Stability of multiple solutions in inclined gas/shear-thinning fluid stratified pipe flow

This work provides an investigation on multiple solutions in gas/shear-thinning fluid inclined stratified pipe flows. Multiple solution operative conditions are studied investigating the effect of the interfacial shear stress modeling and the rheology of the shear-thinning fluid. The modeling of the interfacial shear stress in counter-current has a strong influence of multiple solutions regions. The stability of multiple hold-up solutions is studied considering the structural stability, the interfacial stability, and the minimization of the dissipation approaches. The results of the three different approaches are commented both for concurrent and counter-current flows, giving the same conclusions only for upward inclined flows.     

Performance of a flapping foil flow energy harvester in shear flows

Through numerical simulations, we investigate the energy harvesting performance of a heaving/pitching foil in shear flow. With two-dimensional Navier–Stokes simulations, we examined the energy harvesting efficiencies of such a system in linear shear flows and compared the results with those in uniform flows. It is found that in low shear rates, the performance of the system in linear shear flow is slightly higher than that in uniform flow, whereas the energy harvesting efficiency is greatly diminished if the shear rate is sufficiently high (this effect is more pronounced in higher frequencies). This is attributed to the effects of linear shear on the vorticity generation and the synchronization between fluid forcing and foil motion – when a strong shear flow is introduced the lift force induced by the leading edge vortex that is in phase with the heaving motion of the foil is diminished. Furthermore, by studying the instability of the wake behind the foil, we confirm that the optimal performance of the foil in linear shear flows is associated with the same physical mechanism that controls the performance of the foil in uniform flows, i.e. the excitation of the most unstable modes in the wake when the oscillation frequency of the foil is close to the frequencies of these modes.     

Two-dimensional free boundary layer in a stratified medium

Expressions for the fluctuation characteristics of shear flow in a stratified medium are obtained on the basis of the equations for the single-point second-order moments of the velocity and temperature fields and then closure of those equations by means of semiempirical hypotheses. The Prandtl equation, with the influence or Archimedean forces taken into account, is used to analyze plane jet flows and wake flows of a body, Numerical computations are carried out for a plane wake, and the results are compared with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 71–79, July–August, 1977.     

The Opposing Effect of Viscous Dissipation Allows for a Parallel Free Convection Boundary-Layer Flow Along a Cold Vertical Flat Plate

External free convection boundary-layer flows are usually treated by neglecting the effect of viscous dissipation. This assumption always results in a non-parallel flow, besides a strong parallel component also a weak transversal component of the (steady) velocity field occurs. The present paper shows, however, that the weak opposing effect of the buoyancy forces due to heat release by viscous dissipation, can give rise along a cold vertical plate adjacent to a fluid saturated porous medium to a strictly parallel steady free convection flow. This boundary-layer flow is described by an algebraically decaying exact analytical solution of the basic balance equations.     

Combined Forced and Free Convective Flow in a Vertical Porous Channel: The Effects of Viscous Dissipation and Pressure Work

Buoyant flow is analysed for a vertical fluid saturated porous layer bounded by an isothermal plane and an isoflux plane in the case of a fully developed flow with a parallel velocity field. The effects of viscous dissipation and pressure work are taken into account in the framework of the Oberbeck–Boussinesq approximation scheme and of the Darcy flow model. Momentum and energy balances are combined in a dimensionless nonlinear ordinary differential equation solved numerically by a Runge–Kutta method. Both cases of upward pressure force (upward driven flows) and of downward pressure force (downward driven flows) are examined. The thermal behaviour for upward driven flows and downward driven flows is quite different. For upward driven flows, the combined effects of viscous dissipation and pressure work may produce a net cooling of the fluid even in the case of a positive heat input from the isoflux wall. For downward driven flows, viscous dissipation and pressure work yield a net heating of the fluid. A general reflection on the roles played by the effects of viscous dissipation and pressure work with respect to the Oberbeck–Boussinesq approximation is proposed.     

Cross-stream forces and velocities of fixed and freely suspended particles in viscoelastic Poiseuille flow: Perturbation and numerical analyses

The cross-stream migration of a circular particles (or infinitely long cylinder) in two dimensional, inertia-less viscoelastic pressure-driven flows is examined through complementary finite element simulations and second-order fluid perturbation analyses for small Deborah number (De), where De is defined as the fluid relaxation time divided by the characteristic flow time. A neutrally buoyant, freely suspended particle migrates toward the center of the channel for all particle sizes and cross-stream positions due to the coupled effects of the linear and quadratic variations of the imposed velocity. A particle that is held at a fixed position, in contrast, experiences a cross-stream force directed toward the wall as a result of the coupled effects of the local shear flow and the flow relative to the particle.     

A numerical method for flows in porous and homogenous fluid domains coupled at the interface by stress jump

A numerical method was developed for flows involving an interface between a homogenous fluid and a porous medium. The numerical method is based on the finite volume method with body‐fitted and multi‐block grids. A generalized model, which includes Brinkman term, Forcheimmer term and non‐linear convective term, was used to govern the flow in the porous medium region. At its interface, a shear stress jump that includes the inertial effect was imposed, together with a continuity of normal stress. Furthermore, the effect of the jump condition on the diffusive flux was considered, additional to that on the convective part which has been usually considered. Numerical results of three flow configurations are presented. The method is suitable for coupled problems with regions of homogeneous fluid and porous medium, which have complex geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

Mixing in a stably stratified shear layer: two- and three-dimensional numerical experiments

A direct method for analyzing diapycnal mixing in a stably stratified fluid (Winters et al., 1995) has been applied to the stably stratified shear layer. The diapycnal flux and mixing efficiency are computed as functions of time, whatever the turbulent activity in the fluid. The mixing properties of two- and three-dimensional numerical simulations of the Boussinesq equations are analyzed and compared. The interest of the former simulations is to emphasize the fundamental role of three-dimensional effects in fluid mixing and to quantify it. We focus on the influence of stratification (measured by the minimum Richardson number J) and changes in Prandtl number on the overall mixing that occurs as the computed flows evolve from unstable initial conditions.

In three dimensions, the flow dynamics exhibit three successive stages, each with different mixing properties. During the first stage, a primarily two-dimensional Kelvin–Helmholtz instability develops and the mixing efficiency is high (the flux Richardson number Rf_b ranges between 0.37 and 0.68, decreasing as J increases). The second stage is characterized by the development of small-scale three-dimensional instabilities. These motions result in significantly higher diapycnal flux than during the first stage but in only moderate mixing efficiency (Rf_b0.32), as the rate of kinetic energy dissipation is also high during this stage. Finally, the turbulent activity is progressively expulsed toward the outer regions of the shear layer and decays in time while the central region relaminarizes. During this final stage, Rf_b approaches an asymptotic value close to 0.25 and the diapycnal diffusivity displays a clear functional dependence on a gradient Richardson number Ri_b of the form Ri_b⁻².

As expected, the two-dimensional flows are unable to reproduce the mixing properties of the flow, except during the first stage. During the subsequent turbulent regime, both the diapycnal flux and the dissipation rate of kinetic energy are too small (because, for the latter quantity, of the nonlinear enstrophy conservation constraint). The final stage consists in a quasi-stationary weakly turbulent regime, for which the diapycnal diffusivity behaves as Ri_b⁻¹. It should be noted that, despite these differences, Rf_b relaxes toward the 0.25 value found in three dimensions.     

Strong flow criteria based on microstructure deformation

The dynamics of fluid systems which consist of a suspended material in a Newtonian continuous phase is investigated theoretically. Criteria are derived to predict conditions under which the strength of a flow, i.e. a measure of the form and magnitude of the velocity gradient tensor, is sufficient to induce significant deformation and/or orientation of the fluid microstructure, that is, the elements which collectively comprise the suspended phase. The development relies upon the choice of a model to describe the microstructure, and the form of the criteria reflects this choice. Once the choice is made, however, the detailed material properties of a particular fluid system enter only as parameters in the resulting equations, and thus, the results encompass a large class of systems, including particulate suspensions and macromolecular solutions. Two microstructure models are investigated here. When the microstructure is characterized by a vector, the flow strengths of all linear flows are displayed in a single figure from which the strength of a particular flow can be evaluated directly. A comparison is then made for selected flows between these results and those for the case where an irreducible second order tensor is employed to describe the microstructure. A significant difference between the two models derives from the fact that the “volume” of the microstructure must be conserved in the second-order tensor case. The criteria are finally used to predict the degree of macromolecular stretching in a model turbulent flow and the breakup of immiscible liquid drops in simple shear flow. A comparison between the flow strength predictions and experimental data yields good qualitative agreement in the latter case.     

Rheological characterization of continuous fiber—reinforced viscous fluid

The present paper describes a micromechanical technique to determine rheological properties of viscous fluid reinforced with unidirectional continuous fibers. Fluid viscosity is described by a shear thinning model and high viscosity is considered for continuous fibers having considerable rigidity compared to net fluid. The microstructure is identified by a representative volume element that is subjected to equivalent macroscopic deformation fields. The energy balance and periodicity conditions are considered to relate deformation and stress in macro and micro-levels. It is shown that response of viscous fluid reinforced with rigid fibers depends on deformation history as well as rate-of-deformation in the transverse intraply shear and transverse squeeze flows. An orthotropic viscous constitutive equation is derived to describe response of such materials. The material viscosities are evaluated for viscous fluid reinforced with different fiber volume fractions during deformation applied in different rates of deformation. The results are used to derive the functions predicting effective anisotropic viscosities of reinforced fluid.     

Thermal dispersion and viscous dissipation effects on non-darcy mixed convection in a fluid saturated porous medium

Mixed convection flow and heat transfer about an isothermal vertical wall embedded in a fluid saturated porous medium with uniform free stream velocity is considered and the effects of thermal dispersion and viscous dissipation in both aiding and opposing flows are analysed. Similarity solution is not possible due to the inclusion of the viscous dissipation term, series solution is obtained, first and second order effects of dissipation revealed that viscous dissipation lowers the heat transfer rate. Observations also revealed that the thermal dispersion effect enhances the heat transfer rate and the effect of viscous dissipation is observed to increase with increasing values of the dispersion parameter. Received on 21 March 1997     

Non-modal instabilities of two-dimensional disturbances in plane Couette flow of a power-law fluid

Instabilities of fluid flows have traditionally been investigated by normal mode analysis, i.e. by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem. However, the results of eigenvalue analysis agree poorly in many cases with experiments, especially for shear flows. In this paper we study the instabilities of two-dimensional Couette flow of a polymeric fluid in the framework of non-modal stability theory rather than normal mode analysis. A power-law model is used to describe the polymeric liquid. We focus on the response to external excitations and initial conditions by examining the pseudospectra structures and the transient energy growths. For both Newtonian and non-Newtonian flows, the results show that there can be a rather large transient growth even though the linear operator of Couette flow has no unstable eigenvalue. The effects of non-Newtonian viscosity on the transient behaviors are examined in this study. The results show that the “shear-thinning/shear-thickening” effect increases/decreases the amplitude of responses to external excitations and initial conditions.     

Magnetic field applied to thermochemical non-equilibrium reentry flows in 2D – five species

In this work, a study involving magnetic field actuation over reentry flows in thermochemical non-equilibrium is performed. The Euler and Navier–Stokes equations are studied. The proposed numerical algorithm is centred and second-order accurate. The hypersonic flow around a blunt body is simulated. Three time integration methods are tested. The reactive simulations involve Earth atmosphere of five species. The work of Gaitonde is the reference to couple the fluid dynamics and Maxwell equations of electromagnetism. The results have indicated that the Maciel scheme, using the Mavriplis dissipation model, yields the best prediction of the stagnation pressure.     

Effect of Viscous Dissipation on the Boundary Layer Flow and Heat Transfer Past a Permeable Stretching Surface Embedded in a Porous Medium with a Second-Order Slip Using Chebyshev Finite Difference Method

This article investigates a theoretical and numerical study for the effect of viscous dissipation on the steady flow with heat transfer of Newtonian fluid toward a permeable stretching surface embedded in a porous medium with a second-order slip and thermal slip. The governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting ODEs are successfully solved numerically with the help of Chebyshev finite difference method. Graphically results are shown for non-dimensional velocities and temperature. The effects of the porous parameter, the suction (injection) parameter, Eckert number, first- and second-order velocity slip parameter, the thermal slip parameter and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. A comparison of numerical results is made with the earlier published results under limiting cases.     

Dynamic bulk and shear moduli due to grain-scale local fluid flow in fluid-saturated cracked poroelastic rocks: Theoretical model

Grain-scale local fluid flow is an important loss mechanism for attenuating waves in cracked fluid-saturated poroelastic rocks. In this study, a dynamic elastic modulus model is developed to quantify local flow effect on wave attenuation and velocity dispersion in porous isotropic rocks. The Eshelby transform technique, inclusion-based effective medium model (the Mori–Tanaka scheme), fluid dynamics and mass conservation principle are combined to analyze pore-fluid pressure relaxation and its influences on overall elastic properties. The derivation gives fully analytic, frequency-dependent effective bulk and shear moduli of a fluid-saturated porous rock. It is shown that the derived bulk and shear moduli rigorously satisfy the Biot-Gassmann relationship of poroelasticity in the low-frequency limit, while they are consistent with isolated-pore effective medium theory in the high-frequency limit. In particular, a simplified model is proposed to quantify the squirt-flow dispersion for frequencies lower than stiff-pore relaxation frequency. The main advantage of the proposed model over previous models is its ability to predict the dispersion due to squirt flow between pores and cracks with distributed aspect ratio instead of flow in a simply conceptual double-porosity structure. Independent input parameters include pore aspect ratio distribution, fluid bulk modulus and viscosity, and bulk and shear moduli of the solid grain. Physical assumptions made in this model include (1) pores are inter-connected and (2) crack thickness is smaller than the viscous skin depth. This study is restricted to linear elastic, well-consolidated granular rocks.     

Dissipative instability of fluid flows with piecewise linear velocity profiles

The viscous dissipative instability of two flows with continuous spectrum of neutrally-stable perturbations in the absence of dissipation is investigated. Ranges of wave numbers in which viscosity leads to flow destabilization are determined for a shear discontinuity in a smoothly-stratified fluid. A shear flow with a velocity in the transition layer that depends linearly on the coordinate has a continuum of neutral modes even in the case of an unstratified fluid. When viscosity is present in one of the layers with constant velocity, one of the branches of the spectrum becomes unstable. When the viscosity is the same above and below the shear layer, dissipation only leads to the damping of the perturbations. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 14–19, November–December, 1986.     

Flow Characteristics of a “Multiconfigurational”, Shear Thinning Viscoelastic Fluid with Particular Reference to the Orthogonal Rheometer

This paper deals with the flow characteristics of a class of nonsimple viscoelastic fluid models developed by Rajagopal and Srinivasa (1999). The central feature of these models is that the stress response is lastic from a changing natural configuration with the viscous dissipation occurring due to changes in the natural state. The class of models considered are characterized by three independent parameters that represent respectively the elasticity, the viscosity and the shear thinning index. The stress relaxation response of the material is compared with experimental data reported by Bower et al. (1987) for polyisobutelene in cetane, and parameters that fit the data are calculated. The flow of such a fluid between parallel disks rotating about noncoincident axes (the orthogonal rheometer) is then studied. It is shown that the assumed velocity field leads to a system of second-order nonlinear ordinary differential equations (Rajagopal, 1982). A parametric study is then undertaken to see the effect of the various material, geometrical, and flow parameters on the flow characteristics. It is observed that inertial effects and shear thinning effects are roughly complementary in the range of parameters considered. While it is well known that boundary layers occur in these flows due to inertial effects, it is demonstrated that these boundary effects are insensitive to the Reynolds number but rather are determined by the absorption number. Finally, in the range of parameters that are commonly observed in such rheometers, it is shown that neglect of inertia causes significant discrepancies in the calculation of the boundary shear rates. Received 3 June 1999 and accepted 2 October 1999     

Computation of backward-facing step flows by a second-order Reynolds stress closure model

This paper scrutinizes the predictive ability of the differential stress equation model in complex shear flows. Two backward-facing step flows with different expansion ratios are solved by the LRR turbulence model with an anisotropic dissipation model and the near-wall regions of the separated side resolved by a near-wall model. The computer code developed for solving the transport equations is based on the finite-volume-finite-difference method. In the numerical solution of the time-averaged momenum equations the Reynolds stresses are treated partially as a diffusion term and partially as a source term to avoid numerical instability. Computational results are compared with experimental data. It is found that the near-wall region of the separated side resolved by the near-wall model, the LRR model with a simple modification of an anisotropic dissipation model can predict backward step flows well.     

The Rayleigh and Stokes problems with an incompressible non-Newtonian fluid

Summary The Rayleigh problem or impulsive motion of a flat plate has been solved using a perturbation scheme when the surrounding fluid is representable by the constitutive equations of Oldroyd or Coleman and Noll. The shear stress and normal stress at the wall were expressed analytically for this unsteady motion. Further, an exact solution of the equations was found for a special case of the constitutive equations.The motion of the fluid above a harmonically oscillating plate or the Stokes problem has been determined for a special non-Newtonian fluid. The penetration of the shear wave into the fluid, the energy dissipation, the velocity profiles and the shear and normal stresses at the wall were expressed and compared to an equivalent Newtonian fluid.Some of the features of these non-Newtonian fluids were examined in simple shearing flows, and techniques to calculate some of the material constants discussed.