Impulsive displacement of liquid in a pipe

The solution of a viscous liquid, originally at rest in a pipe, which is set impulsively into motion by a pressure gradient is well known. Analyzing this solution shows that the displacement time – the time it takes to completely displace the liquid originally at rest in the pipe – falls primarily within an inviscid window. Assuming this result remains essentially unchanged when the displaced and displacing liquids are different we apply an inviscid analysis to determine the displacement time as a function of the density ratio of the liquids and the pressure difference. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Small-time Evolution of a Non-linear Diffusion Equation

The small-time evolution of the equation . t 0, – -$$\infty $$ < y < $$\infty $$ is consideredfor initial conditions appropriate in physical terms to botha concentration jump and a finite discharge. It is shown thatthe strong non-linear similarity solution is not uniformly validfor all y but is singular along an unknown moving boundary inthe (y, t) plane which has to be determined as part of the solution.The nature of this singularity and its resolution is investigatedby the introduction of a transition region between the strongnon-linear and linear regimes.     

On a stefan problem involving a spherical region: application of a Bergman-type expansion

A Bergman-type series expansion method is used in the analysis of a spherical reaction problem. The small-time solution so derived is employed as the starting point for the numerical solution, and the time for complete reaction is calculated.     

Marching schemes for inverse acoustic scattering problems

Summary For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n² log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.     

Comparison of turbulence models in simulating swirling pipe flows

Swirling flow is a common phenomenon in engineering applications. A numerical study of the swirling flow inside a straight pipe was carried out in the present work with the aid of the commercial CFD code fluent. Two-dimensional simulations were performed, and two turbulence models were used, namely, the RNG k–ε model and the Reynolds stress model. Results at various swirl numbers were obtained and compared with available experimental data to determine if the numerical method is valid when modeling swirling flows. It has been shown that the RNG k–ε model is in better agreement with experimental velocity profiles for low swirl, while the Reynolds stress model becomes more appropriate as the swirl is increased. However, both turbulence models predict an unrealistic decay of the turbulence quantities for the flows considered here, indicating the inadequacy of such models in simulating developing pipe flows with swirl.     

A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for steady compressible viscous flow

In this paper, the authors consider the Navier–Stokes equations for steady compressible viscous flow in three-dimensional cylindrical domain. A differential inequality for appropriate energy associated with the solutions of the Navier–Stokes isentropic flow in semi-infinite pipe is derived, from which the authors show a Phragmén–Lindelöf alternative result, i.e. the solutions for steady compressible viscous N–S flow problem either grow or decay exponentially as the distance from the entry section tends to infinity. In the decay case, the authors indicate how to bound explicitly the total energy in terms of data.     

Decay estimates for homogeneous Boussinesq equations in a semi-infinite pipe

In this paper, we establish the spatial decay bounds for homogeneous Boussinesq equations in a semi-infinite pipe flow. Assuming that the entrance velocity and magnetic field data are restricted appropriately, and it converges to laminar flow as the distance down the pipe tends to infinity, we derive a second order differential inequality that leads to an exponential decay estimate for the energy E(z,t) defined in (27). We also indicate how to establish the explicit bound for the total energy.     

A singular perturbation approach to integral equations occurring in poroelasticity

Singular perturbation theory is used to solve the integral equationswhich occur when treating finite-length crack problems in porouselastic materials. The method provides the stress intensityfactors which characterize the near crack tip stress and displacementfields for small times. The method also gives the stress andpore pressure fields on the fracture plane for small times relativeto the diffusive time scale. In this paper, the authors treatcrack problems which are unmixed in the pore pressure boundarycondition on the fracture plane. The Abelian result that smalltimes correspond, in Laplace transform space, to large valuesof the transform variable is used to formulate the problemsin terms of a small parameter. Rescaling on this small parameterleads to inner problems which are eigensolutions of the semi-infiniteproblems treated earlier by the authors. The outer solutionsare given by elastic eigensolutions together with appropriatefluid dipole responses. These outer solutions give the completestress and pore pressure fields except in the neighbourhoodof the crack tips; in this region the outer solutions are asymptoticallymatched with inner solutions. The full outer solutions are givenhere as an asymptotic expansion for small times and enable thedevelopment of the outer fields to be followed in real time.A reciprocal theorem in Laplace transform space is used to checkthe small-time solutions. The inner problem is rescaled to asemi-infinite crack problem, so eigensolutions of this semi-infiniteproblem are used together with the known asymptotic behaviourof the real solution to identify the stress intensity factor.The stress intensity factor is then related to an integral involvingthe inner limit of the outer solution together with the eigensolutionof the semi-infinite problem. Using this integral, we recoverthe result for the stress intensity factor found using singularperturbation theory. A ‘nearly’ invariant integralanalogous to the invariant M integral used in elastostaticsis derived. Unfortunately, the poroelastic analogue is not invariant,although it is used to verify the small-time results.     

Spatial decay bounds for time dependent magnetohydrodynamic geophysical flow

This article is concerned with spatial decay bounds for the time dependent magnetohydrodynamic geophysical flow in an infinite pipe when homogeneous lateral surface boundary conditions are applied. Assuming that the entrance velocity and magnetic field data are small enough and the fluid flow converges to laminar flow as the distance down the pipe tends to infinity, we derive a second order differential inequality that leads to an exponential decay estimate for the “energy” associated with the velocity and magnetic field represented by the difference between the entrance flow and fully developed laminar flow. We also show how to establish the explicit decay bounds for the total energy.     

He's homotopy perturbation method for solving the shock wave equation

In this article, we discuss the analytic solution of the fully developed shock waves. The homotopy perturbation method is used to solve the shock wave equation, which describes the flow of gases. Unlike the various numerical techniques, which are usually valid for short period of time, the solution of the presented equation is analytic for 0 < t < ∞. The results presented converge very rapidly, indicating that the method is reliable and accurate.     

Linearized control systems and small-time reachable sets

Using linear approximations of nonlinear systems has long beena practice to design control laws. In this paper, an analysisis given involving linear approximation of the nonlinear controlsystem and small-time reachable sets in R². A useful concept,the swing–out, which is a measure of nonlinearity, isdefined. This is used to examine the relationship between thesmall-time reachable sets of the nonlinear control system andits linear approximation. Behaviour of the nonlinear systemunder a control law is examined within this context. More factsare given about the swing-out for some special cases.     

Time-periodic Poiseuille flow in a pipe for some classes of fluids

Abstract  We consider a fully developed time-periodic pipe flow (Poiseuille flow) for some classes of fluids (micropolar fluids, mixtures of fluids). Such physical cases lead to a parabolic system in which the pressure gradient Γ is a time-periodic function with either only one non vanishing component or the components proportional to a single time-periodic function Γ. For such situations we generalize the results of [7] concerning the Newtonian case. Keywords: Flow in a pipe, Time-periodic Poiseuille flow, Micropolar fluids, Mixtures of fluids Mathematics Subject Classification (2000): 76D03, 76A05, 76T05, 35Q30 An erratum to this article is available at .     

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.

---

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.

     

Digital simulation of thermal reactions

Simulations of the equation for thermal expansion of a reacting gas have been carried out, exploring both the (possible) steady states and time-marching solutions. The critical Frank-Kamenetskii parameter δ_cr has been evaluated to seven decimal places for the slab, cylinder and spherical geometries and the role of the critical activation parameter ? was explored. It was found that there exist one or more mathematical steady states for any δ if ?>0, the curves for steady temperature at the center of the geometry plotted against δ tending to a straight line at large δ. Critical values of ?, the values above which this plot has a single solution for a given δ, have been computed to eight decimals. Time marching simulations showed that the Crank-Nicolson method, applied consistently, produces very accurate results, compared with the implementation in which the nonlinear term is rendered explicit. Where for a given δ there are several mathematical steady states, a time march usually settles on the lowest such state (if it settles at all), regardless of where the simulation is started, within the possible limits. The mathematical multiple steady states are not attained by time marching simulations, and are also physically unlikely.     

Decay of the 3D inviscid liquid–gas two-phase flow model

We establish the optimal \({L^{p}-L^{2}(1 \leq p < 6/5)}\) time decay rates of the solution to the Cauchy problem for the 3D inviscid liquid–gas two-phase flow model and analyze the influences of the damping on the qualitative behaviors of solution. Compared with the viscous liquid–gas two-phase flow model (Zhang and Zhu in J Differ Equ 258:2315–2338, 2015), our results imply that the friction effect of the damping is stronger than the dissipation effect of the viscosities and enhances the decay rate of the velocity. Our proof is based on Hodge decomposition technique, the \({L^{p}-L^{2}}\) estimates for the linearized equations and an elaborate energy method.     

Analytical solutions of laminar swirl decay in a straight pipe

In this work, the laminar swirl flow in a straight pipe is revisited and solved analytically by using prescribed axial flow velocity profiles. Based on two axial velocity profiles, namely a slug flow and a developed parabolic velocity profiles, the swirl velocity equation is solved by the separation of variable technique for a rather general inlet swirl velocity distribution, which includes a forced vortex in the core and a free vortex near the wall. The solutions are expressed by the Bessel function for the slug flow and by the generalized Laguerre function for the developed parabolic velocity. Numerical examples are calculated and plotted for different combinations of influential parameters. The effects of the Reynolds number, the pipe axial distance, and the inlet swirl profiles on the swirl velocity distribution and the swirl decay are analyzed. The current results offer analytical equations to estimate the decay rate and the outlet swirl intensity and velocity distribution for the design of swirl flow devices.     

Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.     

Aerodynamics of an isolated slot-burner from a tangentially-fired boiler

The aerodynamic development of fully turbulent isothermal jets issuing from rectangular slot-burners was modelled by obtaining a solution to the Reynolds averaged Navier–Stokes equations. A finite-volume method was used with the standard k–ε, RNG k–ε and Reynolds stress turbulence models. The slot-burners were based on physical models, which were designed to be representative of typical burner geometries found in tangentially-fired coal boilers. Two cases were investigated, in which jets from three vertically stacked rectangular nozzles discharged at 90° and then 60° to the wall containing the burner. The nozzle angle had little effect on jet centreline velocity decay, with the 60° nozzle showing a marginally higher rate of decay. The jets from the 60° nozzles were found to deviate slightly from their geometric axis slightly due to internal pressure redistribution in the flow at the nozzles. The simulations were validated against the physical models and were found to reproduce the flow field of the jets accurately with the Reynolds stress model producing the best results.     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.     

Simplified Navier-Stokes equations for computing viscous gas flow past long bodies

Simplified Navier-Stokes equations are applied to analyze the flow of supersonic viscous gas at moderately large Reynolds numbers near the lateral surface of long bodies. Numerical integration of the equations is performed by the marching method, stabilizing in each step the solution of the nonstationary system of equations in the longitudinal coordinates. We consider the flow past cylinders and cones with a spherical blunt nose. We investigate the effect of the Reynolds number and the body shape on the flow field, the drag coefficient, and the heat flux. The numerical solutions of simplified and complete Navier-Stokes equations are compared.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 231–239, 1985.