Semi-infinite Tube Research Articles

Existence, uniqueness, and decay estimates for solutions in the nonlinear theory of elastic, edge-loaded, circular tubes

Solutions to the two, coupled, ordinary nonlinear differential equations for a semi-infinite circular elastic tube subjected to edge loads and undergoing small axisymmetric strains, but arbitrarily large axisymmetric rotations—the simplified Reissner equations—are analyzed. First, with the aid of a Green’s function, the differential equations and boundary conditions are transformed to a complex-valued integral equation. From this equation existence, uniqueness, boundedness, and rate of decay are extracted for a dimensionless stress function, f f , and an angle of rotation, β \beta , for sufficiently small edge data. It is shown that these unique solutions must decay at least as fast as the linear solution. Second, it is shown that any solution that decays to zero must, at a sufficiently large distance from the edge, decay at the linear rate. Third, rates of decay are established for any solution for which the L 2 {L_2} or the sup norm of β \beta has certain bounds. Finally, an energy (or Lyapunov) function E E , defined on solutions of the differential equations, is constructed and under certain a priori restrictions on the angle of rotation, three different upper bounds on E E are obtained. These also provide exponential decay estimates for solutions. The energy approach is examined with a view to more general shells where a Green’s function may not be readily available.

On the Nondimensionalization of Constriction Resistance for Semi-infinite Heat Flux Tubes

Technical Briefs On the Nondimensionalization of Constriction Resistance for Semi-infinite Heat Flux Tubes K. J. Negus, K. J. Negus Thermal Engineering Group, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Search for other works by this author on: This Site PubMed Google Scholar M. M. Yovanovich, M. M. Yovanovich Thermal Engineering Group, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada Search for other works by this author on: This Site PubMed Google Scholar J. V. Beck J. V. Beck Heat Transfer Group, Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824 Search for other works by this author on: This Site PubMed Google Scholar Author and Article Information K. J. Negus Thermal Engineering Group, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada M. M. Yovanovich Thermal Engineering Group, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada J. V. Beck Heat Transfer Group, Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824 J. Heat Transfer. Aug 1989, 111(3): 804-807 (4 pages) https://doi.org/10.1115/1.3250755 Published Online: August 1, 1989 Article history Received: March 5, 1986 Online: October 20, 2009

Flame-speed sensitivity to temperature changes and the deflagration-to-detonation transition

To better understand how the dynamics of a folded flame propagating from the closed end of a semiinfinite tube is affected by the blast flow of unburned mixture and the flame-driven acoustics, we study a one-dimensional, unsteady model in which (1) the folded flame is modeled as a planar, markedly subsonic, reactive discontinuity separating the fresh and burned gases, both being chemically inert, and (2) the burning speed of the discontinuity has the same, exponential dependence on the reaction temperature as that of the corresponding laminar flame, but it exceeds the latter by a large factor σ( t) (degree of folding) which is prescribed an assumed known dependence on time. Upon use of asymptotic techniques in the limit of large activation energies, a nonlinear, σ-dependent, difference equation is obtained for the burning speed history. It shows that (1) two regimes of steady propagation are obtained when σ is constant and less than a critical value σ c that we compute in terms of the mixture properties (only the slower regime is stable), (2) if σ( t) exceeds σ c during a long enough period, the reactive discontinuity accelerate limitlessly, and (3) if σ( t) never exceeds σ c, and the initial state lies on the lower branch of steady states, the burning speed remains bounded. The results are discussed in terms of the DDT phenomenon.

Solutions for Steady and Nonsteady Entrance Flow in a Semi-Infinite Circular Tube at Very Low Reynolds Numbers

Analytic solutions are found to the time-dependent Stokes and continuity equations describing entrance flow of an incompressible fluid into a circular tube at very low Reynolds numbers. Fourier transform methods are used, and entrance-flow solutions are obtained in the form of normal-mode expansions. Complex dispersion relations are obtained for each mode, and analytic expressions are determined for the normal-mode coefficients in terms of entrance boundary conditions. Results for velocity and pressure near the entrance are given. The generalized method is applied to obtain solutions to specific nonsteady-flow problems with a sinusoidal time dependence. In the last section of the paper, the steady-flow solutions are obtained by taking the limits of the nonsteady-flow solutions as the frequency approaches zero.

The acoustic impedance of a semi-infinite tube fitted with a conical flange: Part II

The acoustic characteristics of a cylindrical tube of radius ρ0, fitted with a conical flange, of semi-angle O0, are investigated. Let the acoustic potential in the tube be given by O = Ae−ik(x−α) − A′eik(x−α). In this expression the first term represents an incident wave travelling along the tube in the direction of increasing x towards the open end x = 0 and the second term represents the reflected wave. It is shown that, apart from the introduction of the phase shift α = 1/4ρ0 sin O0, the acoustic impedance of the tube is given, to a good approximation, by (A − A′)/(A + A′) = (ikρ0 cos2½O0)/(ikρ0 + sin O0), provided O0 is moderately small. The result is a generalization of a previous investigation [1, Chester 1983]. The earlier restriction to long waves is relaxed in the present work.

A New Model on Transient Wave Propagation in Fluid-Filled Tubes

A new theory was developed for the fluid-filled semi-infinite tube model subjected to pressure impulse. The new model takes into account the shear and flexural waves of the elastic tube. The tube displacement field is axisymmetric. Axial deformations are based on the uniform extensional symmetric mode, and antisymmetric flexural mode along the tube thickness. Radial deformation is based on the uniform symmetric mode about the tube axis. Walker-Phillips’ theory can be recovered as a special, limiting case.

Point charge passing a resonator with beam tubes

Charged particle beams interact with the surroundings because of the co-travelling electromagnetic fields. This interaction limits the beam. One of the most interesting cases is a point charge travelling with arbitrary velocity along the axis of a cylindrical resonator with two semi-infinite tubes. Without the cylindrical wall of the resonator, the structure consists of two tubes connected by an infinite radial line. In the tube region the fields are given by the source fields plus a continuous spectrum of waveguide modes. In the resonator region a discrete set of modes is used. Matching the fields at the common interface yields a system of linear equations for the expansion coefficients. The occurring integrals contain fractions of the Bessel function and its derivative, which were expanded into algebraic series of Bessel function zeros, and are solved by means of the residuum calculus. Numerical results are given for the coupling impedance between charge and structure and for the energy loss of a Gaussian charge distribution.

Wave propagation and reflection in liquid filled distensible tube systems exhibiting dissipation and dispersion

In a previous paper the present authors developed a model describing wave propagation in liquid filled distensible tubes and tested it against impulse experiments involving water filled latex rubber tubes. This model incorporates both dissipative and dispersive mechanism which are absent from the commonly employed linear long wave-length (LLW) theory of haemodynamics. This higher order theory is here employed to study propagation of an impulse in a semi infinite tube, reflection of an impulse from the distal end of a finite length tube, and reflection and transmission of impulses impinging on a function connecting dissimilar liquid filled tubes. Both open and closed type reflections are treated and numerical results presented graphically. To the best of our knowledge this is the first time that such a higher order theory has been employed to treat reflection and transmission of waves in tube systems.

Asymptotic solution of boundary value problems for weakly perturbed wave equations

A method that is asymptotic with respect to the small parameter ε is proposed for solving boundary value problems for weakly linear equations with partial derivatives. Linear travelling waves, defined when x ⩾ 0, t ⩾ 0 and which only interact on the boundary x = 0, are the solution of the unperturbed problems. An asymptotic solution which is uniformly suitable for t, x = 0 ( e −1) is constructed for the perturbed problem using the method of averaging along the characteristics. A model problem of gas dynamics is considered — the problem of the motion of a piston in a semi-infinite tube. The problems considered in this paper are usually solved by the method of regular expansion with respect to the parameter ε /1, 2/. However this method leads to secular terms appearing in the asymptotic solution, which makes the latter unsuitable for values of the arguments t, x = 0 ( e −1). But large values of t and x are more interesting when analysing weakly linear waves, since the non-linear, dissipative and other factors, which are usually disregarded in the simplest linear models, begin to develop. Asympotic methods enabling us to solve Cauchy's problem for the equations considered below were proposed in /3–6/. Boundary value problems of the “resonator” type were solved in /7, 8/. The technique proposed previously is modified below for problems in which x ⩾ 0.

The stokes flow from half-space into semi-infinite circular cylinder

The infinite-series solutions for the creeping motion of a viscous incompressible fluid from half-space into semi-infinite circular cylinder are presented. The results show that inside the cylinder beyond a distance equal to 0.5 times the radius of the tube from the pore opening, the deviation of the velocity profile from Poiseuille flow is less than 1%. The inlet length in this case is comparable to that computed for a finite circular cylinder pore by Dagan et al.[1]. In the half-space outside the cylinder pore region, the flow is strongly affected by the wall. Beyond one radius of the tube from the orifice, the solutions match almost exactly the flow through an orifice of zero thickness given by Sampson[2]. The relationship between the pressure drop and the volumetric flow rate is also computed in the present paper for the semi-infinite tube.

Similar Solutions of Gas Pressure Within Semi-infinite Capillary Tubes

The similar solutions of Ducoffe's equation are presented. Through use of the solutions, the time variation of the gas pressure within thin semi-infinite capillary tubes is solved when the pressure at one end is changed discontinuously from an initial uniform value to a new constant one. The rate of pressure variation propagation decreases as the initial tube pressure becomes lower, and a finite velocity pressure variation front is formed if the initial tube pressure is a vacuum. Very close to the open end, the order of the propagation rate is reversed.

Wave propagation in a thin-walled liquid-filled initially stressed tube

Wave propagation through a thin-walled cylindrical orthotropic viscoelastic initially stressed tube filled with a Newtonian fluid is discussed. Special attention is drawn to the influence of the initial stretch on the wave propagation. It is shown that initial stretching of real arteries enhances the propagation of blood pressure pulses in mammalian arteries. The dispersion equation for the initial-value problem of a semi-infinite tube is also derived. It is shown that the speed of propagation and the attenuation vary with the distance from the support. The results obtained for the axial wave mode provide an explanation for the experimental observations, which is not possible with the results obtained for the infinite tube.

Unsteady motion of viscous fluid in an expanding tube

The problem of unsteady motion of a viscous compressible fluid in a semiinfinite tube with horizontal axis is solved by successive approximation. The circular cross section of the tube depends exponentially on the coordinate measured along the tube axis. At the end of the tube there is a unit such as a sliding valve, compressor, reciprocating pump, or turbine that changes the flow rate. The process is assumed to be barotropic.

The acoustic impedance of a semi-infinite tube fitted with a conical flange

The acoustic characteristics of a semi-infinite tube of constant cross sectionσ, fitted with a semi-infinite conical flange of solid angleω, are investigated. Let the acoustic potential in the tube be given by $$\varphi = e^{ - ikx} - Ae^{ikx}$$ where exp(−ikx) represents the incident wave travelling along the tube towards the open endx=0 and −A exp(ikx) represents the reflected wave. It is shown that for long waves,A can be written in the form $$A = \exp \{ - 2ik\alpha (\sigma /\omega )^{1/2} - 2k^2 \sigma /\omega \}$$ whereα is a parameter which depends only on the solid angleω. From existing work the value ofα is known forω=2π (the infinite plane flange) andω=4π (the unflanged tube). In this paper, the variation ofα for small values ofω is calculated. Because 1≦α≦1.2266 over the whole range 0≦ω≦4π, it is suggested that general values ofα can be interpolated from the existing results. An explicit interpolation formula is proposed.

On the theory of woodwind finger holes

An analysis is given for the effect of side holes in woodwind acoustics. The simplified model studied is that of a solid semi-infinite tube joined to a hollow semi-infinite tube with N side holes. The problem is reduced, without approximation, to that of N coupled integral equations. When the aperture radius is much less than tube radius, wavelength and hole spacing, the equations simplify considerably. If one further restrictive constraint is imposed, the asymptotic solution is then equivalent to that obtained by earlier authors, who have posed a one-dimensional interior wave motion with impedance conditions to characterize the influence of the side holes. A modified value is obtained here for the imaginary part of the impedance, which accounts for radiation losses out of the apertures.

Unsteady Flow in a Porous, Elastic, Circular Tube : Part 3. The Tube Contracting or Expanding in Axial and Radial Directions

Linear theory for unsteady flow of a viscous, incompressible fluid through a semi-infinite circular tube with porous, elastic wall whose radius together with its length contracts or expands elastically, is considered together with the effects of injection into the wall.Laminar similar solutions with respect to space and time are obtained. Velocities, frictional drags, power and work of fluid are analyzed. There are three critical conditions of injection into the wall. The conditions of injection together with the conditions of motion of the tube either contracting or expanding, diversify the induced flow fields.

The Propagation of Sound in Cylindrical Ducts with Mean Flow and Bulk-reacting Lining II. Bifurcated Ducts

The transmission and reflexion of sound in a bifurcated coaxial cylindrical duct is investigated. The inner tube carries a uniform flow and consists of two semi-infinite tubes: one is hard and the other is perforated. The space between the coaxial cylinders is filled with a sound-absorbing material. Transmission and reflexion matrices are calculated for the causal solution. It is found that causality and the boundary conditions require an instability wave when the perforated tube is downstream its junction to the hard tube. When the perforate is situated upstream the junction the analysis permits incident waves that are unstable. This is important for applications to multiple reflexions. It follows from the analysis that, in addition to giving rise to the instability wave, the gas flow has several other important effects on the acoustic properties of the junction.

Unsteady Flow in a Porous, Elastic Circular Tube : Part 1, The Wall Contracting or Expanding in an Axial Direction

A linear theory for unsteady flow of a viscous, incompressible fluid in a semi-infinite circular tube with a porous, elastic wall is developed. The tube radially contracts or expands keeping its axial length constant. The fluid is assumed to pass through the porous wall under a certain defined condition of injection. Laminar similar solutions with respect to space and time are obtained. Velocities, pressures, frictional drags, powers and works of fluid are analyzed. A rapid contracting motion of a tube results in separating the flow into a core region and the boundary layer, while a rapid expanding motion brings about a remarkable compensating flow in the center of the tube. The conditions of elastic motions together with the conditions of injection give a classification of pumping action of a porous, elastic tube.

A mathematical theory of shock-wave formation in arterial blood flow

Theoretical and experimental evidence is available for the occurrence of shock waves in arterial blood flow. A sudden increase of pressure at the entrance of a fluid-filled semi-infinite elastic tube is considered as a model to investigate mathematically the possibility of shock-wave formation. By use of the method of wavefront expansions explicit results are obtained about the circumstances under which shocks may form and about the time and distance at which this may occur.

Axially symmetric vortical flow of a nonviscous liquid in the semiinfinite gap between two coaxial circular cylinders

In the framework of the Hromek-Lamb equations we investigate the axially symmetric vortical flow of a nonviscous incompressible liquid in both semiinfinite and infinite gaps between two coaxial circular cylinders. The investigation is carried out for two circulation and flow functions and two different Bernoulli constants which are chosen in the form of a third-order polynomial in the flow function. This makes it possible to determine the effect of the azimuthal velocity component on the flow in an axial plane with radial and axial components of the velocity. It is shown that under certain circumstances wave oscillations in the flow are possible, in agreement with the results of [1–3] which investigated the flow in an infinite tube [1], in a semiinfinite tube with simpler circulation functions and Bernoulli constants [2], and in the two-dimensional case [3]. We determine the dependence of the formation of wave perturbations on the third term of the Bernoulli constant and on the azimuthal velocity component. The results of this work agree with investigations by other authors [1–4].