On the motion of liquid in a spheroidal cavity of a precessing rigid body

1963 ◽  
Vol 17 (1) ◽  
pp. 1-20 ◽  
Author(s):  
K. Stewartson ◽  
P. H. Roberts
Keyword(s):  
Boundary Layer ◽  
Angular Velocity ◽  
Rigid Body ◽  
Steady Motion ◽  
Viscous Boundary Layer ◽  
Constant Vorticity ◽  
Spheroidal Cavity ◽  
Axis Of Symmetry ◽  
Set Up ◽  
Viscous Boundary

The flow set up in an oblate cavity of a precessing rigid body is examined under the assumptions that the ellipticity of the spheroidal boundary of the fluid is large compared with Ω/ω and that the boundary-layer thickness is small compared with the deviations of the boundary from sphericity (ωis the angular velocity of the rigid body about the axis of symmetry,Ωis the angular velocity with which this axis precesses).The motion of the fluid is found by considering an initial-value problem in which the axis of rotation of the spheroid is impulsively moved at a timet= 0; before that time this axis is supposed to be fixed in space, the fluid and envelope turning about it as a solid body. The solution is divided into a steady motion and transients, and, by evaluating the effects of the viscous boundary layer, the transients are shown to decay with time. The steady motion which remains consists of a primary rigid-body rotation with the envelope, superimposed on which is a circulation with constant vorticity in planes perpendicular toω× (ω×Ω), the streamlines being similarly situated ellipses.The possible effects of the luni-solar precession on the fluid motions in the Earth's core are discussed.

scholarly journals Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration

2012 ◽  
Vol 692 ◽  
pp. 420-445 ◽  
Author(s):  
Keke Zhang ◽  
Kit H. Chan ◽  
Xinhao Liao
Keyword(s):  
Numerical Simulation ◽  
Boundary Layer ◽  
Asymptotic Solution ◽  
Slip Boundary Condition ◽  
Viscous Boundary Layer ◽  
Slip Boundary ◽  
Oblate Spheroidal ◽  
Spheroidal Cavity ◽  
Viscous Boundary

AbstractWe consider a homogeneous fluid of viscosity $\nu $ confined within an oblate spheroidal cavity, ${x}^{2} / {a}^{2} + {y}^{2} / {a}^{2} + {z}^{2} / ({a}^{2} (1\ensuremath{-} {\mathscr{E}}^{2} ))= 1$, with eccentricity $0\lt \mathscr{E}\lt 1$. The spheroidal container rotates rapidly with an angular velocity ${\mbit{\Omega} }_{0} $, which is fixed in an inertial frame and defines a small Ekman number $E= \nu / ({a}^{2} \vert {\mbit{\Omega} }_{0} \vert )$, and undergoes weak latitudinal libration with frequency $\hat {\omega } \vert {\mbit{\Omega} }_{0} \vert $ and amplitude $\mathit{Po}\vert {\mbit{\Omega} }_{0} \vert $, where $\mathit{Po}$ is the Poincaré number quantifying the strength of Poincaré force resulting from latitudinal libration. We investigate, via both asymptotic and numerical analysis, fluid motion in the spheroidal cavity driven by latitudinal libration. When $\vert \hat {\omega } \ensuremath{-} 2/ (2\ensuremath{-} {\mathscr{E}}^{2} )\vert \gg O({E}^{1/ 2} )$, an asymptotic solution for $E\ll 1$ and $\mathit{Po}\ll 1$ in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial–temporal structure of the librating flow. In this case, the librationally driven flow is non-axisymmetric with amplitude $O(\mathit{Po})$, and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When $\vert \hat {\omega } \ensuremath{-} 2/ (2\ensuremath{-} {\mathscr{E}}^{2} )\vert \ll O({E}^{1/ 2} )$, the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes $O(\mathit{Po}/ {E}^{1/ 2} )$ at $E\ll 1$ and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for $E\ll 1$ describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at $E\ll 1$.

scholarly journals Moisture gradients form a vapor cycle within the viscous boundary layer as an organizing principle to worker termites

2018 ◽  
Vol 66 (2) ◽  
pp. 193-209 ◽  
Author(s):  
R. Soar ◽  
G. Amador ◽  
P. Bardunias ◽  
J. S. Turner
Keyword(s):  
Boundary Layer ◽  
Viscous Boundary Layer ◽  
Viscous Boundary

Viscous boundary layers in reversing flow

1976 ◽  
Vol 74 (1) ◽  
pp. 59-79 ◽  
Author(s):  
T. J. Pedley
Keyword(s):  
Boundary Layer ◽  
Shear Rate ◽  
Leading Edge ◽  
Wall Shear Rate ◽  
Viscous Boundary Layer ◽  
Large Molecules ◽  
Displacement Thickness ◽  
The Mean ◽  
Wall Shear ◽  
Viscous Boundary

The viscous boundary layer on a finite flat plate in a stream which reverses its direction once (at t = 0) is analysed using an improved version of the approximate method described earlier (Pedley 1975). Long before reversal (t < −t1), the flow at a point on the plate will be quasi-steady; long after reversal (t > t2), the flow will again be quasi-steady, but with the leading edge at the other end of the plate. In between (−t1 < t < t2) the flow is governed approximately by the diffusion equation, and we choose a simple solution of that equation which ensures that the displacement thickness of the boundary layer remains constant at t = −t1. The results of the theory, in the form of the wall shear rate at a point as a function of time, are given both for a uniformly decelerating stream, and for a sinusoidally oscillating stream which reverses its direction twice every cycle. The theory is further modified to cover streams which do not reverse, but for which the quasi-steady solution breaks down because the velocity becomes very small. The analysis is also applied to predict the wall shear rate at the entrance to a straight pipe when the core velocity varies with time as in a dog's aorta. The results show positive and negative peak values of shear very much larger than the mean. They suggest that, if wall shear is implicated in the generation of atherosclerosis because it alters the permeability of the wall to large molecules, then an appropriate index of wall shear at a point is more likely to be the r.m.s. value than the mean.

On one-dimensional unsteady flow at infinite Mach number

1968 ◽  
Vol 304 (1478) ◽  
pp. 255-273 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Blast Wave ◽  
Outer Part ◽  
The Body ◽  
Viscous Boundary Layer ◽  
Outer Flow ◽  
Hypersonic Speeds ◽  
Set Up ◽  
Viscous Boundary ◽  
Infinite Set

The use of the blast-wave analogy, as an aid to the interpretation of experimental data on the motion of a fluid past an obstacle at hypersonic speeds, has led to the theoretical study of its role in an asymptotic expansion of the solution to the governing equations at large distances downstream of the body. In all attempts to set up such an expansion it has proved necessary to divide the flow régime into two parts, an outer part dominated by the blast wave and an inner part consisting of streamlines which, originally, pass close by the body. The matching of these two regions is apparently only possible if a certain integral vanishes. In the present paper a numerical integration, in one particular set of circumstances, is carried out to test the validity of the asymptotic expansion proposed. Formally, an unsteady problem is tackled, for ease of computation, but the steady analogue follows immediately and is of exactly the form discussed in the earlier investigations. It is found that the main results are in line with the theory and that the integral in question is indistinguishable from zero. However, a deeper investigation of the asymptotic expansion shows that, for an expansion of the type envisaged, an infinite set of integrals must each vanish. The next integral does not appear to be zero according to our computations but this result is not believed to be conclusive. Assuming that all the integrals do vanish, then it appears that the inner layer, which although inviscid, has many of the characteristics of a viscous boundary layer, has the addi­tional, surprising property that it can exert no direct influence on the outer flow at large distances downstream of the body.

Jet Impingement of a Non-Newtonian Fluid

2006 ◽  
Author(s):  
Jiangang Zhao ◽  
Roger E. Khayat
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Hydraulic Jump ◽  
Similarity Solutions ◽  
Surface Velocity ◽  
Free Surface Velocity ◽  
Viscous Boundary Layer ◽  
Boundary Layer Region ◽  
Layer Region ◽  
Viscous Boundary

The similarity solutions are presented for the wall flow which is formed when a smooth planar jet of power-law fluids impinges vertically on to a horizontal plate, and spreads out in a thin layer bounded by a hydraulic jump. This problem is formulated analogous to radial jet flow problem and the solution procedure is accounted for by means of similarity solution of the boundary-layer equation [1] for Newtonian fluids. For the convenience of analysis, the flow may be divided into three regions, namely a developing boundary-layer region, a fully viscous boundary-layer region, and a hydraulic jump region. The similarity solutions of the film thickness and free surface velocity in fully viscous boundary-layer region include unknown constant L, which is solved numerically and approximately in the developing boundary-layer flow region. Comparison between the numerical and approximate solutions leads generally to good agreement, except for severely shear-thinning fluids. The boundary-layer solution depends on two parameters: power-law index n and α, the dimensionless flow parameters. The effect of α on film thickness and free surface velocity is investigated. The relations between the position of the hydraulic jump and dimensionless flow parameter are obtained and the effect of α on the position of the jump is presented.

Mass transport in water waves over an elastic bed

1995 ◽  
Vol 450 (1939) ◽  
pp. 371-390 ◽  
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Mass Transport ◽  
Water Waves ◽  
Viscous Boundary Layer ◽  
Curvilinear Coordinate ◽  
Transport Velocity ◽  
Mass Transport Velocity ◽  
Orthogonal Curvilinear Coordinate System ◽  
Viscous Boundary

The mass transport velocity in water waves propagating over an elastic bed is investigated. Water is assumed to be incompressible and slightly viscous. The elastic bed is also incompressible and satisfies the Hooke’s law. For a small amplitude progressive wave perturbation solutions via a boundary-layer approach are obtained. Because the wave amplitude is usually larger than the viscous boundary layer thickness and because the free surface and the interface between water and the elastic bed are moving, an orthogonal curvilinear coordinate system (Longuet-Higgins 1953) is used in the analysis of free surface and interfacial boundary layers so that boundary conditions can be applied on the actual moving surfaces. Analytical solutions for the mass transport velocity inside the boundary layer adjacent to the elastic seabed and in the core region of the water column are obtained. The mass transport velocity above a soft elastic bed could be twice of that over a rigid bed in the shallow water.

The Reflection of a Stationary Gravity Wave by a Viscous Boundary Layer

2007 ◽  
Vol 64 (9) ◽  
pp. 3363-3371 ◽  
Author(s):  
François Lott
Keyword(s):  
Boundary Layer ◽  
Gravity Wave ◽  
Richardson Number ◽  
Critical Level ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Viscous Boundary Layer ◽  
Lee Waves ◽  
The Mean ◽  
Viscous Boundary

Abstract The backward reflection of a stationary gravity wave (GW) propagating toward the ground is examined in the linear viscous case and for large Reynolds numbers (Re). In this case, the stationary GW presents a critical level at the ground because the mean wind is null there. When the mean flow Richardson number at the surface (J) is below 0.25, the GW reflection by the viscous boundary layer is total in the inviscid limit Re → ∞. The GW is a little absorbed when Re is finite, and the reflection decreases when both the dissipation and J increase. When J &gt; 0.25, the GW is absorbed for all values of the Reynolds number, with a general tendency for the GW reflection to decrease when J increases. As a large ground reflection favors the downstream development of a trapped lee wave, the fact that it decreases when J increases explains why the more unstable boundary layers favor the onset of mountain lee waves. It is also shown that the GW reflection when J &gt; 0.25 is substantially larger than that predicted by the conventional inviscid critical level theory and larger than that predicted when the dissipations are represented by Rayleigh friction and Newtonian cooling. The fact that the GW reflection depends strongly on the Richardson number indicates that there is some correspondence between the dynamics of trapped lee waves and the dynamics of Kelvin–Helmholtz instabilities. Accordingly, and in one classical example, it is shown that some among the neutral modes for Kelvin–Helmholtz instabilities that exist in an unbounded flow when J &lt; 0.25 can also be stationary trapped-wave solutions when there is a ground and in the inviscid limit Re → ∞. When Re is finite, these solutions are affected by the dissipation in the boundary layer and decay in the downstream direction. Interestingly, their decay rate increases when both the dissipation and J increase, as does the GW absorption by the viscous boundary layer.

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