Fluid flow through a helical pipe

In this paper we study the flow of incompressible Newtonian fluid through a helical pipe with prescribed pressures at its ends. Pipe’s thickness and the helix step are considered as the small parameter ɛ. By rigorous asymptotic analysis, as ɛ→ 0 , the effective behaviour of the flow is found. The error estimate for the approximation is proved.     

Investigation of micropolar fluid flow in a helical pipe via asymptotic analysis

In this paper we study the flow of incompressible micropolar fluid through a pipe with helical shape. Pipe’s thickness and the helix step are considered as the small parameter ε. Using asymptotic analysis with respect to ε, the asymptotic approximation is built showing explicitly the effects of fluid microstructure and pipe’s distortion on the velocity distribution. The error estimate for the approximation is proved rigorously justifying the obtained model.     

Fluid flow through a vertical slot under gravity

The flow of fluid from a horizontal uniform channel of finite depth into a vertical slot under the influence of gravity is considered. The shapes of a top and bottom free surface are computed for a range of parameter values. Solutions in which the top free surface attaches smoothly to the vertical wall were only found for Froude numbers greater than or equal to one, though solutions were found for all gap sizes.     

Non-isothermal fluid flow through a thin pipe with cooling

The stationary flow of a Boussinesquian fluid with temperature-dependent viscosity through a thin straight pipe is considered. The fluid in the pipe is cooled by the exterior medium. The asymptotic approximation of the solution is built and rigorously justified by proving the error estimate in terms of domain thickness. The boundary layers for the temperature at the ends of the pipe are studied.     

On the flow of a cosserat fluid through a curved pipe

Summary A theoretical analysis is made of the flow of a Cosserat fluid in a curved pipe under a pressure gradient. It is assumed that the curvature of the pipe is small, that is the radius of the circle in which the central line of the pipe is coiled is large in comparison with the radius of the cross-section. Following Dean [2] a solution is developed by the method of successive approximation. The paths of the particles in the central plane and the projection of the streamlines on the cross-section of the pipe are compared with those of a Newtonian fluid. It is observed that in the theory of Cosserat fluids the curvature of the streamlines in the central plane increases and the motion is slower in the cross-section of the pipe. It is also shown that the rate of flow of a Cosserat fluid through a curved pipe is decreased due to the curvature of the pipe.

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Résumé On fait une analyse théorique de l'écoulement d'un fluide de Cosserat dans un tube sous un gradient de pression. On suppose que la courbature du tube est faible, c'est-à-dire que le radius du cercle qui fait la ligne du centre du tube est fort par rapport au radius de la coupe transversale.D'après Dean [2], on développe une résolution par approximations successives. On fait la comparaison des trajectoires des particules dans le plan central et la projection des lignes d'écoulement sur la coupe transversale du tube avec celles d'un fluide de Newton.On note que dans la théorie des fluides de Cosserat, la courbature des lignes d'écoulement dans le plan central augmente, et que la motion est ralentie dans la coupe transversale du tube. On démontre ensuite que le taux d'écoulement d'un fluide de Cosserat dans un tube courbe se diminue à raison de la courbature.

     

On the flow of a dusty viscous liquid through a circular pipe

Analytical solution of the flow problem of a dusty viscous liquid through a circular pipe in case of axial symmetry is obtained when pressure gradient varies harmonically with time. It is found that the effect of the fine dust is to make the velocity of sedimentation zero and when dust is sufficiently coarse, the effect of the dust is equivalent to an extra frictional force proportional to the fluid velocity.     

Asymptotic analysis of the nonsteady micropolar fluid flow through a curved pipe

ABSTRACT

We consider the nonsteady flow of a micropolar fluid in a thin (or long) curved pipe via rigorous asymptotic analysis. Germano's reference system is employed to describe the pipe's geometry. After writing the governing equations in curvilinear coordinates, we construct the asymptotic expansion up to a second order. Obtained in the explicit form, the asymptotic approximation clearly demonstrates the effects of pipe's distortion, micropolarity and the time derivative. A detailed study of the boundary layers in space is provided as well as the construction of the divergence correction. Finally, a rigorous justification of the proposed effective model is given by proving the error estimates.     

Rarefied gas flow through a pipe of variable square cross section into vacuum

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.     

Dispersion of solid particles in turbulant flow through pipe expansions

Studies of the effects upon gas-liquid two-phase flows of pipefittings such as expansions, contractions, bends, and valveshave usually concentrated upon pressuredrop correlations andhave not attempted to determine changes in the distributionsof the gas and liquid phases caused by the fitting. However,it is known that such information is important if, for example,flow separators, which divide the gas and liquid phases in avariety of industrial processes, are to function efficiently.It is therefore important to gain an understanding of the influenceupon phase distributions of the common pipe fittings mentionedabove, which will be found in almost any industrial pipeworksystem. As a first step, the dispersion of solid particles carriedby turbulent gas flows through a pipe expansion has been modellednumerically. The commercial fluid-flow code CFDS-FLOW{smalltilde}hDas been used to model the gas flow, together with aneddy interaction model for determination of the motion of thesolid particles. Mean particle velocities and root-mean-squarevalues of the particle velocity fluctuations, as well as particleconcentrations, are evaluated and compared with recent experimentalresults. The influence of different eddy-length and eddy-lifetimespecifications upon the dispersion of particles of various sizesis investigated. It is found that the different eddy characteristicshave little effect on predicted mean particle velocities, whereasfluctuations in particle velocities and particle concentrationare sensitive to the changes made. By comparing the resultswith experimental data, it is possible to draw conclusions aboutthe relative merits of the different eddy specifications.     

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation.     

Steady laminar flow with suction or injection and heat transfer through porous pipe

Following the method due to Bhatnagar (P. L.) [Jour. Ind. Inst. Sic., 1968, 1, 50, 1], we have discussed in this paper the problem of suction and injection and that of heat transfer for a viscous, incompressible fluid through a porous pipe of uniform circular cross-section, the wall of the pipe being maintained at constant temperature. The method utilises some important properties of differential equations and some transformations that enable the solution of the two-point boundary value and eigenvalue problems without using trial and error method. In fact, each integration provides us with a solution for a suction parameter and a Reynolds number without imposing the conditions of smallness on them. Investigations on non-Newtonian fluids and on other bounding geometries will be published elsewhere.     

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation. G. P. Galdi: Partially supported by the NSF grant DMS–0404834. K. Pileckas: Supported by EC FP6 MCToK program SPADE2, MTKD–CT–2004–014508 A. L. Silvestre: Supported by FCT-Project POCI/MAT/61792/2004     

On the modeling and simulation of boundary flow through partially open pipe ends

The determination of boundary conditions for the Euler equations of gas dynamics in a pipe with partially open pipe ends is considered. The boundary problem is formulated in terms of the exact solution of the Riemann problem and of the St. Venant equation for quasi-steady flow so that a pressure-driven calculation of boundary conditions is defined. The resulting set of equations is solved by a Newton scheme. The proposed algorithm is able to solve for all inflow and outflow situations including choked and supersonic flow.Received: August 7, 2002; revised: November 11, 2002     

On reactive solute transport through a curved pipe

The transport of a reactive solute by diffusion and convection in a thin (or long) curved pipe is considered. Using asymptotic analysis with respect to the pipe’s thickness, the effective model for solute concentration is formally derived. A simple approximation is computed, showing explicitly the effects of the pipe’s geometry in nature and magnitude.     

Axial dispersion in pressure perturbed flow through an annular pipe oscillating around its axis

A method of moment is employed to study the axial dispersion of passive tracer molecules released in an unsteady pressure-driven flow through an annular pipe which is oscillating around its longitudinal axis. The flow unsteadiness is caused by the oscillation of the tube around its axis as well as by a periodic pressure gradient. A finite difference implicit scheme is adopted to solve the Aris integral moment equations arising from the unsteady convective-diffusion equation for all time periods. The main objective is to study the nature of the dispersion coeffcient and mean concentration distribution under the sole as well as combined oscillation of the two driving forces. The behaviour of the dispersion coeffcient due to the variation of the aspect ratio, the absorption parameter for purely periodic flow has been examined and the sound response from dispersion coeffcient is found with the variation of these parameters in the sole presence of pressure pulsation. There is a remarkable difference in the behavior of the dispersion coeffcient depending on whether the ratio of two frequencies arising from the oscillations of the tube and the pressure gradient possesses a proper fraction or not. Oscillation of the tube produces much more dispersion than the pulsation of the pressure gradient and their combined effect leads to a further increase in dispersion. Tube oscillation shows a stronger effect on the dispersion coeffcient than the pressure pulsation though the effect of physical parameters are pronounced in the presence of pressure pulsation. The effect of the frequency parameter on the axial distribution of mean concentration is insensible when the oscillation of the annular tube is the only forcing. However this effect is much noticeable under the combined action of both forcing and much more effective under the sole influence of pressure pulsation.     

Axial dispersion in pressure perturbed flow through an annular pipe oscillating around its axis

A method of moment is employed to study the axial dispersion of passive tracer molecules released in an unsteady pressure-driven flow through an annular pipe which is oscillating around its longitudinal axis. The flow unsteadiness is caused by the oscillation of the tube around its axis as well as by a periodic pressure gradient. A finite difference implicit scheme is adopted to solve the Aris integral moment equations arising from the unsteady convective-diffusion equation for all time periods. The main objective is to study the nature of the dispersion coeffcient and mean concentration distribution under the sole as well as combined oscillation of the two driving forces. The behaviour of the dispersion coeffcient due to the variation of the aspect ratio, the absorption parameter for purely periodic flow has been examined and the sound response from dispersion coeffcient is found with the variation of these parameters in the sole presence of pressure pulsation. There is a remarkable difference in the behavior of the dispersion coeffcient depending on whether the ratio of two frequencies arising from the oscillations of the tube and the pressure gradient possesses a proper fraction or not. Oscillation of the tube produces much more dispersion than the pulsation of the pressure gradient and their combined effect leads to a further increase in dispersion. Tube oscillation shows a stronger effect on the dispersion coeffcient than the pressure pulsation though the effect of physical parameters are pronounced in the presence of pressure pulsation. The effect of the frequency parameter on the axial distribution of mean concentration is insensible when the oscillation of the annular tube is the only forcing. However this effect is much noticeable under the combined action of both forcing and much more effective under the sole influence of pressure pulsation.     

An exact solution for flow in a porous pipe

Summary An exact solution of the Navier-Stokes equations for flow in a porous pipe is presented. This solution allows the suction or injection at the wall to vary with axial distance and will provide new insight into flows through porous pipes.

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Resumé Une solution exacte d'équation de Navier-Stokes est présentée pour l'écoulement d'un liquide visqueux dans un tube perméable. Ce liquide est aspiré ou injecté avec une vélocité variable et la solution donne une nouvelle optique quant aux tubes poreuses.

     

On axisymmetric Stokes flow past a finite circular pipe

The axisymmetric streaming Stokes flow past a hollow boundary of arc cross-section leads to a mixed boundary value problem that has only been solved for a hollow sphere with equal caps removed. Here a finite circular cylinder is considered and the solution is obtained via dual integral equations, involving modified Bessel functions, arising from Fourier transforms. Numerical values for the flux and drag are obtained, with particular interest accruing to the case of small pipelength for comparison with the spherical hollow boundary.

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Zusammenfassung Die axisymmetrische Stokes-Strömung um (und durch) eine hohle Begrenzung mit bogenförmigem Querschnitt führt auf eine Randwertaufgabe vom gemischten Typus, welche bis jetzt nur für eine Hohlkugel (mit zwei identischen Kugelkappen entfernt) gelöst wurde. Hier wird ein endlicher Kreiszylinder behandelt und die Lösung wird mit Hilfe von dualen Integralgleichungen erhalten; diese enthalten modifizierte Besselfunktionen, die sich durch Fourier-Transformationen ergeben. Numerische Ergebnisse werden für den Fluss und den Widerstand angegeben, wobei das sehr kurze Rohr besonders interessant ist für den Vergleich mit dem Fall der Hohlkugel.

     

Numerical study of transition in a cylindrical pipe flow

We present numerical results of transition in a smooth cylindrical pipe by small periodical suction and blowing (PSB) at the inlet of a laminar pipe flow at Reynolds number 3000 based on the maximum velocity of the laminar flow and radius of the pipe. The spatial development of the PSB disturbance is simulated by means of pseudo-spectrum element method. The transitional process is described in the paper that the disturbances are growing rapidly in a short part of the pipe after they develop gradually in sufficient long distance. When the rapid growth of disturbances occurs the time step of integration should be decreased and then the flow transits to turbulent.