Showing papers on "Velocity gradient published in 2007"

A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium

Abstract: We develop and analyze a mixed finite element method for the solution of an elliptic system modeling a porous medium with large cavities, called vugs. It consists of a second-order elliptic (i.e., Darcy) equation on part of the domain coupled to a Stokes equation on the rest of the domain, and a slip boundary condition (due to Beavers–Joseph–Saffman) on the interface between them. The tangential velocity is not continuous on the interface. We consider a 2-D vuggy porous medium with many small cavities throughout its extent, so the interface is not isolated. We use a certain conforming Stokes element on rectangles, slightly modified near the interface to account for the tangential discontinuity. This gives a mixed finite element method for the entire Darcy–Stokes system with a regular sparsity pattern that is easy to implement, independent of the vug geometry, as long as it aligns with the grid. We prove optimal global first-order L 2 convergence of the velocity and pressure, as well as the velocity gradient in the Stokes domain. Numerical results verify these rates of convergence and even suggest somewhat better convergence in certain situations. Finally, we present a lower dimensional space that uses Raviart–Thomas elements in the Darcy domain and uses our new modified elements near the interface in transition to the Stokes elements.

P-to-S and S-to-P imaging of a sharp lithosphere-asthenosphere boundary beneath eastern North America

Abstract: [1] S-to-P (Sp) scattered energy independently confirms the existence of a seismic velocity discontinuity at the lithosphere-asthenosphere boundary that was previously imaged using P-to-S (Ps) scattered energy in eastern North America. Exploration of the different sensitivities of Ps and Sp scattered energy suggests that the phases contain independent yet complementary high-resolution information regarding velocity contrasts. Combined inversions of Ps and Sp energy have the potential to tightly constrain associated velocity gradients. In eastern North America, inversions of Sp and Ps data require a strong, 5–10% velocity contrast that is also sharp, occurring over less than 11 km at 87–105 km depth. Thermal gradients alone are insufficient to create such a sharp boundary, and therefore another mechanism is required. A boundary in composition, hydration, or a change in anisotropic signature could easily produce a sufficiently localized velocity gradient. Taken separately, the magnitudes of the effects of these mechanisms are too small to match our observed velocity gradients. However, our observations may be explained by a boundary in hydration coupled with a boundary in depletion and/or anisotropy. Alternatively, a small amount of melt in the asthenosphere could explain the velocity gradient. The tight constraints on velocity gradients achieved by combined modeling of Ps and Sp energy offer promise for defining the character of the lithosphere-asthenosphere boundary globally.

Sling Effect in Collisions of Water Droplets in Turbulent Clouds

Abstract: The effect of turbulence on the collision rate between droplets in clouds is investigated Because of their inertia, water droplets can be shot out of curved streamlines of the turbulent airflow The contribution of such a “sling effect” in the collision rate of the same-size water droplets is described and evaluated It is shown that already for turbulence with the dissipation rate 103 cm2 s−3, the sling effect gives a contribution to the collision rate of 15-μm droplets comparable to that due to the local velocity gradient That may explain why the formulas based on the local velocity gradient consistently underestimate the turbulent collision rate, even with the account of preferential concentration

Asymptotic exponents from low-Reynolds-number flows

Abstract: The high-order statistics of fluctuations in velocity gradients in the crossover range from the inertial to the Kolmogorov and sub-Kolmogorov scales are studied by direct numerical simulations (DNS) of homogeneous isotropic turbulence with vastly improved resolution. The derivative moments for orders 0 n 8 are represented well as powers of the Reynolds number, Re, in the range 380 Re 5275, where Re is based on the periodic box length Lx. These low-Reynolds-numberflows give no hint of scaling in the inertial range even when extended self-similarity is applied. Yet, the DNS scaling exponents of velocity gradients agree well with those deduced, using a recent theory of anomalous scaling, from the scaling exponents of the longitudinal structure functions at infinitely high Reynolds numbers. This suggests that the asymptotic state of turbulence is attained for the velocity gradients at far lower Reynolds numbers than those required for the inertial range to appear. We discuss these findings in the light of multifractal formalism. Our numerical studies also resolve the crossover of the velocity gradient statistics from Gaussian to non-Gaussian behaviour that occurs as the Reynolds number is increased.

Modeling the pressure Hessian and viscous Laplacian in Turbulence: comparisons with DNS and implications on velocity gradient dynamics

Abstract: Modeling the velocity gradient tensor A = ∇u along Lagrangian trajectories in turbulent flow requires closures for the pressure Hessian and viscous Laplacian of A. Based on an EulerianLagrangian change of variables and the so-called Recent Fluid Deformation closure, such models were proposed recently (Chevillard & Meneveau, Phys. Rev. Lett. 97, 174501 (2006)). The resulting stochastic model was shown to reproduce many geometric and anomalous scaling properties of turbulence. In this work, direct comparisons between model predictions and Direct Numerical Simulation (DNS) data are presented. First, statistical properties of A are described using conditional averages of strain skewness, enstrophy production, energy transfer and vorticity alignments, conditioned upon invariants of the velocity gradient. These conditionally averaged quantities are found to be described accurately by the stochastic model. More detailed comparisons that focus directly on the terms being modeled in the closures are also presented. Specifically, conditional statistics associated with the pressure Hessian and the viscous Laplacian are measured from the model and are compared with DNS. Good agreement is found in strain-dominated regions. However, some features of the pressure Hessian linked to rotation dominated regions are not reproduced accurately by the model. Geometric properties such as vorticity alignment with respect to principal axes of the pressure Hessian are mostly predicted well. In particular, the model predicts that an eigenvector of the rate-of-strain will be also an eigenvector of the pressure Hessian, in accord with basic properties of the Euler equations. The analysis identifies under what conditions the Eulerian-Lagrangian change of variables with the Recent Fluid Deformation closure works well, and in which flow regimes it requires further improvements.

Ubiquitous low-velocity layer atop the 410-km discontinuity in the northern Rocky Mountains

Abstract: Receiver functions from three 30-station IRIS-PASSCAL small-aperture arrays (2–15 km station spacing) operated for 10 months each in the northern Rocky Mountains show a ubiquitous negative polarity P to S conversion just preceding the 410-km discontinuity arrival. Data from the three arrays were sorted into NW, SE, and SW back-azimuth quadrants and stacked to form nine quadrant stacks. Remarkably, the negative polarity arrival (NPA) is apparent in 8 of the 9 quadrant stacks, with 7 of the 8 having well-correlated waveforms. Each quadrant stack also contains clear P to S conversions from the 410- and 660-km discontinuities. Moveout analysis shows that all the major phases display the correct moveout for forward scattered P-S phases. The waveshapes for the seven similar NPA-410 km discontinuity arrivals are modeled with a five-parameter “double gradient slab” model that is parameterized as follows: a top gradient thickness and shear velocity decrease; a constant velocity layer; bottom gradient thickness; and shear velocity increase. Model misfit is assessed via a grid search over the model space using a reflectivity code to calculate synthetic seismograms. Model likelihood is determined by calculating 1- and 2-D marginal probability density functions (PDF) for the five parameters. The 1-D marginals display a range of peak values, although significant overlap is observed for the top gradient thickness and its associated velocity decrement. From the peak value of the summary PDF, we find the top velocity gradient to be sharp (<6.4 km) and the shear velocity decrement to be large (8.9% Vs). Defining an effective thickness of the low-velocity layer as the mean layer thickness plus half the mean gradient thicknesses, the 410 low-velocity layer thickness is found to be 22 km. A review of changes in the physical state required to match our new 410-LVL constraints suggests that the water-filter model remains an operative hypothesis to test.

Accounting for lateral variations of the upper mantle gradient in Pn tomography studies

Abstract: [1] The effect of an upper mantle velocity gradient on regional arrival times has been approximated by a cubic distance term, which can be extended to two dimensions for use in tomographic studies. To demonstrate this, we add a laterally varying upper mantle gradient to the standard Pn time-term tomography technique, and apply to a data set from Asia compiled using ground truth, event location criteria. We observe strong lateral variations in the gradient, ranging from −0.001 to 0.003 s−1, with high gradients associated with the Tethys convergence zone. The gradient patterns may reflect lateral variations in the thermal gradient of the mantle lid. Variance reduction is 63% with respect to Pn tomography without gradients. Adding gradients allows the use of longer path lengths, improving velocity image definition in high-gradient regions with sparse station distribution, such as Tibet.

Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature

Abstract: In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation, (2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values of ξ ( = Gr x /Re 2 , defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied. We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity gradient is higher than that of temperature.

Constraints on velocity-depth trends from rock physics models

Abstract: Estimates of depth, overpressure and amount of exhumation based on sonic data for a sedimentary formation rely on identification of a normal velocity–depth trend for the formation. Such trends describe how sonic velocity increases with depth in relatively homogeneous, brine-saturated sedimentary formations as porosity is reduced during normal compaction (mechanical and chemical). Compaction is ‘normal’ when the fluid pressure is hydrostatic and the thickness of the overburden has not been reduced by exhumation. We suggest that normal porosity at the surface for a given lithology should be constrained by its critical porosity, i.e. the porosity limit above which a particular sediment exists only as a suspension. Consequently, normal velocity at the surface of unconsolidated sediments saturated with brine approaches the velocity of the sediment in suspension. Furthermore, porosity must approach zero at infinite depth, so the velocity approaches the matrix velocity of the rock and the velocity–depth gradient approaches zero. For sediments with initially good grain contact (when porosity is just below the critical porosity), the velocity gradient decreases with depth. By contrast, initially compliant sediments may have a maximum velocity gradient at some depth if we assume that porosity decreases exponentially with depth. We have used published velocity–porosity–depth relationships to formulate normal velocity–depth trends for consolidated sandstone with varying clay content and for marine shale dominated by smectite/illite. The first relationship is based on a modified Voigt trend (porosity scaled by critical porosity) and the second is based on a modified time-average equation. Baselines for sandstone and shale in the North Sea agree with the established constraints and the shale trend can be applied to predict overpressure. A normal velocity–depth trend for a formation cannot be expressed from an arbitrary choice of mathematical functions and regression parameters, but should be considered as a physical model linked to the velocity–porosity transforms developed in rock physics.

Granular flows in a rotating drum: the scaling law between velocity and thickness of the flow

Abstract: The flow of dry granular material in a half-filled rotating drum is studied. The thickness of the flowing zone is measured for several rotation speeds, drum sizes and beads sizes (size ratio between drum and beads ranging from 47 to 7400). Varying the rotation speed, a scaling law linking mean velocity vs. thickness of the flow, v approximately h(m), is deduced for each couple (beads, drum). The obtained exponent m is not always equal to 1, the value previously reported for a drum in litterature, but varies with the geometry of the system. For small size ratios, exponents higher than 1 are obtained due to a saturation of the flowing zone thickness. The exponent of the power law decreases with the size ratio, leading to exponents lower than 1 for high size ratios. These exponents imply that the velocity gradient of a dry granular flow in a rotating drum is not constant. More fundamentally, these results show that the flow of a granular material in a rotating drum is very sensible to the geometry, and that the deduction of the "rheology" of a granular medium flowing in such a geometry is not obvious.

Green's tensor and radiation patterns of point sources in general anisotropic inhomogeneous elastic media

Abstract: SUMMARY i (Marshall McLuhan, 1958) The elastodynamic ‘ray-theoretical’ Green's tensor for general anisotropic and inhomogeneous media (GANIMED) is derived in the far-field high-frequency approximation. This allows the quantification of the amplitude and phase charac-teristics of arbitrary point sources. The displacement field is controlled by the phase velocity and the Gaussian curvature of the wave surface. Exploiting the already existing computer codes for dynamic ray tracing in GANIMED, we demonstrate how one may couple it to source theory in order to calculate displacement-radiation patterns for unipolar and dipolar sources in various earth crustal materials, such as shales, sandstones, and olivine. The existence of a velocity gradient at the source causes asymmetry in the radiation patterns. In contradistinction, anisotropy at the source tends to distort the lobes and in some cases can beam excessive energy in specific directions. Our calculated radiation patterns for shear waves are similar to those observed from some explosions in soil.

Velocity gradient singularity and structure of the velocity profile in the Knudsen layer according to the Boltzmann equation.

Abstract: Rarefied gas flow modeling presents significant challenges in the characterization of nanoscale devices and their applications. An important feature of such flows is the Knudsen layer, which is known to exhibit non-Newtonian viscosity behavior. Significantly, recent research has suggested that the effective viscosity at the surface is about half the standard dynamic viscosity. We examine these claims using numerical solutions of the linearized Boltzmann equation and direct simulation Monte Carlo calculations and discover that (i) the flow exhibits a striking power-law dependence on distance from the solid surface and (ii) the velocity gradient is singular at this surface. This finding contradicts these recent claims and has direct implications for gas flow modeling and the design of nanoscale devices.

The Merger in Abell 576: A Line-of-Sight Bullet Cluster?

Abstract: Using a combination of Chandra and XMM-Newton observations, we confirmed the presence of a significant velocity gradient along the northeast-southwest direction in the intracluster gas of the cluster Abell 576. The results are consistent with a previous ASCA SIS analysis of this cluster. The error-weighted average over the ACIS-S3 and EPIC MOS1 and MOS2 spectrometers for the maximum velocity difference is >3.3 × 103 km s-1 at the 90% confidence level, similar to the velocity limits estimated indirectly for the Bullet Cluster (1E 0657-56). The probability that the velocity gradient is generated by standard random gain fluctuations with Chandra and XMM-Newton is <0.1%. The regions of maximum velocity gradient are in CCD zones that have the lowest temporal gain variations. It is unlikely that the velocity gradient is due to Hubble distance differences between projected clusters (probability 0.01%). We mapped the distribution of elemental abundance ratios across the cluster and detected a strong chemical discontinuity using the abundance ratio of silicon to iron, equivalent to a variation from 100% SN Ia iron mass fraction in the west-northwest regions to 32% in the eastern region. The "center" of the cluster is located at the chemical discontinuity boundary, which is inconsistent with the radially symmetric chemical gradient found in some regular clusters, but consistent with a cluster merging scenario. We predict that the velocity gradient as measured will produce a variation of the cosmic microwave background (CMB) temperature toward the east of the core of the cluster that will be detectable by current and near-future bolometers. The measured velocity gradient opens up the possibility that this cluster is passing through a near line-of-sight merger stage where the cores have recently crossed.

Electroosmotic flow mixing in zigzag microchannels

Abstract: In this study we performed numerical and experimental investigations into the mixing of EOFs in zigzag microchannels with two different corner geometries, namely sharp corners and flat corners. In the zigzag microchannel with sharp corners, the flow travels more rapidly near the inner wall of the corner than near the outer wall as a result of the higher electric potential drop. The resulting velocity gradient induces a racetrack effect, which enhances diffusion within the fluid and hence improves the mixing performance. The simulation results reveal that the mixing index is approximately 88.83%. However, the sharp-corner geometry causes residual liquid or bubbles to become trapped in the channel at the point where the flow is almost stationary, when the channel is in the process of cleaning. Accordingly, a zigzag microchannel with flat-corner geometry is developed. The flat-corner geometry forms a convergent–divergent type nozzle which not only enhances the mixing performance in the channel, but also prevents the accumulation of residual liquid or bubbles. Scaling analysis reveals that this corner geometry leads to an effective increase in the mixing length. The experimental results reveal that the mixing index is increased to 94.30% in the flat-corner zigzag channel. Hence, the results demonstrate that the mixing index of the flat-corner zigzag channel is better than that of the conventional sharp-corner microchannel. Finally, the results of Taguchi analysis indicate that the attainable mixing index is determined primarily by the number of corners in the microchannel and by the flow passing height at each corner.

Effect of velocity gradient on propagation speed of tribrachial flames in laminar coflow jets

Abstract: The effect of velocity gradient on the propagation speed of tribrachial flame edge has been investigated experimentally in laminar coflow jets for propane fuel. It was observed that the propagation speed of tribrachial flame showed appreciable deviations at various jet velocities in high mixture fraction gradient regime. From the similarity solutions, it was demonstrated that the velocity gradient varied significantly during the flame propagation. To examine the effect of velocity gradient, detail structures of tribrachial flames were investigated from OH LIF images and Abel transformed images of flame luminosity. It was revealed that the tribrachial point was located on the slanted surface of the premixed wing, and this slanted angle was correlated with the velocity gradient along the stoichiometric contour. The temperature field was visualized qualitatively by the Rayleigh scattering image. The propagation speed of tribrachial flame was corrected by considering the direction of flame propagation with the slanted angle and effective heat conduction to upstream. The corrected propagation speed of tribrachial flame was correlated well. Thus, the mixture fraction gradient together with the velocity gradient affected the propagation speed.

Stretching of Polymers in Isotropic Turbulence: A Statistical Closure

Abstract: We present a new closure for the mean rate of stretching of a dissolved polymer by homogeneous isotropic turbulence. The polymer is modeled by a bead-spring-type model (e.g., Oldroyd B, FENE-P, Giesekus) and the analytical closure is obtained assuming the Lagrangian velocity gradient can be modeled as a Gaussian, white-noise stochastic process. The resulting closure for the mean stretching depends upon the ratio of the correlation time for strain and rotation. Additionally, we derived a second-order expression for circumstances when strain and rotation have a finite correlation time. Finally, the base level closure is shown to reproduce results from direct numerical simulations by simply modifying the coefficients.

Lagrangian particle paths and ortho-normal quaternion frames

Abstract: Experimentalists now measure intense rotations of Lagrangian particles in turbulent flows by tracking their trajectories and Lagrangian-average velocity gradients at high Reynolds numbers. This paper formulates the dynamics of an orthonormal frame attached to each Lagrangian fluid particle undergoing three-axis rotations, by using quaternions in combination with Ertel's theorem for frozen-in vorticity. The method is applicable to a wide range of Lagrangian flows including the three-dimensional Euler equations and its variants such as ideal magneto-hydrodynamics. The applicability of the quaterionic frame description to Lagrangian averaged velocity gradient dynamics is also demonstrated.

Taylor‐Couette unit with a lobed inner cylinder cross section

Abstract: A new mixing unit, based on a modification of the classical Taylor-Couette (TC) unit is proposed, where a lobed profile of the inner cylinder cross section is used. The shear rate distribution of the lobed Taylor-Couette (LTC) unit have been computed through computational fluid dynamic simulations and compared with those of the TC unit and a standard stirred tank (ST). It is found that since the flow pattern of the LTC units becomes temporal-periodic at each point with respect to the nonrotational outer cylinder, it reduces the formation of the low velocity gradient (low shear rate) region, typically generated in the vortex core of the TC units. The obtained distributions of the shear rate are substantially narrower than those of the TC and the ST units. Variation of the ratio between the maximum and the minimum gap widths can lead to significant changes in the shear rate distribution, and there exists an optimal range of such ratio, where the shear rate distribution is not only very narrow but also insensitive to the variation of the gap widths.

Reynolds shear stress distributions in a gradually varied flow in a roughened channel Distributions du cisaillement de Reynolds dans un écoulement graduellement varié dans un canal rugueux

Abstract: The Reynolds shear stress distribution in non-uniform flows has been investigated. The theoretical results show that the wall-normal velocity causesthe deviation of Reynolds shear stress from the standard linear distribution, i.e. − uv/u 2∗ = 1 − y/h , but the sum of Reynolds shear stress andthe momentum flux, i.e. − (uv + u v)/u 2∗ remains a linear distribution. By connecting the velocity gradient with Reynolds shear stress, the studydemonstrates theoretically that the linear distribution of Reynolds shear stress and semi-logarithmic distribution of velocity (i.e., log-law) can beobserved when and only when the wall-normal velocity is zero; the concave distribution of Reynolds shear stress and dip-phenomenon can beobserved when and only when the wall-normal velocity is downward; the convex distribution of Reynolds shear stress can be observed and thewake-law correction is needed when and only when the upflow occurs. The theoretical results are in good agreement with experimental data.RESUMELa distribution du cisaillement de Reynolds dans des ecoulements non uniformes a ete etudiee. Les resultats theoriques montrent que la vitesse normalea la partoi produit une deviation du cisaillement de Reynolds par rapport a la distribution lineaire standard, i.e. −

Role of pressure in nonlinear velocity gradient dynamics in turbulence.

Abstract: To identify and understand the effect of pressure on turbulence velocity gradient dynamics, we study the velocity gradient evolution with the inviscid three-dimensional Burgers equation and the restricted Euler equation (REE). While the REE represents the incompressible limit of turbulence, the Burgers equation is taken to be the infinite Mach number model for the Navier-Stokes equation wherein the time scale of flow inertia is very small compared to that of pressure. Analytical fixed-point solutions for the velocity gradient tensor are obtained in the two cases. The results are compared and contrasted to isolate the role of pressure in shaping velocity gradient behavior. Of particular interest is the influence of pressure on (i) the strain rate eigenvalues; (ii) the sign of the intermediate principal strain rate; (iii) the tendency of vorticity to align with the intermediate principal strain rate; and (iv) the energy cascade direction. Importantly, the study provides valuable insight into the velocity gradient dynamics in highly compressible turbulence.

Effect of sheared flow on magnetic islands

Abstract: The effect of sheared flow on a magnetic island is examined. In contrast to the density and temperature gradients which are flattened for sufficiently wide islands, it is found that the velocity gradient persists inside the separatrix whenever the constant-ψ approximation is satisfied. It follows that velocity shear has a negligible effect on island amplitude in that approximation. The effect of the violation of the constant-ψ approximation is explored by using the Kelvin-Stuart family of islands, and it is found that flattening is modest even when the separatrix encloses virtually all the current.

Formation of density singularities in ideal hydrodynamics of freely cooling inelastic gases: A family of exact solutions

Abstract: We employ granular hydrodynamics to investigate a paradigmatic problem of clustering of particles in a freely cooling dilute granular gas. We consider large-scale hydrodynamic motions where the viscosity and heat conduction can be neglected, and one arrives at the equations of ideal gas dynamics with an additional term describing bulk energy losses due to inelastic collisions. We employ Lagrangian coordinates and derive a broad family of exact nonstationary analytical solutions that depend only on one spatial coordinate. These solutions exhibit a new type of singularity, where the gas density blows up in a finite time when starting from smooth initial conditions. The density blowups signal formation of close-packed clusters of particles. As the density blow-up time tc is approached, the maximum density exhibits a power law ∼(tc−t)−2. The velocity gradient blows up as ∼−(tc−t)−1 while the velocity itself remains continuous and develops a cusp (rather than a shock discontinuity) at the singularity. The gas ...

The Shear Layer above and in Urban Canopies

Abstract: The nature and role of the shear layer, which occurs at the level of the average building height in urban canopies, are poorly understood. Velocity data are analyzed to determine the characteristics of the shear layer of the urban canopy, defined as the broad, linear segment of the mean velocity profile in a region of high shear. Particle image velocimetry measurements in a water tunnel were undertaken to resolve velocity profiles for urban canopies of two geometries typical of Los Angeles, California, and New York City, New York, for which the aspect ratios (average building height-to-width ratio) H/wb are 1 and 3, respectively. The shear layers evolve with distance differently: For H/wb = 1 the urban canopy shear layer extends quickly from above the building height to ground level, whereas for H/wb = 3 the urban canopy shear layer remains elevated at the vicinity of the building height, only reaching to a depth of z/H ∼ 0.5 far downstream. Profiles of the mean velocity gradient also differ from...

Structure and Kinematics of CO?J?=?2-1 Emission in the Central Region of NGC?4258

Abstract: We present 12CO J = 2-1 observations toward the central region of the type 2 Seyfert galaxy NGC 4258 with the Submillimeter Array (SMA). Our interferometric maps show two armlike elongated components along the major axis of the galaxy, with no strong nuclear concentration. The CO(2-1) morphology and kinematics are similar to previous CO(1-0) results. The velocity field of the components agrees with the general galactic rotation, except for the east elongated component, which shows a significant velocity gradient along the east-west direction. In order to account for the velocity field, we propose a kinematical model in which the warped rotating disk is also expanding. The line ratio of CO(2-1)/CO(1-0) reveals that the eastern component with the anomalous velocity gradient appears to be warmer and denser. This is consistent with the gas in this component being closer to the center, being heated by the central activities, and possibly interacting with expanding motions from the nuclear region.

Finite element modeling of laminar flow of a third grade fluid in a Darcy-Forcheimmer porous medium with suction effects

Abstract: A mathematical model to simulate the steady laminar flow of an incompressible, third grade, nonNewtonian fluid past an infinite porous plate embedded in a Darcy-Forcheimmer porous medium is presented. A number of special cases are examined for the governing nonlinear differential equation. The model is solved with appropriate boundary conditions using the finite element method. Velocity and velocity gradient are plotted graphically for variation in permeability   k , Forcheimmer parameter   b , third grade material parameter   3  , and suction effect (Vo). It is shown that velocities are generally decreased transverse to the plate surface with increasing Forcheimmer parameter; increasing permeability conversely boosts the velocities, as this corresponds to an increasingly fluid (i.e., progressively less porous) regime. The third grade material parameter is also seen to substantially increase the velocities in the direction normal to the plate surface. The special case of a second order viscoelastic flow is also studied. The flow scenario finds applications in polymer extrusion processes, and other important industrial rheology systems.