scispace - formally typeset
Search or ask a question

Showing papers on "Streamlines, streaklines, and pathlines published in 1983"


Journal ArticleDOI
TL;DR: It is concluded that in the human carotid bifurcation, regions of moderate to high shear stress, where flow remains unidirectional and axially aligned, are relatively spared of intimal thickening.
Abstract: The distribution of nonstenosing, asymptomatic intimal plaques in 12 adult human carotid bifurcations obtained at autopsy was compared with the distribution of flow streamline patterns, flow velocity profiles, and shear stresses in corresponding scale models. The postmortem specimens were fixed while distended to restore normal in vivo length, diameter, and configuration. Angiograms were used to measure branch angles and diameters, and transverse histological sections were studied at five standard sampling levels. Intimal thickness was determined at 15 degrees intervals around the circumference of the vessel sections from contour tracings of images projected onto a digitizing plate. In the models, laser-Doppler anemometry was used to determine flow velocity profiles and shear stresses at levels corresponding to the standard specimen sampling sites under conditions of steady flow at Reynolds numbers of 400, 800, and 1200, and flow patterns were visualized by hydrogen bubble and dye-washout techniques. Intimal thickening was greatest and consistently eccentric in the carotid sinus. With the center of the flow divider as the 0 degree index point, mid-sinus sections showed minimum intimal thickness (0.05 +/- 0.02 mm) within 15 degrees of the index point, while maximum thickness (0.9 +/- 0.1 mm) occurred at 161 +/- 16 degrees, i.e., on the outer wall opposite the flow divider. Where the intima was thinnest, along the inner wall, flow streamlines in the model remain axially aligned and unidirectional, with velocity maxima shifted toward the flow divider apex. Wall shear stress along the inner wall ranged from 31 to 600 dynes/cm2 depending on the Reynolds number. Where the intima was thickest, along the outer wall opposite the flow divider apex, the pattern of flow was complex and included a region of separation and reversal of axial flow as well as the development of counter-rotating helical trajectories. Wall shear stress along the outer wall ranged from 0 to -6 dynes/cm2. Intimal thickening at the common carotid and distal internal carotid levels of section was minimal and was distributed uniformly about the circumference. We conclude that in the human carotid bifurcation, regions of moderate to high shear stress, where flow remains unidirectional and axially aligned, are relatively spared of intimal thickening. Intimal thickening and atherosclerosis develop largely in regions of relatively low wall shear stress, flow separation, and departure from axially aligned, unidirectional flow. Similar quantitative evaluations of other atherosclerosis-prone locations and corresponding flow profile studies in geometrically accurate models may reveal which of these hemodynamic conditions are most consistently associated with the development of intimal disease.

1,451 citations


Journal ArticleDOI
TL;DR: In this article, the homogenization of a passive tracer in a flow with closed mean streamlines is studied, where the initial values of the tracer are replaced by their (generalized) average about a streamline.
Abstract: The homogenization of a passive ‘tracer’ in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time ≈P1/3(L/U), where the Peclet number P=LU/κ (L,U and κ are lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time ≈ L2/κ. The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to κ, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.

355 citations


Journal ArticleDOI
TL;DR: In this paper, numerical solutions to multicellular natural convection in a vertical enclosure were presented, where the calculated streamlines faithfully represent what has been seen in the laboratory by smoke traces in air and particle traces in oils.
Abstract: In this article we present numerical solutions to multicellular natural convection in a vertical enclosure The calculated streamlines faithfully represent what has been seen in the laboratory by smoke traces in air and particle traces in oils The calculated isotherms for air correspond to reported interferometric patterns Solutions exhibiting travelling waves for water were calculated near conditions where they should occur according to linear stability theory Heat-transfer results for air are given and their dependence on the aspect ratio of the enclosure exhibited

193 citations


Journal ArticleDOI
TL;DR: In this paper, the plane, free overfall is treated by using extended energy and momentum equations taking account of the streamline inclination and curvature, and the end-depth ratio is estimated by momentum considerations in the brink and the upstream sections.
Abstract: The plane, free overfall is treated by using extended energy and momentum equations taking account of the streamline inclination and curvature. The end‐depth‐ratio is estimated by momentum considerations in the brink and the upstream sections. Distinction is made between pressure head and flow depth which coincide only for parallel, horizontal streamlines. The surface profiles for the up‐ and downstream zones of the brink section are investigated. Compared with known, laborious calculation procedures, the present computations allow fair and simple determination of all flow quantities.

111 citations


Journal ArticleDOI
TL;DR: In this paper, a numerical scheme for the determination of the fiber orientation state in dilute suspensions is developed, based on the numerical integration of Jeffery's orienta tion equation along the streamlines obtained on the basis of the finite element method.
Abstract: A numerical scheme for the determination of the fiber orientation state in dilute suspensions is developed. Fiber orientations are calculated from the numerical integration of Jeffery's orienta tion equation along the streamlines obtained on the basis of the finite element method. Numerical solutions of the fiber orientation state are presented for the simple shear flow, the fountain flow and the flow in an infinite expansion.

96 citations


Journal ArticleDOI
TL;DR: In this article, an experimental study on the interaction of two two-dimensional turbulent parallel jets is reported, which includes measurements of mean velocity, turbulence intensities, and Reynolds shear stress.
Abstract: Results of an experimental study on the interaction of two two-dimensional turbulent parallel jets are reported. The investigation includes measurements of mean velocity, turbulence intensities, and Reynolds shear stress. The structure of the combined flow is compared with that of a single jet. The results show that the velocity profiles of the combined flow are similar and agree well with that of the single jet. Up to 120 slot widths downstream from the nozzles, true similarity is not found. For a spacing of 12.5 slot widths between the axes of the two nozzles, the half-width of the combined flow grows linearly with downstream distance, but its spread angle is slightly lower than that of a single jet. The centerline velocity decays with the same rate as the single jet, but with a higher value of UM/ U0. Nomenclature 5 = spacing between the two nozzles tp =

85 citations


Journal ArticleDOI
TL;DR: In this paper, the motion of two bubbles along their line of centers in an imposed thermal gradient is considered, and the governing equations are solved in the quasistatic limit using bipolar coordinates.

80 citations



Journal ArticleDOI
TL;DR: In this paper, the mean temperature profiles at the jet exit are found to be considerably nonuniform at low velocity ratios, indicating an early mixing between the cold and hot fluid, and the extra rate of strain due to streamline curvature and the longitudinal and normal temperature gradients affect the rate of generation of temperature fluctuations and turbulent heat fluxes.
Abstract: Measurements are presented of mean temperature in a nonbuoyant heated jet issuing perpendicularly into a cold stream at velocity ratios R=0.25, 0.5, 1, and 2, and of velocity‐temperature fluctuation statistics at R=0.5. The mean temperature profiles at the jet exit are found to be considerably nonuniform at low velocity ratios, indicating an early mixing between the cold and hot fluid. The extra rate of strain due to streamline curvature and the longitudinal and normal temperature gradients affect the rate of generation of temperature fluctuations and turbulent heat fluxes. There are regions of the flow where the streamlines are unstably curved but the generation of ∼(ϑ2) is damped. The eddy diffusivity is well behaved across the flow only at downstream positions where the turbulent Prandtl number has rather high values close to the wall and reaches values around 0.9 away from the wall.

73 citations


Journal ArticleDOI
TL;DR: In this paper, a function ϕ is derived which is constant along the orthogonal trajectories of streamlines in two-dimensional flow, and explicit extra terms appear in the transformed equations embodying the effects of streamline curvature and mean flow acceleration.
Abstract: A function ϕ is derived which is constant along the orthogonal trajectories of streamlines in two-dimensional flow. In irrotational flows, ϕ reduces to the velocity potential. The pair of functions ϕ and ψ, where ψ is the stream function, are used to define a coordinate system in rotational fluid flows. Tensor methods are used to transform the equations of motion of a turbulent fluid and the equations for second moments of turbulent fluctuations to this coordinate system. Explicit extra terms appear in the transformed equations embodying the effects of streamline curvature and mean flow acceleration. These extra terms are characterized by two lengthscales which arise naturally from the transformation: the local radius of curvature of the streamline and the ‘e-folding’ distance of the mean streamwise velocity.

62 citations


Journal ArticleDOI
Kitaro Adachi1
TL;DR: In this paper, integral equations for the relative deformation gradient tensors were solved to give analytical expressions which involve velocities and velocity gradients along streamlines, and the expressions were used to obtain the analytical expressions for the deformation tensors.
Abstract: Integral equations for the relative deformation gradient tensors are solved to give analytical expressions which involve velocities and velocity gradients along streamlines.

Journal ArticleDOI
TL;DR: In this paper, the interference of flow around two circular cylinders forming a cross can be considered as an element of a gauze screen or an offshore structure and the most important feature of the interference was found to be a significant increase of local drag coefficient due to the symmetric formation of secondary flow patterns.
Abstract: The interference of flow around two circular cylinders forming a cross can be considered as an element of a gauze screen or an offshore structure. The most important feature of the interference was found to be a significant increase of local drag coefficient due to the symmetric formation of secondary flow patterns. The pressure distribution measured around both cylinders at various spanwise stations was distinctly different for the two cylinders in the region of interference. A strong secondary flow was found behind the upstream cylinder and in front of the downstream cylinder. The secondary flow distorted and displaced the separation streamlines on the upstream and downstream cylinders. This has been verified by oil-film flow visualization.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the three dimensional hydrodynamic interaction of the sphere with the entrance geometry of the pore in the limit of zero inertia, and the results provided a basic mechanism to explain the onset of Fahraeus phenomenon for red cells entering small blood vessels and glass tubes.

Journal ArticleDOI
TL;DR: In this paper, a swarm theory for flow through a system of fibers is combined with trajectory calculations for colloidal particles to predict single-pass retention efficiencies, and results obtained using other hydrodynamic models to assess the sensitivity of predicted efficiencies with respect to the type of flow model used.

Proceedings ArticleDOI
01 Jul 1983
TL;DR: In this article, a detailed investigation of the unsteadiness in a reattaching, turbulent shear layer is reported, which is loosely described as a "flapping" of the shear layers.
Abstract: A detailed investigation of the unsteadiness in a reattaching, turbulent shear layer is reported. Laser-Doppler velocimeter measurements were conditionally sampled on the basis of instantaneous flow direction near reattachment. Conditions of abnormally short reattachment and abnormally long reattachment were considered. Ensemble-averaging of measurements made during these conditions was used to obtain mean velocities and Rreynolds stresses. In the mean flow, conditional streamlines show a global change in flow pattern which correlates with wall-flow direction. This motion can loosely be described as a 'flapping' of the shear layer. Tuft probes show that the flow direction reversals occur quite randomly and are shortlived. Streses shown also vary with the change in flow pattern. Yet, the global'flapping' motion does not appear to contribute significantly to the stress in the flow. A second type of unsteady motion was identified. Spectral analysis of both wall static pressure and streamwise velocity shows that most of the energy in the flow resides in frequencies that are significantly lower than that of the turbulence. The dominant frequency is at a Strouhal number equal to 0.2, which is the characteristic frequency of roll-up and pairing of vortical structure seen in free shear layers. It is conjectured that the 'flapping' is a disorder of the roll-up and pairing process occurring in the shear layer.

Journal ArticleDOI
TL;DR: In this article, it is shown that Tirrell's diffusion flux can be deduced from a macroscopic modelling which involves second-order gradients and a vectorial internal variable related to the microstructure.
Abstract: The present study concerns the phenomenon of flow-induced polymer migration. It is shown that Tirrell's diffusion flux can be deduced from a macroscopic modelling which involves second-order gradients and a vectorial internal variable related to the microstructure. In contrast to Tirrell's model, however, a migration may also occur across straight streamlines. The flow down an inclined plane is examined by way of example. The roles played by the microstructure and the second gradients, respectively, are thus exhibited.

Journal ArticleDOI
TL;DR: In this paper, a least-square procedure is used to find an interaction solution in which the boundary-layer ap- proximation remains uniformly valid and no additional vortex layers are introduced in the interaction region, thus avoiding the difficulties inherent in Sychev's proposal.
Abstract: Recently, Sychev addressed the question of unsteady interactive separation moving upstream along the wall. Here, by a least-square procedure, an interaction solution is shown to exist in which the boundary-layer ap- proximation remains uniformly valid. No additional vortex layers are introduced in the interaction region, thus avoiding the difficulties inherent in Sychev's proposal. Remarkably, according to the solution, the principal mechanics of separation is a bifurcation-free divergence of streamlines. NSTEADY boundary-layer separation is one of the major fundamental, yet unresolved questions of fluid mechanics. Thus in experimental or numerical work, it remains unclear what are the distinctive features to look for in the results, or how to extrapolate to higher Reynolds numbers than may reliably be determined. The influence of a multitude of factors cannot confidently be estimated, and criteria are missing to judge results where they may mutually contradict each other. As emphasized by Sears and Telionis,1 "separation" is here understood to mean a physically significant departure of the boundary layer away from the wall. The location of zero wall shear is also often loosely denoted as separation, but Moore,2 Rott,3 and Sears4 have all pointed out that for unsteady separation, or for steady separation from moving walls, zero wall shear does not indicate the same physical separation process as for the steady, fixed wall case. Neither are stagnation points a physical characteristic of the flow, since they depend on the choice of the coordinate system for a deformable body in arbitrary motion. But when for sufficiently high Reynolds number the concept of a thin boundary layer can meaningfully be in- troduced, it becomes possible to define separation in the sense that this boundary layer is no longer in the immediate vicinity of the wall as compared to its thickness. And it is this distinction in the location of the boundary layer that will make the physically crucial difference in other flow characteristics as well: wall pressure gradient, wall shear, Kelvin-Helmholtz instability, roll-up, etc. If a description of steady separation from a downstream moving wall can be derived, then the invariance of the flow equations under the Galileo transform may be used to map this one steady separation onto a family of unsteady ones, in all of which the separation point is in upstream motion compared to the wall. Sears and Telionis1 argue that this family should be representativ e of the general case of separation points moving upstream along the wall; certainly the family should be an important example. Thus the object becomes to study steady separation from a downstream moving wall. For this moving wall case, Moore, Rott, and Sears proposed independently that the condition of separation is no longer zero wall shear, but its generalization that the velocity component in the direction of the wall and

Journal ArticleDOI
01 Jan 1983-Tellus A
TL;DR: In this paper, the authors examined the response of ocean currents to forced atmospheric cooling by a simplified two-layer model and found that upon encountering a region of cooling, the interface steepens and the whole current is displaced horizontally.
Abstract: The response of steady ocean currents to forced atmospheric cooling is examined by a simplified two-layer model. The study focuses on currents, such as the Gulf Stream, which lose their heat to the atmosphere as they flow from one region to another. The model is inviscid and includes no coupling between the ocean and the atmosphere. Approximate solutions for specified cooling processes acting steadily on currents whose undisturbed speed is uniform are obtained analytically using a uniformly valid power series expansion. It is found that upon encountering a region of cooling, the interface steepens and the whole current is displaced horizontally. The streamlines in the upper portion of the light layer are displaced to the right (looking downstream) whereas the streamlines in the lower portion of the upper layer are displaced to the left. These movements result from a combined effect of advection and “thermal wind” motion. For actual currents and heat losses in the ocean, the predicted interface steepening is of the same order as the slope upstream and the predicted horizontal displacements during strong cooling processes can be as high as ˜ 100 km. Possible application of this theory to the separated Gulf Stream which loses heat to the atmosphere as it flows from Cape Hatteras toward the northeast is discussed. The model predicts that during the late winter the position of the Gulf Stream front will be farther to the south than it is during the summer and that the slope of the interface will be larger in winter. Both processes agree qualitatively with the observed seasonal variability of the Stream. DOI: 10.1111/j.1600-0870.1983.tb00185.x

Journal ArticleDOI
TL;DR: In this article, the flow between a stationary small diameter needle and a moving drum of much larger diameter, seen as a moving plane, is compared to that of a Newtonian fluid used under the same kinematic conditions.
Abstract: This experimental study is related to the flow between a stationary small diameter needle and a moving drum of much larger diameter, seen as a moving plane. Our visualisation experimental set-up adds new results regarding the flow behaviour of a polymer solution. This flow is compared to that of a Newtonian fluid used under the same kinematic conditions. Important differences between Newtonian and polymeric flows concern mainly streamlines, stagnation point positions, and reverse flow. A second experimental set-up enabled us to obtain normal stress profiles along the needle wall and the flow rate through the gap. Reslts related to Newtonian flows are compared to the well-known analytical solutions for a bidimensional laminar flow.

Journal ArticleDOI
TL;DR: In this paper, a high-specific-speed axial-flow pump impeller was analyzed on the basis of measured streak lines of air bubbles in the impeller, and the results were compared with those obtained by a numerical solution of the bubble motion equations for three dimensional flow.
Abstract: Motion of air bubbles in a high-specific-speed axial-flow pump impeller was analyzed on the basis of measured streak lines of air bubbles in the impeller. The results were compared with those obtained by a numerical solution of the bubble motion equations for three dimensional flow. Governing factors of the bubble motion are the drag force due to the surrounding water and the force due to the pressure gradient. Trajectories of the bubbles deviate somewhat from the streamlines of water, and the amount of the deviation is dependent on the bubble diameter and also on specific-speeds of the pumps and flow rate of water.

Journal ArticleDOI
TL;DR: In this article, the quasi-static problem of the motion of a bubble contained inside a drop in free fall due to a prescribed axisymmetric temperature field on the drop surface is treated theoretically.

Journal ArticleDOI
01 Sep 1983
TL;DR: In this paper, the deformation of a deformed spherical liquid drop has been studied theoretically and the streamlines inside and outside the drop and its deformation have been calculated for the family of linear two-dimensional flows, members of which are determined by a dimensionless parameter −1 ⩽ α ⌽ 1.
Abstract: The creeping flow of a Newtonian fluid around a neutrally buoyant, immiscible spherical liquid drop has been studied theoretically. The streamlines inside and outside the drop and its deformation have been calculated for the family of linear two-dimensional flows, members of which are determined by a dimensionless parameter −1 ⩽ α ⩽ 1. These flows include pure shear flow (α = 1) as one limit and pure rotation (α = −1) as the other, with simple shear (α = 0) as an intermediate case. In the bulk medium, it is found that both an open and a closed streamline region exists for 0 ⩽ α < 1, with the distance from the drop to the limiting streamline dividing the two regions being determined by the ratio of the drop viscosity to that of the medium, λ. All streamlines are open in pure shear flows (α = 1) regardless of λ and for all flows when 0 ⩽ α ⩽ 1 and λ = 0. For flows having −1 ⩽ α < 0, streamlines are always closed. The circulation established in the drop due to the external flow varies according to α and λ. In particular, as many as four pockets of circulation are found for flows with 0 < α ⩽ 1 while for −1 < α ⩽ 0 only two such “pockets” exist. In the trivial case α = −1 there is a single pocket in the drop. Equations giving the axis ratio and orientation of a deformed drop are also derived.

Journal ArticleDOI
TL;DR: In this paper, the pathlines around oblate and prolate spheroids freely rotating in shear flow according to Jeffery's equations have been calculated numerically, and the relevance of pathlines to problems of heat and mass transfer and particle deposition in flowing sols is discussed.

Journal ArticleDOI
TL;DR: In this article, the velocity field around a vortex shedding strut mounted in a circular pipe has been observed in detail with a laser Doppler velocimeter (LDV) at a pipe Reynolds number equal to 90,000.
Abstract: The velocity field around a vortex shedding strut mounted in a circular pipe has been observed in detail with a laser Doppler velocimeter (LDV) at a pipe Reynolds number equal to 90,000. The instantaneous velocity is decomposed into mean, periodic, and random components. Only the first two harmonics are large enough to be detected; the large-scale structure can be characterized by just these two these and the mean. Profiles of the different velocity terms are given upstream of, downstream of, and close to the strut. The two-dimensional velocity vector field of the mean flow on the transverse diametral plane of symmetry is presented along with its streamlines. Finally, for each spatial component, profiles of vortex visibility, the ratio of the energy of a periodic component to the total fluctuating energy in a narrow frequency band, are given.

Journal ArticleDOI
TL;DR: In this paper, a one-fluid model is employed to study the global expansion of the solar wind from a two-hole corona, under the assumptions that the holes are confined to polar caps within 30 deg of heliographic colatitude.
Abstract: A one-fluid model is employed to study the global expansion of the solar wind from a two-hole corona, under the assumptions that the holes are confined to polar caps within 30 deg of heliographic colatitude, the flow is steady and axisymmetric, and the geometry of streamlines is prescribed. The boundary conditions are adjusted in such a way that the calculated solar-wind properties at 1 AU are in reasonable agreement with observational results. A series of numerical solutions are obtained, the series produces a maximum terminal speed of 829 km/s at the pole. The calculated solar-wind speeds are strongly latitude-dependent and are positively correlated with local divergence factor of a stream tube. The solutions imply that most plasma properties are highly inhomogeneous at the polar caps. The flow velocity, the temperature, the proton-number flux and the conduction-heat flux all increase towards the hole center.

Journal ArticleDOI
TL;DR: In this article, the authors used streak photographs and stress birefringence techniques to analyze the flow of poly(ethylene terephthalate) through potential chain-ordering die geometries.
Abstract: Streak-photographic and stress birefringence techniques were used to analyze the flow of poly(ethylene terephthalate) through potential chain-ordering die geometries. The streak photographs were used to determine velocity distributions and streamlines in various convergent dies. The different contours were seen to have a significant effect on the polymer streamlines and velocity distributions. The measured velocities were used to develop empirical equations, specific for each geometry, which relate velocity to position within the die and to throughput rate. Flow birefringence was used to determine the extent of molecular ordering. The optimum chain-ordering die geometry was determined to be one which included a rapid initial decrease in cross-sectional area. Birefringence was also used to monitor polymer flow instability. An unusual mechanism for instability was observed at intermediate throughput rates.

Journal ArticleDOI
TL;DR: In this paper, the three-dimensional flow of a hyersonic stream of ideal gas round bodies of arbitrary thickness allowing for radiation at high temperatures is investigated using the method of a thin optically transparent shock layer, which is a generalization of the well-known method of thin boundary layer /1/.

Proceedings ArticleDOI
27 Mar 1983
TL;DR: In this article, a computational model was developed for calculating the flow field, in the presence of solid particles, through a two-dimensional compressor cascade, and it was found that the total pressure loss associated with the particulate flow is very high for very small particles as compared with large particles.
Abstract: A computational model is developed for calculating the flow field, in the presence of solid particles, through a two-dimensional compressor cascade. Results show that the effect of solid particles on the flow field contributes to the bending of the streamlines toward the blade suction surface. It is determined that the difference in the pressure coefficient for particulate flow, with 165 micron diameter for the particles, is of the order of 3 percent over the air only flow. The change in the pressure coefficients is shown to be much larger for very small particles. It was found that the total pressure loss associated with the particulate flow is very high for very small particles as compared with large particles. The total pressure loss is also shown to be higher for accelerating flow than for decelerating flow. The total pressure loss is shown to be directly proportional to the particle concentration. It is concluded that significant reductions in performance can occur in a real multistage machine due to the changes in the pressure ratios, particularly if the suspended matter is small particles.

Journal ArticleDOI
TL;DR: The boundary-value problem posed is based on the model developed by Keller & Skalak and is solved by adapting Jeffrey's general solution for the Stokes flow about a rigid, freely rotating ellipsoid immersed in an unbounded viscous flow.

Journal ArticleDOI
TL;DR: In this article, the theory of steady transonic winds from condensed bodies is extended to general, two-dimensional, axisymmetric systems, and a stream function is used to reduce the gasdynamics equations to a single, second-order differential equation plus an algebraic equation for the density.
Abstract: The theory of steady transonic winds from condensed bodies is extended to general, two-dimensional, axisymmetric systems A stream function is used to reduce the gasdynamics equations to a single, second-order differential equation plus an algebraic equation for the density The approach extends Parker's (1958) quasi-one-dimensional theory, which uses Bernoulli's theorem, to a complete two-dimensional calculation which includes vorticity and rotation The conceptual basis of the stream function approach is described in detail, and a numerical method for solving the resulting equations is presented The applications illustrate solutions for stellar (spherical source surface) and galactic (oblate spheroidal source surface) models Among other things, it is found that for rapidly rotating stellar models the Coriolis force dominates the centrifugal terms and the streamlines bend toward the rotation axis as a consequence of the axisymmetry For both the rotating and nonrotating galaxy models, the streamlines bend toward the equator