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.     

The optimal shape of a pipe

In shape optimization, recently the question arose, whether or not the cylindrical pipe has the optimal shape for the transport of an incompressible fluid. In this short note, a proof will be presented that a cylindrical pipe with Poiseuille’s flow inside indeed is optimal for the transportation of an incompressible fluid under the criterion “energy dissipated by the fluid.” The proof reduces the problem to the minimization of a two-dimensional Dirichlet’s integral. This simpler problem can be solved with a symmetrization argument.     

On the effects of curved geometry on heat conduction through a distorted pipe

Starting from the Boussinesq system in thin (or long) curved pipe, we derive a simplified model via rigorous asymptotic analysis with respect to the pipe’s thickness. We are particularly interested in finding explicitly the effects of distortion of the pipe on the heat conduction.     

Vibration analysis of conveying fluid pipe via He’s variational iteration method

In this paper, a recently new semi-analytical method, i.e., He’s variational iteration method is developed to apply to free vibration analysis of conveying fluid pipe. The critical flow velocity and frequency of pipe conveying fluid are obtained with considering the various boundary conditions. The results are compared with the ones of different transform method, and prove VIM that has the same precision and efficient with DTM. The mode shapes of cantilevered pipe and both ends with elastic support pipe are shown under different flow velocity.     

A network model for stemflow solute transport

While the role of stemflow in directing and concentrating water and nutrients at the tree base is rarely in dispute, its mathematical representation remains a subject of inquiry and research. A network model that seeks to estimate stemflow solute concentration and leaching is proposed. The model accommodates the physico-chemical properties of individual furrows embedded within the tree bark and their interconnections. The within-furrow equations for water and solute transport that include leaching are first developed and integrated along a rough-bark network topology to describe solute concentration and fluxes out of the network. The model is parameterized using published data on stemflow, field measurements of bark geometry, and laboratory experiments on bark leaching for potassium, magnesium, and calcium. The parameterization is intended to impose plausibility constraints and not to test model predictions at a particular site, a single event, or an individual experiment. The outflow concentration is then analyzed as a function of the network complexity that includes asymmetry in the lengths or subpaths connecting network nodes. For a symmetric network, an effective ’channel-flow’ analogy may be used to represent solute concentration at the outflow. However, as the asymmetry increases in subpath lengths, the efficiency of the bark network at moving solutes diminishes for the same rainfall input onto the stem. The network representation featured here is by no means offering a ’finality’ to the stemflow mathematical representation. It must be viewed as an embryonic step that opens up the possibility of using modern advances in network theories to link rainfall properties to stemflow water and solute input from a variety of tree species with differing bark microrelief configurations into the soil.     

Heat Conduction Problem in a Dilated Pipe: Existence and Uniqueness Result

We study the heat conduction through a pipe filled with incompressible viscous fluid. The goal of this paper is to take into account the effects of the spipe’s dilatation due to the heating. In view of that, we assume that the longitudinal dilatation of the pipe is described by a linear heat expansion law. We prove the existence and uniqueness theorems for the corresponding boundary value problem. The main difficulty comes from the fact that the flow domain changes depending on the solution of the heat equation leading to a nonlinear coupled governing problem.     

Direct numerical simulation of the turbulent flow in an elliptical pipe

The turbulent flow in a pipe with an elliptical cross section is directly simulated at Re = 4000 (where the Reynolds number Re is calculated in terms of the mean velocity and the hydraulic diameter). The incompressible Navier-Stokes equations are solved in curvilinear orthogonal coordinates by using a central-difference approximation in space and a third-order accurate semi-implicit Runge-Kutta method for time integration. The discrete equations inherit some properties of the original differential equations, in particular, the neutrality of the convective terms and of the pressure gradient in the kinetic energy production. The distributions of the mean and fluctuation characteristics of the turbulent motion over the pipe’s cross section are computed.     

Pipe roughness identification of water distribution networks: The full turbulent case

This paper proposes a technique to identify individual pipe roughness parameters in a water distribution network by means of the inversion of the steady-state hydraulic network equations. By enabling the reconstruction of these hydraulic friction parameters to be reliable, this technique improves the conventional model’s accuracy and thereby promises to enhance model-based leakage detection and localization. As it is the case in so-called fireflow tests, this methodology is founded on the premise to measure the pressure distributed at a subset of nodes in the network’s graph while assuming the nodal consumption to be known. Beside of the proposed problem formulation, which is restricted to only allow turbulent flow in each of the network’s pipes initially, developed algorithms are presented and evaluated using simulation examples.     

任意多连通截面扭转应力函数的有限元解法

杆件扭转问题的求解,主要有基于扭转理论翘曲函数的边界元法和有限元法、基于薄壁杆件理论的数值解法和基于扭转理论应力函数的有限元法．根据任意多连通截面直杆扭转问题的应力函数理论,讨论并改进了与微分方程及定解条件等效的泛函,在此基础上推导了求解多连通截面扭转应力函数的有限元列式,将扭转问题的翘曲位移单值条件转化为边界节点上的集中荷载．采用主从节点法满足孔洞边界上应力函数的同值条件,实现了任意多连通复杂截面扭转应力函数的有限元直接求解,通过应力函数积分获得截面的扭转常数．算例验证了方法的可行性和有效性．     

Directional fields algebraic non-linear solution equations for mobile robot planning

In this study, a novel approach to robot navigation/planning by using half-cell electrochemical potentials is presented. The half-cell electrode’s potential is modelled by the Nernst equation to yield automatic search/detection of pipeline flaws by using the direct current voltage gradient (DCVG) technique. We introduce a theory of spherical volumetric electric density in the soil to sustain our postulates for navigational potential fields. The Nernst potential is correlated with the distance to a pipe’s flaw by proposing a fitted theoretical-empirical nonlinear regression model. From this, volumetric derivatives are solved as gradient-based fields to control wheeled robot’s motion. A nonlinear system for trajectory planning is proposed, and analytically solved by an algebraic solution. This solution directly adjust robot’s speed kinematic values to lead it toward the flaw. The inverse/forward kinematic constraints are non-holonomic, and are recursively integrated into the general potential equation. Analytical modelling is reported, and a set of numerical simulations are presented to prove the feasibility of the proposed formulations.     

基于有限元法的混凝土输水管内硫酸根离子扩散分析

暴露于硫酸盐环境中的混凝土输水管易遭受硫酸盐化学侵蚀,导致其耐久性退化、提前失效;而环境中硫酸根离子传输进入混凝土是其化学侵蚀的前提.为获得混凝土内硫酸根离子的扩散进程,首先基于Fick定律及质量守恒定律,建立饱和混凝土管内硫酸根离子的扩散-反应模型.其次,将扩散-反应模型的边界条件齐次化,建立其有限元控制方程.然后,开展硫酸钠溶液中水泥砂浆圆柱试件的腐蚀试验,测定试件不同深度处的硫酸根离子浓度,与模型计算结果对比,以验证模型.最后,开展数值模拟研究,分析混凝土输水管外表面、内外表面暴露于浓度恒定或振荡的硫酸盐溶液情况下管内硫酸根离子浓度的时空分布.     

Nonlinear Frequency Analysis of FGM Pipes Based on the Homotopy Method北大核心CSCD

Based on the homotopy analysis method, the nonlinear vibration of porous functionally graded material (FGM) conveying pipes under generalized boundary conditions was studied. Based on the power-law distribution of the FGM and the Voigt model, the physical properties of the porous pipe material were described. Under the Euler-Bernoulli beam theory and the von Kármán nonlinear theory, and by means of Hamilton’s variational principle, the dynamic control equations and generalized boundary conditions for porous FGM conveying pipes were established. The homotopy analysis method was used to solve the nonlinear vibration characteristics of the porous FGM conveying pipe under generalized boundary conditions. The numerical results show that, the translation spring has little effect on the critical velocity of instability, while the rotation spring increases the critical velocity of instability, making the system more stable; in the nonlinear system, the viscoelastic coefficient does not change the critical velocity; the pipe length, the power-law exponent and the porosity all influence the nonlinear free vibration of the porous FGM conveying pipe. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Starting solutions for a viscoelastic fluid with fractional Burgers’ model in an annular pipe

In this work, we have discussed some simple flows of a viscoelastic fluid with fractional Burgers’ model in an annular pipe. The fractional calculus approach is introduced in the constitutive relationship of a Burgers’ fluid model. Exact analytical solutions are obtained by using Laplace and Weber transforms for two types of flows, namely: Poiseuille flow and Axial Couette flow.     

An equivalent pipe network model for free surface flow in porous media

Free surface flow analysis in porous media is challenging in many practical applications with strong non-linearity. An equivalent pipe network model is proposed for the simulation and evaluation of free surface flow in porous media. On the basis of representative elementary volume with homogeneous pore-scale patterns, the pore space of the homogeneous isotropic porous media is conceptualized as a collection of capillary tubes. According to Hagen-Poiseulle's law and flux equivalence principle, equivalent hydraulic parameters and unified governing formulations for the pipe network model are deduced. The two-dimensional free surface flow problem is reduced to a one-dimensional problem of pipe networks and a one-dimensional procedure based on the finite element method is then developed by introducing a continuous penalized Heaviside function. The proposed equivalent pipe network model is verified with results from numerical solutions and laboratory-measured data available in the literature, and good agreements are obtained. The proposed equivalent pipe network model is shown to be effective in analyzing the free surface flow in porous media. The numerical results also indicate that the proposed equivalent pipe network model has weak sensitivity of the mesh size and penalty parameters.     

Hall effect on the pipe flow of a Burgers’ fluid: An exact solution

An analysis is made to see the influences of Hall current on the flow of a Burgers’ fluid. The velocity field corresponding to the flow in a pipe is determined. The closed form analytical solutions for several Newtonian and non-Newtonian fluid models can be obtained from the present analysis as the limiting cases.     

A complete steady state model of solute and water transport in the kidney

The purpose of this paper is to incorporate a detailed model, along with an optimized set of parameters for the proximal tubule, into J. L. Stephenson's current central core model of the nephron. In this model a set of equations for the proximal tubule are combined with Stephenson's equations for the remaining four tubules and interstitium, to form a complete nonlinear system of 34 ordinary differential and algebraic equations governing fluid and solute flow in the kidney. These equations are then discretized by the Crank-Nicholson scheme to form an algebraic system of nonlinear equations for the unknown concentrations, flows, hydrostatic pressure, and potentials. The resulting system is solved via factored secant update with a finite-difference approximation to the Jacobian. Finally, numerical simulations performed on the model showed that the modeled behavior approximates, in a general way, the physiological mechanisms of solvent and solute flow in the kidney.     

Optimization of sample size and order size in an inventory model with quality inspection and return of defective items

To ensure all products as perfect, inspection is essential, even though it is not possible to inspect all products after producing them like some special type products as plastic joint for the water pipe. In this direction, this paper develops an inventory model with lot inspection policy. With the help of lot inspection, all products need not to be verified still the retailer can decide the quality of products during inspection. If retailer founds products as imperfect quality, the products are sent back to supplier. As it is lot inspection, mis-clarification errors (Type-I error and Type-II error) are introduced to model the problem. Two possible cases are discussed for sending back products as defective lots are immediately withdrawn from the system and send back to supplier with retailer’s payment and for second case, retailer sends defective products during receiving next lot from supplier with supplier’s investment, like in food industry or in hygiene product industry. The model is solved analytically and results indicate that optimal order size and sample size are intrinsically linked and maximize the total profit. Numerical examples, graphical representations, and sensitivity analysis are given to illustrate the model. The results suggest that sending defective products maintaining the first case is the more profitable than the second case.     

On flow of a Navier–Stokes fluid in curved pipes. Part I: Steady flow

We consider the steady, fully developed motion of a Navier–Stokes fluid in a curved pipe of cross-section $D$ under a given axial pressure gradient $G$. We show that, if $G$ is constant, this problem has a smooth steady solution, for arbitrary values of the Dean’s number $\kappa$, for $D$ of arbitrary shape and for any curvature ratio $\delta$ of the pipe. This solution is also unique for $\kappa$ sufficiently small. Moreover, we prove that the solution is unidirectional (no secondary motion) if and only if $\kappa =0$. Finally, we show the same properties for the approximations to the Navier–Stokes equations called “Dean’s equations” and provide a rigorous way in which solutions to the full Navier–Stokes equations approach those to this approximation in the limit of $\delta \to 0$.     

On a free piston problem for potential ideal fluid flow

Our goal was to model and analyze a stationary and evolutionary potential ideal fluid flow through the junction of two pipes in the gravity field. Inside the ‘vertical’ pipe, there is a heavy piston that can freely move along the pipe. In the stationary case, we are interested in the equilibrium position of the piston in dependence on the geometry of junction, and in the evolutionary case, we study motion of the piston also in dependence on geometry. We formulate corresponding initial and boundary value problems and prove the existence results. The problem is nonlinear because the domain is unknown. Furthermore, we study some qualitative properties of the solutions and compare them with the qualitative properties of a free piston problem for Newtonian fluid flow. All theoretical results are illustrated with numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

An analytical model for carrier-facilitated solute transport in weakly heterogeneous porous media

Carrier-facilitated solute transport in heterogeneous aquifers is studied within a Lagrangian framework. Dissolved solutes and carriers are advected by steady random groundwater flow, which is modeled by Darcy's law with uncertain hydraulic conductivity that is treated as a stationary random space function. We derive general expressions for the spatial moments of the dissolved concentration and the concentration associated with the carrier phase. In order to reduce the computational effort, we use previously derived solutions for the flow field. This enables us to obtain closed-form solutions for the spatial moments of the two concentration fields. The mass and center of gravity of the two propagating plumes depend only on the mean velocity field and chemical/degradation processes. The higher (second and third) moments are affected by the coupling between reactions (sorption/desorption and degradation) among the three phases (i.e., dissolved, carrier and sorbed concentrations) and the aquifer’s heterogeneity. We investigate the potentially enhancing effect of carriers by comparing spatial moments of the two propagating plumes. The forward/backward mass transfer rates between the liquid and carrier phases, and the degradation coefficients are identified as critical parameters. The carrier's role is most prominent when detachment from carrier sites is slow, provided that degradation on the carriers is smaller than that in the liquid phase.