveys and studies.

On integration of the system of MHD equations modeling wave processes in a rotating liquid with arbitrary magnetic Reynolds number

The paper is concerned with the dynamics of large-scale wave processes in a rotating layer of inviscid conducting incompressible liquid of variable depth. The problem is modelled as a system of partial differential equations with necessary boundary conditions. With the help of auxiliary functions, the above system of partial differential magnetohydrodynamic equations is reduced to a single scalar partial differential equation. An exact analytic solution of the small perturbation problem is obtained. It is shown that if the external magnetic field is parallel to the axis of rotation of the layer, then the magnetic field decays for finite values of the magnetic Reynolds number.

Stability of the compressible viscous fluid around the plane Couette flow in the presence of a transverse uniform magnetic field

In this paper, we consider the stability of three-dimensional compressible viscous fluid around the plane Couette flow in the presence of a uniform transverse magnetic field and show that the uniform transverse magnetic field has a stabilizing effect on the plane Couette flow. Namely, for a sufficiently large Hartmann number, the compressible viscous plane Couette flow is nonlinear stable for small Mach number and arbitrary Reynolds number so long as the initial perturbation is small enough.

Stability of plane Couette flow of Carreau fluids past a deformable solid at arbitrary Reynolds numbers

The linear stability of the plane Couette flow of both power-law and Carreau fluids past a deformable, neo-Hookean solid is analyzed at arbitrary Reynolds numbers. An algebraic error in the mathematical formulation of the earlier studies (for the power-law fluid) is corrected and is shown to result in quantitative differences in the predictions for the stability of the flow. Due to the lack of a proper (zero-shear) viscosity scale and a time scale for the onset of shear thinning in the power-law model, we show that the stability analysis of the flow yields vastly different scalings for the unstable mode depending on the way the problem is scaled to render it dimensionless. When the deformable solid properties are used to non-dimensionalize, we show that for the unstable modes (the so-called “wall modes” at high Re) Γc∝Re−1(2n+1), while when flow properties are used to non-dimensionalize, Γc∝Re−13 much akin to a Newtonian fluid, where Γ=Vm*η*/G*R* is the dimensionless shear rate in the flow, and Γc denotes the minimum value required for instability. Here, Vm* is the velocity of the top plate, G* is the shear modulus of the solid, R* is the fluid thickness, and η* is the (arbitrary) viscosity scale in the power-law model. Within the framework of the power-law model, it is not possible to discriminate between the two predicted scalings. To resolve this in an unambiguous manner, we used the Carreau model to account for shear thinning and to study its role on the stability of flow past deformable solid surfaces. The Carreau model has a well-defined zero-shear viscosity η0* as well as a time scale λ* that characterizes the onset of shear thinning. For fixed λ*η0*/(ρ*R*2), we show that the unstable wall modes scale as Γc∼Re(1−2n)3 at high Re, thus providing a resolution to the ambiguity in the results obtained using the power-law model. The present work thus shows that, at moderate to high Re, shear thinning has a strongly stabilizing effect on the wall mode instability in flow past deformable solid surfaces.