scholarly journals Regions of Positive Vorticity in Steady Axisymmetric Flow Past a Viscous Spherical Drop

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
G. M. Moremedi ◽  
D. P. Mason
Keyword(s):  
Reynolds Number ◽  
Viscous Liquid ◽  
Liquid Drop ◽  
Axisymmetric Flow ◽  
Steady Motion ◽  
Perturbation Solution ◽  
Spherical Drop ◽  
First Order ◽  
Positive Vorticity ◽  
Unbounded Fluid

The vorticity exterior and interior to a viscous liquid drop in steady motion in an unbounded fluid is investigated. The perturbation solution to first order in the Reynolds number derived by Taylor and Acrivos (1964) is used. New analytical results are derived for the attached region of positive vorticity behind the drop and for the region of positive vorticity inside the drop.

The motion of a spherical liquid drop at high Reynolds number

1968 ◽  
Vol 32 (2) ◽  
pp. 367-391 ◽  
Author(s):  
J. F. Harper ◽  
D. W. Moore
Keyword(s):  
Reynolds Number ◽  
Boundary Layers ◽  
Liquid Drop ◽  
Spherical Surface ◽  
Linear Equations ◽  
Steady Motion ◽  
Reynolds Numbers ◽  
Internal Circulation ◽  
High Reynolds ◽  
Experimental Values

The steady motion of a liquid drop in another liquid of comparable density and viscosity is studied theoretically. Both inside and outside the drop, the Reynolds number is taken to be large enough for boundary-layer theory to hold, but small enough for surface tension to keep the drop nearly spherical. Surface-active impurities are assumed absent. We investigate the boundary layers associated with the inviscid first approximation to the flow, which is shown to be Hill's spherical vortex inside, and potential flow outside. The boundary layers are shown to perturb the velocity field only slightly at high Reynolds numbers, and to obey linear equations which are used to find first and second approximations to the drag coefficient and the rate of internal circulation.Drag coefficients calculated from the theory agree quite well with experimental values for liquids which satisfy the conditions of the theory. There appear to be no experimental results available to test our prediction of the internal circulation.

Radial lift on a suspended finite-sized sphere due to fluid inertia for low-Reynolds-number flow through a cylinder

2013 ◽  
Vol 722 ◽  
pp. 159-186 ◽  
Author(s):  
Sukalyan Bhattacharya ◽  
Dil K. Gurung ◽  
Shahin Navardi
Keyword(s):  
Reynolds Number ◽  
Particle Diameter ◽  
Axial Direction ◽  
Perturbation Scheme ◽  
Fluid Inertia ◽  
Regular Perturbation ◽  
Cross Sectional ◽  
First Order ◽  
Reynolds Number Flow ◽  
Freely Suspended

AbstractThis article describes the radial drift of a suspended sphere in a cylinder-bound Poiseuille flow where the Reynolds number is small but finite. Unlike past studies, it considers a circular narrow conduit whose cross-sectional diameter is only $1. 5$–$6$ times the particle diameter. Thus, the analysis quantifies the effect of fluid inertia on the radial motion of the particle in the channel when the flow field is significantly influenced by the presence of the suspended body. To this end, the hydrodynamic fields are expanded as a series in Reynolds number, and a set of hierarchical equations for different orders of the expansion is derived. Accordingly, the zeroth-order fields in Reynolds number satisfy the Stokes equation, which is accurately solved in the presence of the spherical particle and the cylindrical conduit. Then, recognizing that in narrow vessels Stokesian scattered fields from the sphere decrease exponentially in the axial direction, a simpler regular perturbation scheme is used to quantify the first-order inertial correction to hydrodynamic quantities. Consequently, it is possible to obtain two results. First, the sphere is assumed to follow the axial motion of a freely suspended sphere in a Stokesian condition, and the radial lift force on it due to the presence of fluid inertia is evaluated. Then, the approximate motion is determined for a freely suspended body on which net hydrodynamic force including first-order inertial lift is zero. The results agree well with the available experimental results. Thus, this study along with the measured data would precisely describe particle dynamics inside narrow tubes.

Scattering of elastic waves near an explosion

1993 ◽  
Vol 83 (4) ◽  
pp. 1277-1293
Author(s):  
Donald Leavy
Keyword(s):  
Displacement Field ◽  
Elastic Waves ◽  
Near Field ◽  
Static Solution ◽  
Simple Relation ◽  
Perturbation Solution ◽  
First Order ◽  
Scattering Of Elastic Waves ◽  
Harmonic Decomposition ◽  
Complete Set

Abstract We use the method of small perturbation to study the scattered waves generated by an arbitrary 3D inhomogeneous medium around a spherically symmetric compressional source. We consider two models of the medium inside the source: a homogeneous solid and a fluid. The results from these two models differ only when scattering occurs within a few source's radii from the explosion. We find that there is a simple relation between the structure of the first order scattered waves and the structure of the medium, namely that a given harmonic of the medium parameters excites only the same harmonic of the two spheroidal potentials. When scattering occurs within a wavelength from the source, we find that the quadrantal terms in the spherical harmonic decomposition of the field have the lowest frequency dependence. They depend on frequency only through the spectrum of the source. Thus, in the far field, the dominant scattered waves generated near an explosion are similar to the primary waves generated by an earthquake. However, when the displacement field is observed in the near field of the explosion, the static solution reveals that a complete set of harmonics may be required to properly account for the displacement field. We compare the perturbation solution with the exact solution of the scattering by a sphere located within a wavelength from the source. This suggests that the perturbation solution has a fairly wide domain of practical applicability. We attempt to apply these results to the Love wave generated near the Boxcar nuclear explosion.

scholarly journals Deformation of a magnetic liquid drop in an applied non-stationary uniform magnetic field

2018 ◽  
Vol 185 ◽  
pp. 09006
Author(s):  
Alexander Tyatyushkin
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Steady State ◽  
Uniform Magnetic Field ◽  
Liquid Drop ◽  
Magnetic Liquid ◽  
Stationary Approximation ◽  
The Magnetic Field

Small steady-state deformational oscillations of a drop of magnetic liquid in a nonstationary uniform magnetic field are theoretically investigated. The drop is suspended in another magnetic liquid immiscible with the former. The Reynolds number is so small that the inertia can be neglected. The variation of the magnetic field is so slow that the quasi-stationary approximation for the magnetic field and the quasi-steady approximation for the flow may be used.

scholarly journals Steady-State Three-Dimensional Ice Flow over an Undulating Base: First-Order Theory with Linear Ice Rheology

1987 ◽  
Vol 33 (114) ◽  
pp. 177-185 ◽  
Author(s):  
Niels Reeh
Keyword(s):  
Three Dimensional ◽  
Steady Motion ◽  
Simple Shear Flow ◽  
Finite Width ◽  
Two Dimensional ◽  
Internal Layers ◽  
Ice Flow ◽  
Depth Variation ◽  
First Order ◽  
First Order Theory

AbstractThe problem of ice flow over threedimensional basal irregularities is studied by considering the steady motion of a fluid with a linear constitutive equation over sine-shaped basal undulations. The undisturbed flow is simple shear flow with constant depth. Using the ratio of the amplitude of the basal undulations to the ice thickness as perturbation parameter, equations to the first order for the velocity and pressure perturbations are set up and solved.The study shows that when the widths of the basal undulations are larger than 2–3 times their lengths, the finite width of the undulations has only a minor influence on the flow, which to a good approximation may be considered two-dimensional. However, as the ratio between the longitudinal and the transverse wavelengthL/Wincreases, the three-dimensional flow effects becomes substantial. If, for example, the ratio ofLtoWexceeds 3, surface amplitudes are reduced by more than one order of magnitude as compared to the two-dimensional case. TheL/Wratio also influences the depth variation of the amplitudes of internal layers and the depth variation of perturbation velocities and strain-rates. With increasingL/Wratio, the changes of these quantities are concentrated in a near-bottom layer of decreasing thickness. Furthermore, it is shown, that the azimuth of the velocity vector may change by up to 10° between the surface and the base of the ice sheet, and that significant transverse flow may occur at depth without manifesting itself at the surface to any significant degree.

Note on the motion of liquid near a position of separation

1950 ◽  
Vol 46 (2) ◽  
pp. 293-306 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Viscous Liquid ◽  
Stream Function ◽  
Steady Motion ◽  
Double Series ◽  
Constant Density

1. A steady motion of viscous liquid of constant density is considered in §§ 2–7; full allowance is made for the inertia of the motion, and it is assumed that the stream-function can be expanded in a double series in the coordinates x, y.

scholarly journals The acoustic impedance of a laminar viscous jet through a thin circular aperture

2019 ◽  
Vol 864 ◽  
pp. 5-44 ◽  
Author(s):  
David Fabre ◽  
Raffaele Longobardi ◽  
Paul Bonnefis ◽  
Paolo Luchini
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
Classical Problem ◽  
Base Flow ◽  
Large Reynolds Number ◽  
Linear Perturbation ◽  
Circular Aperture ◽  
Vena Contracta ◽  
Numerical Resolution

The unsteady axisymmetric flow through a circular aperture in a thin plate subjected to harmonic forcing (for instance under the effect of an incident acoustic wave) is a classical problem first considered by Howe (Proc. R. Soc. Lond. A, vol. 366, 1979, pp. 205–223), using an inviscid model. The purpose of this work is to reconsider this problem through a numerical resolution of the incompressible linearized Navier–Stokes equations (LNSE) in the laminar regime, corresponding to $Re=[500,5000]$. We first compute a steady base flow which allows us to describe the vena contracta phenomenon in agreement with experiments. We then solve a linear problem allowing us to characterize both the spatial amplification of the perturbations and the impedance (or equivalently the Rayleigh conductivity), which is a key quantity to investigate the response of the jet to acoustic forcing. Since the linear perturbation is characterized by a strong spatial amplification, the numerical resolution requires the use of a complex mapping of the axial coordinate in order to enlarge the range of Reynolds number investigated. The results show that the impedances computed with $Re\gtrsim 1500$ collapse onto a single curve, indicating that a large Reynolds number asymptotic regime is effectively reached. However, expressing the results in terms of conductivity leads to substantial deviation with respect to Howe’s model. Finally, we investigate the case of finite-amplitude perturbations through direct numerical simulations (DNS). We show that the impedance predicted by the linear approach remains valid for amplitudes up to order $10^{-1}$, despite the fact that the spatial evolution of the perturbations in the jet is strongly nonlinear.

scholarly journals Steady-State Three-Dimensional Ice Flow over an Undulating Base: First-Order Theory with Linear Ice Rheology

1987 ◽  
Vol 33 (114) ◽  
pp. 177-185 ◽  
Author(s):  
Niels Reeh
Keyword(s):  
Three Dimensional ◽  
Steady Motion ◽  
Simple Shear Flow ◽  
Finite Width ◽  
Two Dimensional ◽  
Internal Layers ◽  
Ice Flow ◽  
Depth Variation ◽  
First Order ◽  
First Order Theory

AbstractThe problem of ice flow over threedimensional basal irregularities is studied by considering the steady motion of a fluid with a linear constitutive equation over sine-shaped basal undulations. The undisturbed flow is simple shear flow with constant depth. Using the ratio of the amplitude of the basal undulations to the ice thickness as perturbation parameter, equations to the first order for the velocity and pressure perturbations are set up and solved.The study shows that when the widths of the basal undulations are larger than 2–3 times their lengths, the finite width of the undulations has only a minor influence on the flow, which to a good approximation may be considered two-dimensional. However, as the ratio between the longitudinal and the transverse wavelength L/W increases, the three-dimensional flow effects becomes substantial. If, for example, the ratio of L to W exceeds 3, surface amplitudes are reduced by more than one order of magnitude as compared to the two-dimensional case. The L/W ratio also influences the depth variation of the amplitudes of internal layers and the depth variation of perturbation velocities and strain-rates. With increasing L/W ratio, the changes of these quantities are concentrated in a near-bottom layer of decreasing thickness. Furthermore, it is shown, that the azimuth of the velocity vector may change by up to 10° between the surface and the base of the ice sheet, and that significant transverse flow may occur at depth without manifesting itself at the surface to any significant degree.

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