Steady laminar flow of a power-law fluid in a curved pipe of circular cross-section with varying curvature

The steady laminar flow of power-law fluid through pipes of circular cross-section, whose center-line curvature varies locally, is analyzed theoretically. The flows, in three kinds of pipes whose center-lines are specified by

as examples of once, twice, and periodically-curved pipes, respectively, are discussed in comparison with Newtonian flow. The analysis is valid for any other two-dimensionally curved pipes, when the center-line curvature is small.     

椭圆截面曲线管道内二次流动的Galerkin解

推导了任意空间曲线直角坐标系内的张量流体力学方程．采用Galerkin方法和计算机符号运算技术求解了椭圆截面螺旋管道（电括弯管、扭管）内的二次流动．计算结果表明了Galerkin方法的适用和有效性，克服了振动法的小参数局限．对所得结果的分析揭示了椭圆截面螺旋管道内流动的特性，给出了k,r和Re较大情况的研究结论．     

Simplified Navier-Stokes Equations for Internal Mixed Flows and a Numerical Method of Solution

Simplified Navier-Stokes equations, of the elliptic and hyperbolic type in the subsonic and supersonic flow regions, respectively, are derived for viscous flows in channels and nozzles with curved walls whose local radii of longitudinal curvature are comparable with the transverse channel dimensions. A new numerical method is developed for the system of equations obtained. This method is of the evolution type along the longitudinal coordinate and includes global iterations of the streamline direction field and the longitudinal pressure gradient field. The effectiveness of the method is illustrated with reference to the solution of the direct Laval nozzle problem for an air flow at Reynolds numbers Re10⁴ and 10⁶ in conical nozzles with throat curvatures K _w=1.0 and 1.6 (K _w is the curvature divided by the inverse radius of the nozzle throat). Two iterations are sufficient to calculate the nozzle flow rate and power correct to 0.01%.     

Stability analysis of viscoelastic curved pipes conveying fluid

Based on the Hamilton' s principle for elastic systems of changing mass, a differential equation of motion for viscoelastic curved pipes conveying fluid was derived using variational method, and the complex characteristic equation for the viscoelastic circular pipe conveying fluid was obtained by normalized power series method. The effects of dimensionless delay time on the variation relationship between dimensionless complex frequency of the clamped-clamped viscoelastic circular pipe conveying fluid with the Kelvin-Voigt model and dimensionless flow velocity were analyzed. For greater dimensionless delay time, the behavior of the viscoelastic pipe is that the first, second and third mode does not couple, while the pipe behaves divergent instability in the first and second order mode, then single-mode flutter takes place in the first order mode.     

Streamline flow through a curved annulus

Summary Theoretical calculations are carried out for the steady laminar incompressible flow of an ordinary viscous fluid in a curved annulus. At first the flow parameters are evaluated for the general case and then a particular example is solved numerically. The graphical representations of the flowline in the plane of symmetry and the projection of the stream-lines on a normal section are given. Only the case of large radius of curvature of the annulus is considered; more precisely this means that the ratio of the radius of the outer curved pipe to that of the circle in which the common axis of the two curved pipes is coiled is sufficiently small.     

Viscous fluid flow near the line of intersection of curved surfaces

Viscous fluid flow near the line of intersection of curved surfaces at large Re numbers is a topic of considerable interest. The intersection of two fixed planes has been the subject of many experimental and theoretical studies. This case is characterized by very small transverse velocities and by the fact that the corner does not affect the remoter parts of the flow [1–4]. The flows near intersecting curved surfaces have received very little attention, except for the particular case of the intersection of a concave cylindrical surface and a plane in an incompressible fluid flow. With reference to this example it has been shown that the curvature qualitatively affects the flow pattern not only near the line of intersection but also at a distance from it [5]. The present article is concerned with viscous fluid flow at Re1 near the line of intersection of arbitrary, relatively smooth surfaces in the presence of external body forces and, moreover, in the noninertial coordinate system moving with the exposed surfaces (for example, rotating surfaces). On the basis of an analysis of the Navier-Stokes equations and the energy equation as Re sufficient conditions are obtained for the development of intense transverse flows near the line of intersection, which also lead to a qualitative change in the flow pattern; it is shown that depending on the external forces and the geometric parameters of the surfaces various types of flow are possible; the relations determining the occurrence of a particular type of flow and the equations and necessary boundary conditions describing some of these flows are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 16–21, May–June, 1985.     

Numerical simulation of inertial flow of heated and cooled viscoelastic fluids inside a planar sudden expansion channel: investigation of stresses effects on the total dissipation

In this paper, the inertial and non-isothermal flow of viscoelastic fluids in a planar channel with 1:3 sudden expansion has been simulated for Brinkman numbers in the range \( - \,20 \le Br \le 20 \). The mass, momentum and energy conservation equations with the non-linear form of Phan-Thien–Tanner constitutive equation are used to describe the behavior of heated and cooled viscoelastic fluids flow. The properties of fluid are assumed temperature-dependent and the viscous dissipation terms are considered in the energy equation. The object of the current paper is to investigate the stresses and their effects on heat generation via the viscous dissipation terms in the energy equation for inertial flow of heated and cooled viscoelastic fluids. Therefore, plots of streamlines, isothermal lines, normal stress (\( \tau_{xx} \)), normal-transverse stress (\( \tau_{yy} \)) and shear stress (\( \tau_{xy} \)), total dissipation, temperature and local Nusselt numbers have been drawn and examined in the channel expansion. The results show that for the asymmetric flow of heated and cooled viscoelastic fluids, the maximum values of total dissipation are located adjacent to the lower wall and at the centerline of the channel expansion. Also, by incrementing the Brinkman number in the hydrodynamically and thermally developing and fully developed zones, the values of total dissipation are increased.     

Memory effects in a non-uniform flow: A study of the behaviour of a tubular film of viscoelastic fluid

Summary Formal use of constitutive equations such as that ofOldroyd in the mathematical model of a flow leads, in general, to a higher order differential equation than is obtained for a purely viscous fluid, and so we expect to need more boundary conditions in order to specify the problem completely. (These extra boundary conditions may be thought of as arising from the need to specify what the fluid remembers of the flow outside the region of interest.) In flows which are uniform spatially, or uniform with time for a material element, the uniformity will provide the extra information and so no extra conditions are needed. Similarly for confined flows, where no new fluid enters the region of interest, no information about flow outside this region is needed.Here the steady flow of a tubular film of a viscoelastic fluid is studied with the particular aim of examining the effect of these extra boundary conditions in a situation where they may be expected to have some significant influence on the flow as a whole. The flow, while being geometrically complex, is essentially an elongational free-surface flow involving the biaxial stretching of a thin axisymmetric tubular film. Features of the constitutive equations studied are the presence of a non-zero relaxation time and the possibility of a variable viscosity. One effect of the non-zero relaxation time is that a tube of constant radius (possible but unstable for aNewtonian fluid) is not dynamically possible. Preliminary computational results suggest that the effect of the extra upstream boundary conditions is not large, and also have failed to show any major difference between the two generalisations of theMaxwell model which have been used.With 1 figure     

Spatial Motion of a Rod in a Viscous Fluid Flow

Equations of spatial motion of a curved finitelength rod in a viscous fluid flow are derived. Analytical solutions of problems on the motion of a straight rod under conditions of pure shear, simple shear, and uniaxial extension of the fluid are obtained. Longitudinal stability of the straight rod during its spatial motion is considered. Effective viscosity of a suspension filled by rigid straight rods is evaluated.     

Development of secondary flows in viscoelastic curved ducts and the influence of outlet region

This paper presents numerical simulations of Newtonian and viscoelastic flows through a 180° curved duct of square cross section with a long straight outlet region. A particular attention is paid to the development of the flow in the output rectangular region after the curved part. The viscoelastic fluid is modeled using the constitutive equation proposed by Phan–Thien–Tanner (PTT). The numerical results, obtained with a finite-volume method, are shown for three different Dean numbers (125,137,150)$(125,137,150)$ and for three Deborah numbers (0.1,0.2,0.3)$(0.1,0.2,0.3)$. The necessary outlet length to impose boundary conditions is presented and discussed for these cases. Streamlines and vortex formation are shown to illustrate and analyze the evolution of the secondary flow in this region.     

Determination of separated region in a curved tube

The formation of atherosclerosis in a curved aorta is closely related to the existence of separated vortex region. This paper deals with the steady laminar motion of an incompressible Newtonian fluid through a curved tube with circular cross-section whose curvature is small and whose curvature gradient is not too large. Using the momentum integral method and the approximation of quasi-constant curvature, an equation which determines the location of separation and reattachment is derived. From this equation the earliest point of separation and the corresponding critical Reynolds number are obtained, and the relation between the position of separation and reattachment and Reynolds number R_e for different azimuthal angle are revealed. It is concluded that the separation first emerges at the position whose curvature gradient has the maximum absolute value. With increasing R_e, the separation region extends in the direction of mainstream, azimuthal angle and radius vector, and then forms a three-dimensional separated vortex, which gradually enlarges in all three directions with the increase of Reynolds number. The theoretical results also very clearly demonstrate the following striking experimental fact: if a symmetrical curved tube exhibits a separated vortex at the outside of the upstream, then it must have another one symmetrically placed at the inside of its downstream.     

Experimental investigations and numerical simulations for an open channel flow of a weak elastic polymer solution around a T-profile

The present paper is concerned with experimental and numerical investigations of planar complex flows of weak elastic polymer solutions (whose concentration are below the critical overlap concentration), characterised by small relaxation times (<0.1 s) and almost constant shear viscosities for small and medium shear rates. The main aim of the study is to detect to what extent a very small amount of elasticity present in a viscous fluid can influence its behaviour in complex flows, without introducing major modifications of classical rheological tests. The samples are polymer solutions of low PIB molecular weight dissolved in highly viscous Newtonian mineral oil. The analysed motion is steady, and takes place in an open channel around a T profile. Maximum values of the characteristic parameters for the experiments, the Reynolds and Weissenberg numbers, were 45 and 0.1, respectively. The experiments show a decrease of the wake length downstream the profile for weak elastic solutions in comparison to the Newtonian solvent. Actually, the same wake length as in the Newtonian case was obtained for tested polymer solutions, but at higher Re numbers. Numerical simulations using the Giesekus model predict the same behaviour and are consistent with experiments from both qualitative and quantitative point of views. The results of research conclude that, even in small amounts, the presence of elasticity in pure viscous liquids induces quantitative changes from Newtonian flow in complex dominant elongational flows, at elongational rates for which the sudden thickening of extensional viscosity is remarkable. The study is important, since it should enable better understanding and modelling of viscoelastic flows that involve dilute polymer solutions, or fluids with similar rheology; biofluid mechanics being one area of application of this research. Corroboration of experimental flow visualization with numerical simulation is currently a feasible method used to characterise weak elastic polymer solutions, since classical rheological techniques generally fail to obtain realistic values of relaxation time for these particular viscoelastic fluids. Corneliu Balan dedicates this paper to the anniversary of one hundred years from the birth of Academician Dumitru Dumitrescu (1904–1983), charismatic personality of the Romanian school of fluid mechanics.

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Experimental Investigation of the Transverse Self-Oscillations of Circular Cylinders Mounted with a Narrow Clearance in a Plane Channel

The results of an investigation of the flow past and the behavior of free bluff bodies mounted in pipes and channels with a narrow clearance, conducted in the Institute of Mechanics of Moscow State University, are presented. The drag of circular cylinders of different size and mass in a circulation-free water flow in a plane channel of rectangular cross-section was studied in the transverse self-oscillation regime. The experiments were conducted for Reynolds numbers based on the cylinder diameter 1.7 ? 10⁴ ≤ Re ≤ 7.2 ? 10⁴, relative clearances \(\bar S\) based on a cross-sectional area ranging from 0.76 to 0.9, and cylinder-to-water density ratios ρ _c /ρ ranging from 1.29 to 8.2. Only the case of intense transverse self-oscillations accompanied by impact interaction with the channel wall was considered. The dependence of the period-average cylinder drag coefficient C _x on the basic dimensionless relevant parameters is obtained. The dependence of the dimensionless self-oscillation frequency determined in [1] is refined. The kinematic and dynamic features of the flows past spheres in cylindrical pipes and cylinders in plane channels are compared in the transverse self-oscillation regime.     

Optimal numerical flux of power-law fluids in some partially full pipes

Consider the steady state pressure driven flow of a power-law fluid in a partially filled straight pipe. It is known that an increase in flux can be achieved for a fixed pressure by partially filling the pipe and having the remaining volume either void or filled with a less viscous, lubricating fluid. If the pipe has circular cross section, the fluid level which maximizes flux is the level which avoids contact with exactly 25% of the boundary. This result can be proved analytically for Newtonian fluids and has been verified numerically for certain non-Newtonian models.

This paper provides a generalization of this work numerically to pipes with non-circular cross sections which are partially full with a power-law fluid. A simple and physically plausible geometric condition is presented which can be used to approximate the fluid level that maximizes flux in a wide range of pipe geometries. Additional increases in flux for a given pressure can be obtained by changing the shape of the pipe but leaving the perimeter fixed. This computational analysis of flux as a function of both fluid level and pipe geometry has not been considered to our knowledge.

Fluxes are computed using a special discretization scheme, designed to uncover general properties which are only dependent on fluid level and/or pipe cross-sectional geometry. Computations use finite elements and take advantage of the variational structure inherent in the power-law model. A minimization technique for approximating the critical points of the associated non-linear energy functional is used. In particular, the numerical scheme for the non-linear partial differential equation has been proved to be convergent with known error estimates. The numerical results obtained in this work can be useful for designing pipes and canals for transportation of non-Newtonian fluids, such as those in chemical engineering and food processing engineering.     

Structure and rheology of fiber-laden membranes via integration of nematodynamics and membranodynamics

This paper presents a study of the structure and dynamics of rigid fiber-laden deformable curved fluid membranes based on an viscoelastic model that integrates the statics of anisotropic membranes, the planar nematodynamics of fibers and the dynamics of isotropic membranes. Fiber-laden membranes arise frequently in biological systems, such as the plant cell wall and in protein–lipid bilayers. Based on the membrane's force and torque balance equations and the fiber's balance of molecular fields, a viscoelastic anisotropic model that provides the governing equations for the membrane's velocity and curvature and the fiber structure (fiber orientation and order) is found. A Helmholtz free energy that incorporates the tension/bending/and torsion membrane elasticity, the Landau–de Gennes fiber ordering, and fiber order-membrane curvature interactions is used to derive elastic moments, torques, and stresses. The corresponding viscous stresses and moments include the Boussinesq–Scriven contributions as well as bending, torsion, and rotational dissipation. A spectral decomposition leads to the main viscoelastic material functions for anisotropic fluid membranes. Applications of the rheological model to cylindrical growth and cylindrical axial stretching show that competing curvo-phobic, curvo-philic interactions under extensional flow predict transitions between axial and azimuthal fiber arrangements, of interest to cellulose fiber orientation in plant morphogenesis.     

Secondary cells and separation in developing laminar curved-pipe flows

Laminar flows through 180° curved bends of circular cross section are investigated numerically. For small curvature ratio, , defined as pipe radius over mean bend radius, the governing equations could be parabolized. The equations are solved for an range of from 0.04 to 0.143, a Dean number (De) range of from 277.5 to 1360, and for a uniform flow, a potential vortex, and a parabolic flow inlet condition. In all these studies a zero cross-stream flow at the inlet is assumed. A detailed study of the effects of , De, and inlet condition on the secondary flow pattern is carried out. Within the range of parameters investigated, up to three secondary cells are found in the cross-stream half-plane of a curved pipe. They are the Dean-type secondary cell, a secondary separation cell near the inner bend (closest to the center of curvature of the bend), and a third cell near the pipe center. The number of secondary cells in the cross-stream half-plane is greatly influenced by the inlet flow, and to a much lesser extent by and De. For example, only the Dean cell is found in a curved-pipe flow where and De are small and the inlet flow is either uniform or a potential vortex. When the inlet condition of the same case is changed to a parabolic flow, a three-cell structure results. Furthermore, as De increases to 1180, incipient axial flow separation begins at around 23° downstream of the curved-pipe entrance. The formation and extent of the separation and third cells are investigated together with their dependence on the parameters studied. This investigation further shows that, within the range of parameters examined, there is no secondary cell occurring near the outer bend, contrary to some earlier findings on fully developed curved-pipe flows.This work was supported by the Office of Naval Research under Grant No. N0014-81-K-0428 and by DTRC, Annapolis, Maryland, under Contract No. N00167-86-K-0075. Also, support in the form of an IPA awarded to RMCS during his sabbatical leave at DTRC, Annapolis, Maryland, in the spring of 1990 is gratefully acknowledged.