Flow of a liquid with an anisotropic viscosity tensor

This paper presents a simple model of an anisotropic incompressible viscous fluid, whose equations of state involve one anisotropic physical constant tensor (in the sense of Oldroyd [1]). Attention is restricted to the case of a fluid that is everywhere transversely isotropic at some given instant, so that the model is essentially one of a liquid with initially just one privileged direction at each point. Transverse isotropy does not persist, however, in some flow situations.Predictions are made for simple shearing flows and for channel and pipe flows with different initial directions of orientation. In some cases, the volume rate of flow under constant pressure-gradient decreases steadily and tends to zero after a long time.     

Oscillatory flows in straight tubes with an axisymmetric bulge

A laser-Doppler anemometer has been used to study oscillatory flow of a Newtonian viscous fluid in straight circular tube with an axisymmetric bulge of two different sizes. The axial velocities were measured at successive cross-sectional planes for sinusoidal waveforms having Reynolds numbers (based on Stokes layer thickness at the inlet) from 445 to 806 and Womersley numbers ranged from 7.2 to 12.2. The cyclic flow development inside the bulge at different phases within a cycle was determined. Stability analysis obtained by solving the Orr–Sommerfield equation on instantaneous velocity profiles showed instability grows progressively during the acceleration phase and transition to turbulence in the bulge happened shortly after the commencement of the deceleration phase. Depending on the bulge geometry, the turbulent region was initially confined either to the proximal or the distal end of the bulge. This region would spread larger as the deceleration phase furthered and the smaller bulge had a larger spread than the bigger bulge. The differences could be attributed to the vortical structures development inside the bulge. Relaminarisation for the flow appeared in the subsequent acceleration phase. Finally, some comparisons had been made with results obtained from using the physiological waveform.     

Magnetic hydrodynamics of two-dimensional flows in a plasma with an anisotropic pressure

The hydrodynamic equations of Chew, Goldberger, and Low [1] are used to analyze certain types of two-dimensional flows of a plasma with an anisotropic pressure (the pressure along the magnetic field p differs from the pressure across it p). In Sec. 1 the relationships derived in [2] for the transition of plasma state across surfaces of strong discontinuity are invoked to investigate the variation of the hydrodynamic parameters in weak shock waves in the linear approximation. The flow around bodies which only slightly perturb the main flow is investigated in Sec. 2 in the linear approximation. Similar problems for the case of an isotropic pressure are studied in detail in [3–5], for example.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–10, November–December, 1970.     

Calculation of cavity flows in an axisymmetric channel

The results are presented of a calculation of cavity flow in an axisymmetric channel with an annular obstacle. The problem was suggested to the author by G. B. Tsvetnov.The problem is solved by the method published in [1, 2].     

Flow and capacity tensor characteristics of anisotropic porous media: Theory and experiment

Tensors determining and describing the main flow and capacity characteristics of anisotropic media are introduced. A system of laboratorymeasurements for the determination of the tensors of areal porosity, permeability, limiting gradients, and characteristic linear dimensions is considered. The pressure dependences of permeability are generalized to take into account the medium anisotropy. The results of an experiment for the determination of the permeability and limiting (initial) gradient tensors in band sandstone with orthotropic flow characteristics are presented. The measurements were carried out on four cores: two along the principal stratification directions, one perpendicular to the stratification, and one inclined at an angle of 45° to the stratification plane. The latter (fourth) specimen was taken for reference: to test the tensor nature of the introduced mathematical objects and formulas. The good agreement of the theoretical and experimental results makes it possible to recommend for engineering calculations both the formulas proposed and the developed method of laboratory investigation of the flow characteristics of anisotropic reservoirs.     

Compliance and Hill polarization tensor of a crack in an anisotropic matrix

This work aims at developing an efficient method to compute the compliance due to a crack modeled as a flat ellipsoid of any shape in an infinite elastic matrix of arbitrary anisotropy (Eshelby problem) when no closed-form solution seems currently available. Whereas the solution of this problem usually requires the calculation of the so-called fourth-order Hill polarization tensor if the ellipsoid is not singular, it is shown that the crack compliance can be derived from the first-order term in the Taylor expansion of the Hill tensor with respect to the smallest aspect ratio of the ellipsoidal inclusion. For a 3D ellipsoidal crack model, this first-order term is expressed as a simple integral thanks to the Cauchy residue theorem. A similar method allows to express the same term in the case of a cylindrical crack model without any integral. A numerical example is finally treated.     

The elastodynamic Green's tensor for a half-space with an embedded anisotropic layer

The two-dimensional elastodynamic Green's tensor is found in the form of a generalized ray expansion, for an isotropic half-space in which is embedded an anisotropic layer. Particular attention is paid to the displacement of the free surface when the source transmits waves through the layer. The first motion approximation, in which individual terms are replaced by their asymptotic forms close to their arrival times, is shown to provide a fair representation without the massive computation that the full solution requires. An example which lacks symmetry shows that the layer can transmit significant SH waves from a source of P-SV type; this phenomenon is relevant to studies of the earth's upper mantle.     

Incompressible Poiseuille flows of Newtonian liquids with a pressure-dependent viscosity

The pressure-dependence of the viscosity becomes important in flows where high pressures are encountered. Applications include many polymer processing applications, microfluidics, fluid film lubrication, as well as simulations of geophysical flows. Under the assumption of unidirectional flow, we derive analytical solutions for plane, round, and annular Poiseuille flow of a Newtonian liquid, the viscosity of which increases linearly with pressure. These flows may serve as prototypes in applications involving tubes with small radius-to-length ratios. It is demonstrated that, the velocity tends from a parabolic to a triangular profile as the viscosity coefficient is increased. The pressure gradient near the exit is the same as that of the classical fully developed flow. This increases exponentially upstream and thus the pressure required to drive the flow increases dramatically.     

Asymptotic form of the admixture transport equation for a channel flow with an anisotropic diffusion tensor

The problem of the asymptotically correct reduction of a 3-D mass (heat) transfer equation to a 1-D equation in a flow with anisotropic diffusion properties is considered. The convective mass (heat) transfer domain is a cylindrical channel of arbitrary cross section. The diffusion coefficient matrix is assumed to be independent of the spatial coordinates. In the equivalent diffusion equation constructed, a certain effective diffusion (dispersion [1]) coefficient is introduced. Formulas for this coefficient are obtained. A relation between the effective diffusion coefficient calculations and the problem of minimization of a certain functional is established, i. e. the possibility of calculations based on variational methods is noted. An example of an exact calculation of the effective diffusion coefficient is considered. The possibility of a generalization of the problem, in which the effective diffusion (heat conduction) equation is essentially a nonlinear equation of general form for the one-dimensional case, is indicated. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 110–123, March–April, 2000.     

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 40–50, May–June, 2007.     

On the obtaining of axisymmetric flows from plane-parallel flows

The obtaining of axisymmetric flows of incompressible and compressible fluids from plane-parallel flows by means of integral transformations relating harmonic and p-harmonic functions [1] is considered. A transformation is found that carries plane-parallel flows from elementary singularities into axisymmetric flows. It is shown that this transformation makes it possible to obtain the general form of the solution of axisymmetric problems of flow past bodies from the solution of plane-parallel problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1982.     

Nonstationary axisymmetric electroelasticity problem for an anisotropic piezoceramic radially polarized cylinder

We consider an axisymmetric nonstationary electroelasticity problem for an anisotropic piezoceramic radially polarized cylinder of finite size whose lateral surface is subjected to an electric voltage that is an arbitrary function of the axial coordinate and time. A new closed-form solution is constructed by the vector eigenfunction expansion method in the form of a structural finite transform algorithm. This solution permits determining the natural vibration frequencies, the stress-strain state of an element, and the electric field potential and intensity. The results permit analyzing and optimizing the operation of inverse piezoelectric effect devices with cylindrical transducers.     

An experimental study of two flows through an axisymmetric sudden expansion

Two turbulent separated and reattaching flows produced by a sudden expansion in a pipe have been studied. The first was produced by a simple axisymmetric sudden enlargement from a nozzle of diameter 80 mm to a pipe of diameter 150 mm. The second was the flow at the same enlargement with the addition of a centerbody 90 mm downstream of the nozzle exit. Detailed measurements of velocity and skin friction (made primarily using pulsed wires) and of wall static pressure are presented. Without the centerbody the flow structure is similar to that observed in other sudden pipe expansions and over backward-facing steps. A turbulent free shear layer, bearing some similarity to that of a round jet, grows from separation and then reattaches to the pipe wall downstream. Reattachment is a comparatively gradual process, the shear layer approaching the wall at a glancing angle. The introduction of the centerbody causes the shear layer to curve towards the wall and reattach at a much steeper angle. Reattachment is much more rapid; gradients of skin friction and pressure along the wall are many times those without the centerbody. The high curvature of the shear layer strongly influences its turbulent structure, locally suppressing turbulence levels and reducing its growth rate.