Viscous Sphere Research Articles

Impact of a spherical interface on a concentrical spherical droplet

<p>In this paper, an analytical and numerical technique are examined in order to analyse the Stokes flow determination problem due to a viscous sphere droplet moving at a concentric instantaneous position inside a spherical interface separating finite and semi-infinite immiscible fluid phases. Here, when only one of the three phases of the fluid (micropolar fluid) has a microstructure, attention is focused on this case. The motion is considered when Reynolds- and capillary-numbers are low, and the droplet surface and the fluid-fluid interface have insignificant deformation. A general solution is obtained in a spherical coordinate system based on a concentric position to analyse the slow axisymmetric movement of the micropolar fluid, considering microrotation and velocity components. Boundary conditions are initially fulfilled at the fluid-fluid interface and subsequently at the droplet surface. The normalised hydrodynamic drag force applying to a moving viscous droplet appears to be a function of the droplet-to-interface radius ratio, which increases monotonically and becomes unbounded when the droplet surface touches the fluid-fluid interface. The numerical outcomes of the normalised drag force acting on the viscous droplet are derived for different values of the parameters, and are presented in a tabular and graphical framework. A comparison was made between our numerical outcomes for the drag force and the pertinent data for the special cases found in the literature.</p>

A Study on Oscillatory Micropolar Flow Beyond a Contaminated Micropolar Fluid Sphere

In this paper, the hypothesis of the axisymmetric rectilinear oscillatory flow beyond a micropolar tainted fluid sphere particle in an incompressible non-Newtonian fluid and also the axisymmetric rectilinear oscillatory flow over a viscous tainted fluid sphere particle in an incompressible Newtonian fluid with small amplitude oscillations have been investigated. The velocity field is exhibited in terms of stream functions, and a slip condition is considered on the boundary. The fluid velocities and microrotation components were derived through analytical procedure. The drag force acting on the particle was also computed and verified for special cases. The real drag and imaginary drag values are numerically extracted for varying slip parameter i.e., 2≤s≤30, micro polarity i.e., 8≤e≤32 , and viscosity ratio i.e., 5≤μ≤20 at a fixed parameter values k=0.1,ρ=0.6,ω=0.6,t=0.6. Graphs and tables are used to display the numerical results. It was observed that there was an inverse proportion between slip parameter values, real drag and direct proportion between slip parameter and imaginary drag, for different viscosity ratio and micro polarity values.

Motion through a viscous liquid sphere enclosed by a solid core embedded into a Brinkman medium

AbstractThe flow around a solid spherical particle encased in a Newtonian liquid sphere and immersed in a couple stress fluid medium is studied. The problem is expressed by using the Brinkman and Stokes equations, which describe both the flow outside and inside the liquid sphere, respectively. The Gegenbauer polynomials and modified Bessel function are used to express the stream function solution for the internal and external regions. An analytical determination for the flow field in terms of stream function is examined by wielding the method of separation of variables. The drag force on a solid spherical particle placed in a permeable region is calculated. On the drag coefficient, the effects of the permeability κ, the viscosity ratio γ2, and the couple stress parameter λ are investigated. Corresponding dependencies (such as the permeability parameter, couple stress parameter, viscosity ratio, and separation parameter) are graphically represented and discussed. The findings shows when the separation parameter is increased the drag coefficient gradually increases, it refers to a sphere surface with a high level of flow resistance. Passages to the limits are used to describe known specific cases. The present study is essentially significant in the course through a layer developed by penetrable particles and has very important and persuasive applications both in nature and innovation, with various potential outcomes. Thus, the discoveries of this article are comprehensively pertinent to the investigation of the flow of permeable liquids past spherical permeable rocks, aloxite materials, sand beds, earthen soil, petrol supply rocks, and so forth. The present application will support in planning a productive bearing framework.

Couple stress fluid past a contaminated fluid sphere with slip condition

In this article, an exact solution for couple stress fluid flow past a contaminated fluid sphere, filled with couple stress fluid was formed by using interfacial slip boundary condition. It's involved with internal and external cap and no cap regions. The velocity is expressed in terms of stream function. The external velocity and internal velocity are obtained. The Drag coefficient is computed. The special cases of a viscous fluid with a no-slip condition, solid sphere case are evaluated. The obtained results are in good agreement with the results from the literature. Additionally studied viscous fluid past a contaminated couple stress fluid sphere and couple stress fluid past a contaminated viscous sphere, also done. The results of special cases are also agreeing with the results from the literature.

Finite boundary effects on the spherical Rayleigh–Taylor instability between viscous fluids

For the Rayleigh–Taylor unstable arrangement of a viscous fluid sphere embedded in a finite viscous fluid spherical shell with a rigid boundary and a radially directed acceleration, a dispersion relation is developed from a linear stability analysis using the method of normal modes. aR1 is the radially directed acceleration at the interface. ρi denotes the density, μi is the viscosity, and Ri is the radius, where i = 1 is the inner sphere and i = 2 is the outer sphere. The dispersion relation is a function of the following dimensionless variables: viscosity ratio s=μ1μ2, density ratio d=ρ1ρ2, spherical harmonic mode n, B=R1aR1ρ22μ221/3, H=R2R1, and the dimensionless growth rate α=σμ2aR12ρ21/3, where σ is the exponential growth rate. We show that the boundedness provided by the outer spherical shell has a strong influence on the instability behavior, which is reflected not only in the modulation of the growth rate but also in the selection of the most unstable modes that are physically possible. This outer boundary effect is quantified by the relative magnitude of the radius ratio H. We find that when H is close to unity, lower order harmonics are excluded from becoming the most unstable within a vast region of the parameter space. In other words, the effect of H has precedence over the other controlling parameters d, B, and a wide range of s in establishing what the lowest most unstable mode can be. When H ∼ 1, low order harmonics can become the most unstable only for s ≫ 1. However, in the limit when s → ∞, we show that the most unstable mode is n = 1 and derive the dispersion relation in this limit. The exclusion of most unstable low order harmonics caused by a finite outer boundary is not realized when the outer boundary extends beyond a certain threshold length-scale in which case all modes are equally possible depending on the value of B.

Structure scalars with charged dissipative systems in [formula omitted] gravity

The aim of this investigation is to examine the effect of gravitational modification, in particular, the f(G,T) gravitational field, for dynamically relativistic spherically symmetric systems in the existence of electromagnetic field. The theoretical analysis of dynamical variables linked with charged viscous anisotropic sphere with dissipation is formulated in these environments. We consider that with a relativistic matter, the non-static diagonally symmetric spherical structure is coupled. We linked the Weyl scalar, structure parameters including the charged terms with metric scale factors and mass function. To formulate general form of dynamical variables, we implemented Herrera’s methodology for the orthogonally decomposition of the Riemann tensor. The presence of these variables in the behavioral dynamics of radiating spheres is then analyzed. With the existence and exclusion of fixed f(G,T) values, the parameters essential for the occurrence of inhomogeneities have been studied. It is suggested that with generalized forms of scalar variables, the evolutionary periods of the spherical interiors can be well observed.

Erratum: The Rayleigh–Taylor instability in a self-gravitating two-layer viscous sphere

Erratum: The Rayleigh–Taylor instability in a self-gravitating two-layer viscous sphere

Acoustics of finite asymmetric exotic beams: Examples of Airy and fractional Bessel beams

The purpose of this investigation is to examine the properties of finite asymmetric exotic scalar (acoustic) beams with unusual properties using the angular spectrum decomposition in plane waves. Such beams possess intrinsic uncommon characteristics that make them attractive from the standpoint of particle manipulation, handling and rotation, and possibly other applications in particle clearing and separation. Assuming a specific apodization function at the acoustic source, the angular spectrum function is calculated and used to synthesize the radiated pressure field (i.e., excluding evanescent waves that decay away from the source) in the forward direction of wave motion (i.e., away from the source). Moreover, a generalized hybrid method combining the angular spectrum approach with the multipole expansion formalism in spherical coordinates is developed, which is applicable to any finite beam of arbitrary wavefront. The improved approach allows adequate computation of the resonance scattering, radiation force, and spin torque components on an object of arbitrary shape, located on or off the axis of the incident beam in space. Considering the illustrative example of a viscous fluid sphere submerged in a non-viscous liquid and illuminated by finite asymmetric beams such as the Airy and the Bessel vortex beam with fractional order, numerical computations for the scattering, radiation force, and torque components are performed with an emphasis on the distance from the source, the arbitrary location of the particle ,and the asymmetric nature of the incident field. Moreover, beamforming calculations are presented with supplementary animations for the pressure field distribution in space, with an emphasis on the intrinsic properties of the selected beams. The numerical predictions illustrate the scattering, radiation force, and spin torque properties depending on the beam parameters and the distance separating the sphere from the source. This study provides a generalized hybrid method to analyze quantitatively the scattering, radiation force, and spin torque by any finite asymmetric (or symmetric) acoustic beam with potential applications in various fields of applied physics (such as beam-forming, imaging, and mechanical effects of asymmetric sound beams).

The Rayleigh–Taylor instability in a self-gravitating two-layer viscous sphere

The Rayleigh–Taylor instability in a self-gravitating two-layer viscous sphere

Stokes flow inside a sphere in an inviscid extensional flow

We derive the streamfunction solution for flow in and around a viscous sphere suspended in an inviscid extensional flow with matched stress boundary conditions, which is a model for estimating the stresses on a tiny suspended organism by a nearby expanding and collapsing bubble. The boundary conditions are enforced in an easily resolvable form by expressing the surface stresses as sums of Legendre and Gegenbauer functions. The flow inside the sphere reflects a balance of exterior inertia with internal viscous forces, which together are shown to constitute the relevant flow Reynolds number. The solution is evaluated to examine the flow field inside this sphere as a potential source of damage to the organism.

Wall effects on viscous fluid spheroidal droplet in a micropolar fluid spheroidal cavity

The present study examines the steady, axisymmetric Stokes flow past a viscous fluid spheroid whose shape deviates slightly from that of a sphere in a micropolar fluid spheroidal cavity. Inertial effects are neglected for both the inner and outer fluids. The boundary value problem is solved analytically using the cell model technique. The boundary conditions used are the vanishing of the normal velocities, the continuity of the tangential velocities, the continuity of shear stresses and the spin-vorticity relation at the surface of the inner viscous fluid spheroid. On the outer spheroidal surface (filled with micropolar fluid), four known boundary conditions, namely Happel’s, Kvashnin’s, Kuwabara’s and Cunningham’s (Mehta–Morse) are considered. The dependence of the wall correction factor on the spin parameter, viscosity ratio, volume fraction, deformation parameter and the micropolarity parameter is studied numerically and its variation is presented graphically. Both types of spheroids, prolate and oblate are considered. As a limiting case, the drag force acting on viscous fluid spheroid by the micropolar fluid in an unbounded medium and the wall correction factor for slow motion of a viscous fluid sphere located at the center of a spherical cell are obtained from the present analysis.

Photocontrol of Clustering, Retaining, and Releasing of Microbeads Concomitant with Phototransformation of Supramolecular Architecture of Amphiphilic Diarylethene.

Photoinduced clustering of polystyrene microbeads and photocontrol of their diffusion was achieved in water with the assistance of photoinduced transformation of supramolecular architecture of amphiphilic diarylethene between sphere and fiber states. When a suspension of polystyrene beads containing the sphere state of diarylethene was UV-irradiated from beneath, clustering of the polystyrene beads by thermal convection was observed. The velocity of clustering was dependent on the amount of photogenerated nanofibers that determines the viscosity of the water. Diffusion of the clustered polymer beads was suppressed by the surrounding fibers, but was restored to regular Brownian motion upon irradiation with visible light. It was suggested that the diffusion of the microbeads was controlled by the transformation of aggregates between the more viscous fiber state and the less viscous sphere state. These results provide new insight into the photocontrol of particle motion in fluidic media.

Stress relaxation in viscous soft spheres.

We report the results of molecular dynamics simulations of stress relaxation tests in athermal viscous soft sphere packings close to their unjamming transition. By systematically and simultaneously varying both the amplitude of the applied strain step and the pressure of the initial condition, we access both linear and nonlinear response regimes and control the distance to jamming. Stress relaxation in viscoelastic solids is characterized by a relaxation time τ* that separates short time scales, where viscous loss is substantial, from long time scales, where elastic storage dominates and the response is essentially quasistatic. We identify two distinct plateaus in the strain dependence of the relaxation time, one each in the linear and nonlinear regimes. The height of both plateaus scales as an inverse power law with the distance to jamming. By probing the time evolution of particle velocities during relaxation, we further identify a correlation between mechanical relaxation in the bulk and the degree of non-affinity in the particle velocities on the micro scale.

Spin reversal and orbital torques on a viscous fluid Rayleigh sphere located arbitrarily in acoustical Bessel vortex (spiraling) beams

The goal of this work is to demonstrate the emergence of a spin torque singularity (i.e. zero spin torque) and a spin rotation reversal of a small Rayleigh lipid/fat viscous fluid sphere located arbitrarily in space in the field of an acoustical Bessel vortex beam. This counter-intuitive property of negative spin torque generation suggests a direction of spin rotation in opposite handedness of the angular momentum carried by the incident beam. Such effects may open new capabilities in methods of quantitative characterization to determine physical properties such as viscosity, viscoelasticity, compressibility, stiffness, etc., and other techniques for the rotation and positioning using acoustical tractor beams and tweezers, invisibility cloaks, and acoustically-engineered composite metamaterials to name a few examples. Based on the descriptions for the velocity potential of the incident beam and the scattering coefficients of the sphere in the long-wavelength approximation limit, simplified expressions for the spin and orbital radiation torque components are derived. For beams with (positive or negative) unit topological charge (m=±1), the axial spin torque component for a Rayleigh absorptive sphere is maximal at the center of the beam, while it vanishes for |m|>1 therein. Moreover, the longitudinal orbital torque component, causing the sphere to rotate around the center of the beam is evaluated based on the mathematical decomposition using the gradient, scattering and absorption transverse radiation force vector components. It is shown that there is no contribution of the gradient transverse force to the orbital torque, which is only caused by the scattering and absorption transverse force components. Though the incident acoustical vortex beam carrying angular momentum causes the sphere to rotate in the same orbital direction of the beam handedness, it induces a spin torque singularity (i.e. zero spin torque) and subsequent sign reversal. This phenomenon of negative spin torque generation may be exploited from the standpoint of particle sizing, and possibly other applications in particle manipulation and rotation. In addition, an application of the spin and orbital radiation torque formulations derived here in the Rayleigh limit concerns the inverse determination of the host fluid viscosity from the induced sphere spinning and/or orbital rotation.

Kilogram-scale synthesis of Pd-loaded quintuple-shelled Co3O4 microreactors and their application to ultrasensitive and ultraselective detection of methylbenzenes.

We report the kilogram-scale, simple, and cost-effective synthesis of Pd-loaded quintuple-shelled Co3O4 microreactors by spray drying of aqueous droplets containing cobalt nitrate, palladium nitrate, citric acid, and ethylene glycol and subsequent heat treatment. Highly viscous gel spheres containing Co and Pd salts were successfully converted into multi thin-shelled Co3O4 reactors uniformly loaded with Pd catalysts by the sequential combustion of carbon and decomposition of the metal salts from the outer to the inner regions during one-step heat treatment. The responses (resistance ratio) of the Pd-loaded quintuple-shelled Co3O4 microreactors to 5 ppm toluene and p-xylene were 30.8 and 64.2, respectively, and the selectivity values to toluene and p-xylene against ethanol interference (response ratio) were 14.5 and 30.1, respectively. The unprecedented high response and selectivity were attributed to the effective dissociation of less reactive methylbenzenes into more active smaller species assisted both by catalytic Co3O4 and Pd during the prolonged retention within the microreactors. Kilogram-scale preparation of noble metal-loaded multishelled microreactors and their unique gas-sensing characteristics based on a novel microreactor concept can pave a new way to design of high-performance gas sensors for practical applications.

Slow motion of spherical droplet in a micropolar fluid flow perpendicular to a planar solid surface

The Stokes axisymmetrical flow problem of a viscous fluid sphere moving perpendicular to an impermeable bounding surface within a micropolar stagnant fluid as well as the related problem of a micropolar fluid sphere moving perpendicular to an impermeable planar surface within a stagnant viscous fluid are considered. The fluids are considered to be incompressible, and the deformation of the fluid particle is neglected. A general solution is constructed from fundamental solutions in both cylindrical and spherical coordinate systems. As boundary conditions, continuity of velocity, continuity of shear stress and the spin–vorticity relation at the droplet surface are applied. Also the no-slip and no-spin boundary conditions are used at the impermeable plane surface. A combined analytical-numerical procedure based on collocation technique is used. The drag acting, in each case, on the fluid particle is evaluated with good convergence. Numerical results for the normalized hydrodynamic drag force versus the relative viscosity, relative separation distance between the particle and wall, micropolarity parameter (a viscosity ratio characterizing micropolar fluids) and spin parameter (a non-dimensional scalar factor relating the microrotation and vorticity at the droplet surface) are presented both in tabular and graphical forms. The results for the drag coefficient are in good agreement with the available solutions in the literature for the limiting cases.

Cell models for micropolar flow past a viscous fluid sphere

The Stokes axisymmetrical flow of an incompressible micropolar fluid past a viscous fluid sphere and the flow of a viscous fluid past a micropolar fluid sphere are investigated. The appropriate boundary conditions are taken on the surface of the sphere, while the proper conditions applied on the fictitious boundary of the fluid envelope vary depending on the kind of cell-model. These problems are solved separately in an analytical fashion, and the velocity profile and the pressure distribution inside and outside of the droplet are shown in several graphs for different values of the parameters. Numerical results for the normalized hydrodynamic drag force acting, in each case, on the spherical droplet-in-cell are obtained for various values of the parameters representing volume fraction, the classical relative viscosity, the micropolarity and spin parameters are presented both in tabular and graphical forms. Results of the drag force are compared with the previous particular cases.

On the drag force of a viscous sphere with interfacial slip at small but finite Reynolds numbers

We investigate the hydrodynamic drag force on a viscous sphere in a fluid of different viscosities at small but finite Reynolds numbers when interfacial slip is present at the surface of the sphere. The sphere is small enough for it to retain its spherical shape, as is the case with most small droplets. By using a singular perturbation method, the exterior flow field of the droplet is decomposed into an inner region, where the viscous effects dominate, and an outer region, where the inertia is important. The interior flow of the viscous sphere is also solved analytically. By applying appropriate boundary conditions to the surface of the viscous sphere and matching the conditions between the inner and outer flow fields, stream functions up to the order of Re2 log Re for both the exterior and the interior flow are obtained. Thus, an analytical expression for the drag force coefficient of the viscous droplet is derived. This general expression yields, as special cases, several other expressions that are applicable to spheres that translate rectilinearly under more restrictive conditions. One of the practical conclusions from this study is that the presence of interfacial slip can significantly reduce the drag force on a droplet.

Model for the Scaling of Stresses and Fluctuations in Flows near Jamming

We probe flows of soft, viscous spheres near the jamming point, which acts as a critical point for static soft spheres. Starting from energy considerations, we find nontrivial scaling of velocity fluctuations with strain rate. Combining this scaling with insights from jamming, we arrive at an analytical model that predicts four distinct regimes of flow, each characterized by rational-valued scaling exponents. Both the number of regimes and the values of the exponents depart from prior results. We validate predictions of the model with simulations.

Viscosity of the Earth*

Direct and indirect estimates of the variation of viscosity with depth in the mantle indicate that a low viscosity layer exists in the upper mantle. A viscosity varying with depth can be used to reconcile the various estimates of relaxation times. If the seismic anelasticity can be used as a guide the average viscosity of the lower mantle is about 10^(23)P. Combined with previous estimates of the upper mantle viscosity this gives a relaxation time of about 3000 years for the non-equilibrium bulge of the Earth. This is close to the time from the last ice age but is much less than the 10^7 years required if the non-equilibrium bulge is due to the changing rate of rotation which requires an average mantle viscosity of 10^(26) P. If the latter value is correct the activation volume for creep is much larger than for anelasticity or the effect of a phase change in the upper mantle is more effective in suppressing creep than attenuation. The response of a layered viscous sphere to a surface load is calculated for a wide range of parameters including the above range of estimates for lower mantle viscosity. These results can be used to estimate the decay time, or the isostatic time scale, for various sized features.