Reflection of an impulse spherical wave at a plane interface separating two fluids

1992 ◽  
Vol 92 (3) ◽  
pp. 1706-1720 ◽  
Author(s):  
Didier Cassereau ◽  
Daniel Guyomar
Keyword(s):  
Spherical Wave ◽  
Plane Interface ◽  
Two Fluids

scholarly journals Motion of a sphere in the presence of a plane interface. Part 2. An exact solution in bipolar co-ordinates

1980 ◽  
Vol 98 (1) ◽  
pp. 193-224 ◽  
Author(s):  
S. H. Lee ◽  
L. G. Leal
Keyword(s):  
Exact Solution ◽  
General Solution ◽  
Approximate Solutions ◽  
Plane Interface ◽  
Spherical Particles ◽  
Fluid Interface ◽  
Two Fluids ◽  
Freely Suspended ◽  
Velocity Pressure ◽  
Particle Shapes

A general solution for Stokes’ equation in bipolar co-ordinates is derived, and then applied to the arbitrary motion of a sphere in the presence of a plane fluid/fluid interface. The drag force and hydrodynamic torque on the sphere are then calculated for four specific motions of the sphere; namely, translation perpendicular and parallel to the interface and rotation about an axis which is perpendicular and parallel, respectively, to the interface. The most significant result of the present work is the comparison between these numerically exact solutions and the approximate solutions from part 1. The latter can be generalized to a variety of particle shapes, and it is thus important to assess their accuracy for this case of spherical particles where an exact solution can be obtained. In addition to comparisons with the approximate solutions, we also examine the predicted changes in the velocity, pressure and vorticity fields due to the presence of the plane interface. One particularly interesting feature of the solutions is the fact that the direction of rotation of a freely suspended sphere moving parallel to the interface can either be the same as for a sphere rolling along the interface (as might be intuitively expected), or opposite depending upon the location of the sphere centre and the ratio of viscosities for the two fluids.

scholarly journals A novel expression of the spherical-wave reflection coefficient at a plane interface

2017 ◽  
Vol 211 (2) ◽  
pp. 700-717 ◽  
Author(s):  
Jingnan Li ◽  
Shangxu Wang ◽  
Yonghui Tao ◽  
Chunhui Dong ◽  
Genyang Tang
Keyword(s):  
Reflection Coefficient ◽  
Complex Plane ◽  
Wave Reflection ◽  
Incidence Angle ◽  
Velocity Ratio ◽  
Spherical Wave ◽  
Density Ratio ◽  
Plane Interface ◽  
Analytical Equation ◽  
Wave Number

Abstract The spherical-wave reflection coefficient (SRC) describes the reflection strength when seismic waves emanating from a point source impinge on an interface. In this study, the SRC at a plane interface between two infinite half-spaces is investigated. We derive an analytical equation of the SRC when kR → 0 (k is the wave number and R is the wave propagation distance). It only depends on the density ratio; it is independent of the velocity ratio and incidence angle. On the other hand, we find that the SRCs at different kR lie along an elliptical curve on the complex plane (the complex plane is a geometric representation of the complex numbers established by the real axis and perpendicular imaginary axis). Based on this feature, we construct a new analytical equation for the reflected spherical wave with high accuracy, which is applicable to both small and large kR. Using the elliptical distribution of the SRCs for a series of frequencies recorded at only one spatial location, the density and velocity ratios can be extracted. This study complements the spherical-wave reflection theory and provides a new basis for acoustic parameters inversion, particularly density inversion.

A New Approach to Instability Theory in Porous Media

1985 ◽  
Vol 25 (05) ◽  
pp. 765-779 ◽  
Author(s):  
R.G. Bentsen
Keyword(s):  
Porous Medium ◽  
Perturbation Theory ◽  
Velocity Potential ◽  
Plane Interface ◽  
New Approach ◽  
First Order ◽  
Two Fluids ◽  
Instability Theory ◽  
Perturbation Velocity ◽  
First Order Perturbation

Abstract Early work in the area of instability theory is limited in that it is based on first-order perturbation theory and the concept of a velocity potential. Thus, while it can deal with an incipient finger, it cannot deal with the subsequent growth of a finger. This paper develops a new approach to the instability theory that overcomes this limitation. The new approach, like earlier work, is based on the assumption that the immiscible displacement of one fluid by another can be treated as a moving-boundary problem. Therefore, two solutions arise, one for each side of the plane interface that initially separates the two fluids. plane interface that initially separates the two fluids. Because the new approach makes use of a force potential rather than a velocity potential, it is possible to impose several new conditions on these two solutions. As a consequence, further extensions to the stability theory have been obtained. In particular, it is now possible to predict the steady-state velocity at which a finger propagates and, consequently, the breakthrough recovery obtained not only when the displacement is stable, but also when it is pseudostable. pseudostable. Introduction It has been recognized for some time 1 that viscous fingering can play a significant role in the immiscible displacement of one fluid by another. This led a number of people to attempt a theoretical description of viscous fingering. Many of these attempts are based on first-order perturbation theory. When this approach is taken, it is usual to postulate the existence of a velocity potential. Then, if the divergence of the perturbation velocity is identically equal to zero, the postulated velocity potential will satisfy Laplace's equation. There are several problems with this approach. First, a velocity potential exists only for fields of flow involving a fluid of constant density and viscosity and a porous medium that is homogeneous and isotropic throughout. Thus it is debatable whether results based on the concept of a velocity potential have any application when the porous medium is not ideal. Second, the concept of a porous medium is not ideal. Second, the concept of a velocity potential has only kinematical significance and gives no insight whatsoever into the dynamic properties of the flow. As a consequence, when a velocity potential is used, one cannot infer how variations in rock and fluid properties might affect the perturbation velocity. properties might affect the perturbation velocity. Finally, the divergence of the perturbation velocity is zero only when a plane interface separates the displaced fluid from the displacing fluid. As soon as the interface is perturbed, two fluids will occupy a region where there was previously only one, and the divergence of the perturbation velocity will no longer be zero. Rather, the perturbation velocity will no longer be zero. Rather, the divergence of the sum of the perturbation velocities, one for each fluid, will be zero. Thus, while first-order perturbation theory can deal with an incipient finger, it perturbation theory can deal with an incipient finger, it cannot deal with the subsequent growth of the finger. The purpose of this paper is to develop a theory for viscous purpose of this paper is to develop a theory for viscous fingering that does not suffer from the aforementioned limitations that arise from the use of a velocity potential. The new theory, which is based on the concept of a force potential, not only explicitly accounts for the dynamic flow potential, not only explicitly accounts for the dynamic flow properties of both fluids, but also is capable of delineating properties of both fluids, but also is capable of delineating the entire life-span of the growing finger. Theory Fluid Displacement. To develop the theory presented in subsequent sections, it is necessary to have a working model of how one fluid displaces another. Such a model must be simple to keep the mathematics tractable. Consequently, it is usual to assume that the displaced and displacing fluids are separated initially by a plane interface and that the porous medium is ideal. If the porous medium is truly ideal (in the sense that the size and shape of all flow paths are identical), the displaced and displacing fluids will continue to be separated during the course of a displacement by a saturation discontinuity (or shock), provided the displacement is stable. 4 Thus stable provided the displacement is stable. 4 Thus stable displacements of this kind can be treated as a moving boundary problem. However, any real porous medium will contain a distribution of pore sizes. As a consequence, different parts of an initially plane interface will move at different parts of an initially plane interface will move at different velocities through the porous medium; i.e., a distribution of saturations will evolve to separate that part of the porous medium where only displacing fluid is flowing from that where only displaced fluid is flowing. The length and shape of this transition zone will depend not only on the pore-size distribution of the porous medium, but also on pore-size distribution of the porous medium, but also on the balance between viscous, gravity, and capillary forces that exists at each point along the saturation profile during the course of the displacement. Moreover, the shape of the saturation profile can be predicted by conventional displacement theory "if the displacement is stable. Even in an ideal porous medium, the displacement front separating the fluids may become unstable under certain conditions. Consequently, a distribution of saturations will evolve again to separate the two regions where only one fluid is flowing. However, the length and shape of this transition region will differ markedly from that which results from a distribution of pore sizes. SPEJ P. 765

SPHERICAL WAVE CORRECTIONS IN XAFS

1986 ◽  
Vol 47 (C8) ◽  
pp. C8-31-C8-35
Author(s):  
J. J. REHR ◽  
R. C. ALBERS ◽  
C. R. NATOLI ◽  
E. A. STERN
Keyword(s):  
Spherical Wave
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