Tubes Of Arbitrary Cross-section Research Articles

Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section

Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section is explored. A constitutive equation superposing elasticity on the viscoplastic behavior through a linear combination of the non-linear simplified Phan–Thien–Tanner constitutive model of viscoelasticity and the non-linear Bingham constitutive equation of viscoplasticity is developed. The equation of motion is solved analytically for the longitudinal field for steady flow in tubes of non-circular cross-section. The evolution of the plug and stagnant zones, hallmarks of viscoplastic behavior, is studied when elasticity is present, and the rate of flow is determined in terms of the Weissenberg and Bingham numbers. Elasticity tends to enhance the rate of flow for given viscoplastic conditions, and stagnant and plug zone configurations are substantially altered.

A BEM formulation of two dimensional steady state heat conduction in exchanger tubes of arbitrary cross sections

This article focuses on investigating the steady state heat conduction through the walls of heat exchanger tubes of any arbitrary geometry using a numerical technique called boundary element method (BEM) for two different cases. In the first case, the problem is studied for exchanger tubes where both the inner and outer surfaces are maintained at isothermal condition. Secondly, BEM is used for the prediction of heat transfer through exchanger tubes with the inner surface kept at a constant temperature and the outer surface subjected to convective condition. A two dimensional semi-analytic technique called boundary collocation method (BCM) is also used to solve the same two dimensional heat conduction problem for the purpose of comparison. The conduction shape factor (CSF) calculated through BEM for all the shapes in the first case shows excellent agreement with that obtained through BCM and other published results. Further, the heat transfer calculated by BEM is in good agreement with that obtained through the numerical software Ansys Fluent®. For both the cases, the temperature profiles computed by BEM match very closely with those generated through Ansys Fluent®. The problem has been solved for some exchanger tubes having different types of geometries which pose numerical complication in grid-based techniques. The excellent agreement of the results of BEM with the existing results shows the accuracy and efficiency of the method.

The Graetz problem in tubes of arbitrary cross section

The Graetz problem is investigated analytically for the steady laminar flow of Newtonian fluids in tubes of arbitrary cross section. The one-to-one mapping introduced satisfies the no-slip condition and allows the determination of the velocity field in a wide spectrum of arbitrary tube contours. The energy equation is solved for the temperature field in the corresponding tube contours, and the temperature distribution for the triangular, square and circular cross-sectional tubes is presented as particular case. Furthermore, in order to illustrate its relevance for a moderate Peclet number (Pe) regime, the solution applied to the square cross section is compared to numerical simulations for two scenarios, Case I with Pe = 100 and Case II with Pe = 500. It is found that for Case I the relative error does not exceed 2.9 %, being maximum at the center of the tube, while for Case II both analytical and numerical solutions match rather precisely, with less than 1 % of difference near the edges.

Flow of a second grade fluid through a cylindrical permeable tube

In the present work, a second grade fluid flow in a cylindrical tube is investigated. The perturbations were expanded in terms of the second order material coefficient, which means that either the second order parameter is sufficiently small or the characteristic diameter is very large. After obtaining the solutions for the impermeable tube, problem is examined for suction or injection through the surfaces of the tube. The general solutions are obtained for a tube of circular cross-section and also applied to a tube of arbitrary cross-section by using conformal transformation.

Geometric Approach to Laminar Convection

The analytical approach to the solution of a steady-state convection problem for Hagen-Poiseuille flow in tubes of arbitrary cross section was analyzed. The problem was studied in the view of isoperimetric inequalities, that is of inequalities holding for domain functionals, provided that equality is attained. The formal analogy with continuum solid mechanics allowed to develop a semiqualitative theory of laminar convection in tubes of arbitrary cross section. The heat transfer rate per unit length across the wall can be expressed in terms of a geometric parameter for a fluid of constant shear viscosity. The heat transfer rate could be calculated in terms of overall quantities without requiring pointwise solution of the Hagen-Poiseuille problem

Anisotropic Elastic Tubes of Arbitrary Cross Section Under Arbitrary End Loads: Separation of Beamlike and Decaying Solutions

Abstract First approximation analytical solutions are constructed for finite and semi-infinite, fully anisotropic elastic tubes of constant thickness h and arbitrary cross section, subject to purely kinetic or purely kinematic boundary conditions. Final results contain relative errors of O(h∕R), where R is some equivalent cross sectional radius. Solutions are decomposed into the sum of an exact beamlike or Saint-Venant solution, treated in Ladevèze et al. (Int. J. Solids Struct., 41, pp. 1925–1944, 2004) and extended in an appendix; a rapidly decaying edge-zone solution; and a slowly decaying semi-membrane-inextensional-bending (MB) solution. Explicit conditions on the boundary data are given that guarantee decaying solutions. The MB solutions are expressed as an infinite series of complex-valued exponential functions times real-valued one-dimensional eigenfunctions which satisfy a fourth-order differential equation in the circumferential coordinate and depend on the pointwise cross sectional curvature only.

Flow due to a periodic array of point forces, and the motion of small particles within a cylindrical tube of arbitrary cross section

The properties and computation of Stokes flow due to a periodic array of point forces exerted in the interior of a fluid-filled cylindrical tube with an arbitrary cross-sectional shape are discussed. It is shown that the relationship between the pressure drop and the axial flow rate occurring when the point forces have a component parallel to the generators can be deduced immediately from a knowledge of the velocity profile corresponding to unidirectional pressure-driven flow. A boundary-integral method for computing the associated Green’s function of Stokes flow is developed and implemented in a numerical procedure that exploits the cylindrical boundary geometry to improve the accuracy of the results and efficiency of the computations. Streamline patterns of the flow within tubes with circular, elliptical, and nearly square shapes are presented and discussed with reference to flow reversal. In the limit as the separation between the point forces becomes increasingly larger than the typical size of the cross section of the tube, we recover the flow due to a solitary point force, and the numerical result are in agreement with those derived by previous authors for the particular case of a tube with a circular cross-sectional shape. The flow due to the point forces is then coupled with the boundary integral representation to develop asymptotic expansions for the surface stress, force, torque, and higher moments of the traction exerted on a small suspended particle that belongs to a periodic array. Each particle may translate and rotate while the ambient fluid undergoes pressure-driven flow. The coefficients of the asymptotic expansion are related to the non-singular part of the Green’s function and its spatial derivatives, evaluated at the location of the point force. These quantities are computed and plotted for several cross-sectional shapes.

Hydrodynamic flows and heat transfer in slightly noncylindrical channels

Viscous incompressible laminar flow and heat transfer in channels with a small arbitrary deviation from a cylindrical surface are examined. A linear system of equations and boundary conditions for the disturbed dynamic and thermal fields, obtained by linearizing the complete system of Navier-Stokes equations with respect to the solution for developed flows in cylindrical tubes of arbitrary cross section, is presented. In the important practical case in which the perturbations of the channel surface are concentrated on an interval of finite length it is shown that the integral dynamic and thermal characteristics of the channel can be found without solving the three-dimensional equations by going over to effective two-dimensional boundary-value problems which are fundamentally no more difficult to solve than those for developed flows. Extensions of the theory to flows with low-efficiency power sources are given. Applications to plane channels and circular tubes with deformed surfaces are considered. Among the numerous applications requiring information about the integral characteristics of flows in channels whose initially cylindrical surface is slighty deformed, we note the problem of heat transfer intensification by slightly deforming the tube surface with careful estimation of the accompanying increase in resistance [1] and the calculation of the resistance of capillaries and biological transport systems in the form of tubes and channels when the walls are deformed [2]. Below we consider laminar flow in channels with deformed walls. Whereas for the first problem this class of flows is only one of those possible (in general it is necessary to analyze the transition, turbulence and flow separation effects), in the second case, which is characterized by low Reynolds numbers, the laminar flow model is perfectly adequate.

Eigenvalues and eigenfunctions for general graetz problems

A method is derived for the determination of the eigenvalues and the corresponding eigenfunctions which arise in the problem of forced convection of heat through an infinite tube of arbitrary cross-section. The solution is obtained in terms of the eigenvalues and eigenfunctions of the related reduced wave equation, and involves the calculation of the eigenvalues of a suitable symmetric matrix.

Analysis of shear lag in cylindrical tubes of arbitrary cross section

A very general analysis is given of the phenomenon of shear lag in thin-walled cylindrical tubes, with single-cell cross sections of arbitrary shape, containing any number of concentrated longitudinal booms that carry direct stress only, and subjected to any longitudinal distribution of bending moment and torque. Two equations relating the distributions of direct and shearing stresses on the cross section are derived for the most general case where the tube is non-uniform because of an arbitrary longitudinal variation of wall thicknesses and boom areas. These equa­tions, which are remarkably simple in view of their generality, incor­porate all the requirements of equilibrium and compatibility and provide corrections to the stresses, curvature and twist calculated from the engineers’ theory of bending and torsion. They also govern the distri­bution of stresses arising from the application of self-equilibrating systems of tractions to the end cross sections. Exact solutions are ob­tained for the case of a uniform, but otherwise arbitrary, cross section under any polynomial distribution of bending moment and torque, and it is shown how conditions at the end cross sections can be satisfied with the aid of solutions of a simple eigenvalue problem. The equations are in a particularly ideal form for incorporating into a general purpose com­puter program for the automatic numerical solution of any problem of this type.

A charnock-based estimate of interfacial resistance and roughness for internal, fully-developed, stratified, two-phase horizontal flow

Charnock's (1955) relation between interfacial roughness and drag is utilized in order to determine these two quantities for fully-developed flows in tubes of arbitrary cross-section.

Peristaltic Transport of a Fluid-Particle Mixture

An asymptotic method is developed for the solution of the mathematical problem of peristaltic transport of a viscous fluid with solid particles in a flexible tube of arbitrary cross-section. Under long wave approximation, the coupled nonlinear equations governing the fluid-particle mixture are reduced to a sequence of two-dimensional, linear boundary-value problems. The asymptotic method is justified rigorously and the existence and uniqueness of the exact solution are proved.

Asymptotic method for peristaltic transport.

The purpose of this paper is to justify an asymptotic method developed for the study of peristaltic transport in a tube of arbitrary cross section. Within the framework of long wave approximation, the three-dimensional nonlinear Navier-Stokes equations are reduced to a sequence of two-dimensional linear boundary value problems of Laplace and biharmonic operators. It is shown that, if a Reynolds number is less than some constant, the solution of the approximate equations is indeed an asymptotic approximation to the exact solution of the problem as the ratio of the maximum radius of the tube to the wave length of the peristaltic motion of the wall tends to zero, and the error estimates are expressed in L 2 norms. Furthermore, under the same condition the exact solution is shown to be unique and stable under arbitrary perturbation of spatially periodic disturbance. Application of the stability condition to peristaltic transport in a tube of circular cross section is given.

Comments on ’’Asymptotic theory for peristaltic transport in a tube of arbitrary cross section’’

Comments on ’’Asymptotic theory for peristaltic transport in a tube of arbitrary cross section’’

Asymptotic theory for peristaltic transport in a tube of arbitrary cross section

An asymptotic theory within the framework of long wave approximation is proposed for the study of peristaltic transport in a flexible tube of arbitrary cross section. The nonlinear problem of the three-dimensional Navier–Stokes equations is reduced to a sequence of linear boundary-value problems of the two-dimensional Laplace and biharmonic operators. Explicit expressions for the velocity profiles are obtained in terms of the solutions of the boundary value problems. Results for a tube of elliptical cross section are given.

Dispersion in flow through small tubes

A new method is presented for obtaining asymptotic solutions of the now classical problem of the dispersion of a solute in a medium flowing through a tube of small diameter. Features of the method (a non-standard twotiming technique) are: (1) it is capable of treating arbitrary time-dependent flow fields and tubes of arbitrary cross-section; (2) the approximate solutions constructed are valid immediately after the initial time rather than after a time lag; (3) approximations to any desired order may be readily obtained; (4) their asymptotic validity may be proved rigorously. ‘Transient’ terms as well as ‘developed’ terms arise naturally in the treatment; the former decay rapidly in time.

On a generalized multiregion Graetz problem

An analytical study is presented for the solution of a generalizedGraetz problem in a right, annular, cylindrical tube of arbitrary cross-section, which contains a fully-developed, steady, unidirectional, multicomponent flow. The problem incorporates arbitrary inlet-conditions and considerably general, inhomogeneous boundary-conditions. Each component flow is characterized by a prescribed velocity distribution and is under the influence of an arbitrary heat source. The system of differential equations for the temperature distribution in each component is coupled through interface-conditions which involve finite thermal conductance. Through the use of a novel procedure, which appears to be more convenient mathematically than any possible alternative, infinite series expressions are derived for the temperature distribution in each and every component, and various alternative forms are noted. The general expressions, not available hitherto, contain the solutions of many specialGraetz problems of engineering significance. An illustrative example of a two-component flow in an annular, circular cylinder is included.

Alternating flow of non-Newtonian fluids in tubes of arbitrary cross-section

Alternating flow of non-Newtonian fluids in tubes of arbitrary cross-section

On the flow of an electrically conducting fluid in tubes of arbitrary cross-section in the presence of a magnetic field

On the flow of an electrically conducting fluid in tubes of arbitrary cross-section in the presence of a magnetic field

The General Theory of Cylindrical and Conical Tubes under Torsion and Bending Loads

SummaryPart 5 continues the theory given in Parts 1–4 in the February 1947 issue of the Journal (pp. 199–269). The present paper deals with the stresses and deformations of conical and cylindrical tubes under arbitrary loading, thus completing the analysis given in Part 4 which dealt only with pure torque.A remarkable unification of the theory of bending and torsion is achieved. It is shown that the axial constraint stresses, i.e. the corrections to the engineers' stresses, may in general be calculated as if caused by torque about axes different in each mode. The analysis proves that it is in no circumstances correct to calculate the axial constraint stresses from the torque about the flexural axis.Functions giving the cross-wise distribution of stress are fully worked out for the n-boom tube of arbitrary cross-section and for the singly symmetrical trapezoidal tube with continuous direct stress-carrying covers. The analysis is a considerable extension of the results given for the four-boom tube in Part 4.