Viscous Equations Of Motion Research Articles

Benchmarked computation of time-domain viscoelastic Love numbers for adiabatic mantles

SUMMARYThe viscoelastic load Love numbers encapsulate the Earth’s rheology in a remarkably efficient fashion. When multiplied by a sudden increment of spherical harmonic load change, they give the horizontal and vertical surface displacements and gravity change at all later times. Incremental glacial load changes thus need only be harmonically decomposed, multiplied by the Love numbers and summed to predict the Earth’s response to glacial load redistributions. The computation of viscoelastic Love numbers from the elastic, viscous and adiabatic profiles of the Earth is thus the foundation upon which many glacial isostatic adjustment models are based. Usually, viscoelastic Love numbers are computed using the Laplace transform method, employing the correspondence principle to convert the viscoelastic equations of motion into the elastic equations with complex material parameters. This method works well for a fully non-adiabatic Earth, but can accommodate realistic partially adiabatic and fully adiabatic conditions only by changing the Earth’s density profile. An alternative method of Love number computation developed by Cathles (1975) avoids this dilemma by separating the elastic and viscous equations of motion. The separation neglects a small solid-elastic/fluid-elastic transition for compressible deformation, but allows freely defining adiabatic, partially adiabatic or fully non-adiabatic profiles in the mantle without changing the Earth’s density profile. Here, we update and fully describe this method and show that it produces Love numbers closely similar to those computed for fully non-adiabatic earth models computed by the correspondence principle, finite element and other methods. The time-domain method produces Love numbers as good as those produced by other methods and can also realistically accommodate any degree of mantle adiabaticity. All method implementations are available open source.

The Dam Break Problem For A Hyperconcentrated Suspension

Abstract A rheological approach of muddy debris flows is presented for environmental engineering and risk assessment purposes. The debris flow is viewed as a hyperconcentrated suspension of non brownian solid particles. Experimental and theoretical studies are presented versus the concentration. The experimental work generally consists in generating the flow of well characterized suspensions by a dam break. Then, the time evolution of the front as well as that of the stage at given position are accurately determined. On the theoretical framework, the viscous equations of motion of the suspension are written with the shallow water approximation, and the asymptotic solution in a non dimensional form is derived. Finally, the experimental and the theoretical results are successfully compared and the limits of this rheological approach is fixed.

On the Mach- and Reynolds-Number Dependence of the Flat-Plate Turbulent Boundary Layer Wall-Pressure Spectrum

Wall-pressure measurements are performed using an array of pinhole microphones mounted into the test wall of a low noise, low turbulence flow facility. The frequency–wave-number spectra for both streamwise and spanwise directions are obtained through the spatial Fourier transformation of cross-spectral measurements in the said directions. The results of the wind-tunnel measurements reveal an interesting behavior of the low wave-number portion of the wall-pressure spectrum. Namely, this portion of the spectrum does not exhibit any Mach-number dependence for nearly incompressible flows. A low Mach-number analysis accounting for finite Reynolds-number effects is performed. This analysis shows the wall-pressure spectrum to have a leading term independent of the Mach number. The frequency dependence of the theoretical spectrum agrees well with that of the experimental spectrum. The physical mechanism for the Mach-number independence of the spectrum is attributed to a hydrodynamic dipole contribution to the wall pressure resulting from the Stokes layer induced by the turbulent eddies. The higher differential order of the viscous equations of motion allows for the existence of such Stokes layers. They have no counterpart in invisicd theories wherein no such Mach-number independence of the spectrum has been observed. A simple model problem is devised which isolates the viscous mechanism responsible for the modification of the wall pressure and the corresponding sound field. It is argued that the physical process involved is akin (albeit inversely) to that operative in the boundary-layer receptivity problem for nearly incompressible flows.

A viscohyperelastic finite element model for rubber

A recently developed finite deformation internal solid theory for viscoelastic deformations of rubber is reviewed and an axisymmetric finite element implementation of the theory is presented. The theoretical model is a three-dimensional, total Lagrangian, finite deformation generalization of the Maxwell internal solid theory for viscoelasticity. It utilizes time dependent reference shapes for internal hyperelastic solids to approximate the viscous effects found in rubber and is called a viscohyperelastic model. The material constants for the sum of the internal solids' hyperelastic energy functions can be found by performing the classical tensile, biaxial and shear tests as step-strain tests. A nearly incompressible finite element version of the theory is presented. The viscous equations of motion are solved with a predictor-corrector integration scheme. Time dependent pressure loads are applied to the interior of both a thick walled rubber cylinder and sphere to numerically demonstrate the viscohyperelastic finite element algorithm.

The motion of a rising disk in a rotating axially bounded fluid for large Taylor number

The motion of a disk rising steadily along the axis in a rotating fluid between two infinite plates is considered. In the limit of zero Rossby number and with the disk in the middle position, the boundary value problem based on the linear, viscous equations of motion is reduced to a system of dual-integral equations which renders ‘exact’ solutions for arbitrary values of the Taylor number, Ta, and disk-to-wall distance, H (scaled by the radius of the disk). The investigation is focused on the drag and on the flow field when Ta is large (but finite) for various H. Comparisons with previous asymptotic results for ‘short’ and ‘long’ containers, and with the preceding unbounded-configuration ‘exact’ solution, provide both confirmation and novel insights.In particular, it is shown that the ‘free’ Taylor column on the particle appears for H > 0.08 Ta and attains its fully developed features when H > 0.25 Ta (approximately). The present drag calculations improve the compatibility of the linear theory with Maxworthy's (1968) experiments in short containers, but for the long container the claimed discrepancy with experiments remains unexplained.

Viscous flow and non-linear phenomena in non-equilibrium thermodynamics of membrane transport Part 2 — The phenomenological equations

Phenomenological equations are deduced which give the flows of matter, volume, charge and heat for a discontinuous system in which inertial terms in the viscous equations of motion are not negligible. It is found that the equations contain terms in the powers of the first-order affinities, with corresponding phenomenological coefficients. These coefficients can all be derived from first-order coefficients, however, and thus refer to a system which is close to equilibrium. The theory agrees with well-documented specific hydrodynamic calculations, but generalizes these in the framework of now equilibrium thermodynamics.

Stability Analysis and Integration of the Viscous Equations of Motion

Stability Analysis and Integration of the Viscous Equations of Motion

The asymptotic expansions at large Reynolds numbers for steady motion between non-coaxial rotating cylinders

The motion of a viscous fluid contained between two rotating, circular cylinders whose axes are set slightly apart is considered. The equations of viscous motion are linearized by expanding the stream function in the form , the terms ζ(n)1 of the corresponding expansions for the vorticity are shown to be uniform throughout the fluid. It is noted that the asymptotic expansions of ψ1 for the region of the boundary layers and for the region outside the boundary layers may be combined in a single expansion which holds in both regions. The leading terms of this expansion are calculated by boundary layer methods.

The Theory of Air Flow

IT is a common experience, and by no means confined to engineers, to fail to understand the general theory of non‐rectilinear flow of air. An investigation culminates, of course, in the well ‐ known equations of viscous motion. But, as Dr. Prescott remarked recently when introducing his new proof, many derivations of these are cither unsatisfying or else difficult to follow. Indeed, the equations are usually taken for granted. Yet the disadvantage is obvious: even the mathematician can do very little with them in the end, and fundamental ideas on which the proof rests, though likely to be of general service to engineers, lie buried. An example is perhaps worth while. Consider the problem of determining the viscosity of an oil from measurement of the torque required slowly to revolve a circular cylinder in a tank or pot of it. The equations of motion can be solved in this case, but it is not necessary to go so far; there is a well‐known treatment much more to the heart of an engineer. On the other hand, the usual theory of rectilinear viscous flow is without avail, and if the engineer can solve this simple problem without the equations then he understands the bulk of the proof of them.

The Equations of Viscous Motion and the Circulation Theorem

The following derivation of the equations of viscous motion, and, by analogy, the equations of motion of an isotropic elastic solid, appears to have an advantage in brevity over those in general use.