Hybrid Numerical-analytical Solution Research Articles

Integral Transform Solution for Natural Convection Within Horizontal Concentric Annular Cavities

Abstract Natural convection inside horizontal concentric annular cavities is dealt with through the generalized integral transform technique (GITT), offering a hybrid numerical-analytical solution of the continuity, Navier–Stokes, and energy equations in cylindrical coordinates. The flow is in steady-state, laminar regime, two-dimensional, buoyancy-induced, and the governing equations are written in the streamfunction-only formulation. Two strategies of integral transformation are adopted to verify the best computational performance, namely, the usual one with eigenvalue problems for both streamfunction and temperature defined in the radial variable, and a novel alternative with eigenvalue problems defined in the azimuthal angular coordinate. First, the eigenfunction expansions convergence behavior is analyzed to critically compare the two integral transform solution strategies. Then, test cases for different aspect ratios and Rayleigh numbers are validated with experimental data from the classical work of Kuehn and Goldstein. A maximum relative deviation of 5% is found comparing the GITT results for the average Nusselt number against the experimental data, while an 8% maximum relative deviation is found comparing against an empirical correlation by the same authors. It is concluded that the GITT solution with the eigenvalue problem in the angular coordinate yields better convergence rates than the more usual eigenfunction expansion in the radial variable. This is due to the originally homogeneous boundary conditions in the angular direction, which do not require filtering for convergence enhancement, as opposed to the required filter in the radial direction that introduces a source term in the filtered equation for the streamfunction.

Influence of thermophysical and acoustic properties on temperature elevation at muscle/bone interface induced by ultrasound hyperthermia

Therapeutic ultrasound heating is a very common technique in the treatment of muscular and joints injuries. However, insufficient or excessive heating delivered to living tissues can be either innocuous or hazardous to the patient. The present work deals with the heat transfer process in living tissues, with particular concern on the behavior at the muscle-bone interface, when ultrasound irradiation is applied. Based on the Generalized Integral Transform Technique (GITT), a hybrid numerical-analytical solution of the Pennes’ equation using the open-source UNIT code (“UNified Integral Transforms”) is proposed, considering different boundary conditions. The influence of physical properties and parameters, such as beam incidence angle, acoustic attenuation coefficients for refracted longitudinal and shear waves in the bone, and blood perfusion in the muscle, were more closely analyzed. Numerical results of temperature profiles showing variations of up to 2 °C for different situations are presented, illustrating a marked dependence of heating patterns on the perfusion rates as well as on the effect of shear waves. The conclusions are that such type of more detailed analysis is essential for the accurate identification of physical properties in living tissues and phantoms and can contribute to the treatment planning for therapeutic ultrasound.

Bending of variable thickness rectangular thin plates resting on a double-parameter foundation: integral transform solution

PurposeThe purpose of this study is to propose a generalized integral transform technique (GITT) to investigate the bending behavior of rectangular thin plates with linearly varying thickness resting on a double-parameter foundation.Design/methodology/approachThe bending of plates with linearly varying thickness resting on a double-parameter foundation is analyzed by using the GITT for six combinations of clamped, simply-supported and free boundary conditions under linearly varying loads. The governing equation of plate bending is integral transformed in the uniform-thickness direction, resulting in a linear system of ordinary differential equations in the varying thickness direction that is solved by a fourth-order finite difference method. Parametric studies are performed to investigate the effects of boundary conditions, foundation coefficients and geometric parameters of variable thickness plates on the bending behavior.FindingsThe proposed hybrid analytical-numerical solution is validated against a fourth-order finite difference solution of the original partial differential equation, as well as available results in the literature for some particular cases. The results show that the foundation coefficients and the aspect ratio b/a (width in the y direction to height of plate in the x direction) have significant effects on the deflection of rectangular plates.Originality/valueThe present GITT method can be applied for bending problems of rectangular thin plates with arbitrary thickness variation along one direction under different combinations of loading and boundary conditions.

An improved lumped model for freezing of a freely suspended supercooled water droplet in air stream

This work deals with the mathematical modeling of the transient freezing process of a supercooled water droplet in a cold air stream. The aim is to develop a simple yet accurate lumped-differential model for the energy balance for a freely suspended water droplet undergoing solidification, that allows for cost effective computations of the temperatures and freezing front evolution along the whole process. The complete freezing process was described by four distinct stages, namely, supercooling, recalescence, solidification, and cooling. At each stage, the Coupled Integral Equations Approach (CIEA) is employed, which reduces the partial differential equation for the temperature distribution within the spherical droplet into coupled ordinary differential equations for dimensionless boundary temperatures and the moving interface position. The resulting lumped-differential model is expected to offer improved accuracy with respect to the classical lumped system analysis, since boundary conditions are accounted for in the averaging process through Hermite approximations for integrals. The results of the CIEA were verified using a recently advanced accurate hybrid numerical-analytical solution through the Generalized Integral Transform Technique (GITT), for the full partial differential formulation, and comparisons with numerical and experimental results from the literature. After verification and validation of the proposed model, a parametric analysis is implemented, for different conditions of airflow velocity and droplet radius, which lead to variations in the Biot numbers that allow to inspect for their influence on the accuracy of the improved lumped-differential formulation.

Integral transforms for three-dimensional pumping in confined, leaky, and unconfined aquifers

Abstract Analytical or hybrid numerical-analytical solutions based on the Generalized Integral Transform Technique (GITT) are obtained for the transient three-dimensional pumping problem of aquifers with a fully penetrating vertical well between two parallel streams. The problem formulation for confined and leaky aquifers allows for achieving exact analytical solutions through integral transforms, while the unconfined aquifer case introduces a fourth kind boundary condition which leads to a coupled transformed head ordinary differential system, that can be solved either analytically or numerically. A convergence analysis is performed to illustrate the consistency of the numerical results achieved for the head distribution, as well as for the related pumping rates. Results are obtained for selected cases and comparisons with literature results are performed. A solution verification confirms the agreement of the integral transform solutions with available simulations and provides additional confidence for the analysis of a few physical parameters that influence the hydrological behavior.

Hybrid integral transform analysis of supercooled droplets solidification.

The freezing phenomena in supercooled liquid droplets are important for many engineering applications. For instance, a theoretical model of this phenomenon can offer insights for tailoring surface coatings and for achieving icephobicity to reduce ice adhesion and accretion. In this work, a mathematical model and hybrid numerical–analytical solutions are developed for the freezing of a supercooled droplet immersed in a cold air stream, subjected to the three main transport phenomena at the interface between the droplet and the surroundings: convective heat transfer, convective mass transfer and thermal radiation. Error-controlled hybrid solutions are obtained through the extension of the generalized integral transform technique to the transient partial differential formulation of this moving boundary heat transfer problem. The nonlinear boundary condition for the interface temperature is directly accounted for by the choice of a nonlinear eigenfunction expansion base. Also, the nonlinear equation of motion for the freezing front is solved together with the ordinary differential system for the integral transformed temperatures. After comparisons of the solution with previously reported numerical and experimental results, the influence of the related physical parameters on the droplet temperatures and freezing time is critically analysed.

Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling.

The electrical coupling between myocytes and fibroblasts and the spacial distribution of fibroblasts within myocardial tissues are significant factors in triggering and sustaining cardiac arrhythmias, but their roles are poorly understood. This article describes both direct numerical simulations and an asymptotic theory of propagation and block of electrical excitation in a model of atrial tissue with myocyte-fibroblast coupling. In particular, three idealized fibroblast distributions are introduced: uniform distribution, fibroblast barrier and myocyte strait-all believed to be constituent blocks of realistic fibroblast distributions. Primary action potential biomarkers including conduction velocity, peak potential and triangulation index are estimated from direct simulations in all cases. Propagation block is found to occur at certain critical values of the parameters defining each idealized fibroblast distribution, and these critical values are accurately determined. An asymptotic theory proposed earlier is extended and applied to the case of a uniform fibroblast distribution. Biomarker values are obtained from hybrid analytical-numerical solutions of coupled fast-time and slow-time periodic boundary value problems and compare well to direct numerical simulations. The boundary of absolute refractoriness is determined solely by the fast-time problem and is found to depend on the values of the myocyte potential and on the slow inactivation variable of the sodium current ahead of the propagating pulse. In turn, these quantities are estimated from the slow-time problem using a regular perturbation expansion to find the steady state of the coupled myocyte-fibroblast kinetics. The asymptotic theory gives a simple analytical expression that captures with remarkable accuracy the block of propagation in the presence of fibroblasts.

Integral transform analysis of microchannel fluid flow: Irregular geometry estimation using velocimetry data

A hybrid numerical-analytical solution to fluid flow in irregularly-shaped microchannels has been developed. The solution is based on the Generalized Integral Transform Technique (GITT) applied to a trapezoid-like duct to account for fabrication imperfections, as commonly found in commercially manufactured microchannels. With this approach, the eigenseries expansion is done using a 2D eigenfunction basis, constructed from a combination of simple 1D eigenfunctions defined within the original irregular domain. In order to improve convergence, a new filtering scheme based on an analytical solution to the problem in a rectangular channel, which is distorted to match the irregular domain, is employed. The solution is tested on a real microchannel geometry, obtained from a SEM image, and compared with micro-PIV measurements for the flow field in the same channel. In addition, the actual velocimetry data is input to an inverse problem implementation for recovering geometric parameters related to the irregular geometry using the GITT results as the forward solution. The numerical results obtained with the GITT presented a very satisfactory convergence, and the inverse solution was able to recover the channel geometric parameters with very reasonable accuracy.

Integral transforms for flow and transport in discrete and continuum models of fractured heterogeneous porous media

The Generalized Integral Transform Technique (GITT) is employed in combination with a single domain reformulation strategy and a coupled eigenvalue problem expansion base to construct analytical or hybrid numerical-analytical solutions for flow and transport in fractured porous media. The problem formulation and provided formal solutions encompass both continuum (multi-porosity/multi-permeability) and discrete (fracture-matrix interaction) models for heterogeneous porous media. The single domain representation is written with the aid of space variable coefficients, that account for the abrupt physical properties and source terms transitions across the different subregions. The aim is to provide an unified formalism that includes a wide class of problems in fluid flow, heat and mass transfer in heterogeneous media, with the integral transformation process for the potentials (pressure, velocities, temperature, or concentrations) being applied over the whole physical domain at once, thus markedly reducing the mathematical manipulations. The space variable coefficients are then carried on to the eigenvalue problem formulation which provides the base for the eigenfunction expansion. In addition, a coupled eigenvalue problem alternative is proposed, which is particularly useful in dealing with multi-porosity/multi-permeability flow and transport models, collapsing the multiple coupled potentials equations into one single integral transformation process. Two representative applications are briefly considered, one dealing with contaminant transport with discrete fracture-matrix interaction in layered porous media and the other related to dual porosity/dual permeability model of flow in unsaturated soils.

Detection of contact failures employing combination of integral transforms with single-domain formulation, finite differences, and Bayesian inference

This work addresses the hybrid numerical–analytical solution of heat conduction in multilayered media by means of a single-domain formulation, integral transforms, and the finite difference method. The basic concept is to incorporate different materials into spatially variable coefficients with abrupt transitions, representing different subdomains of the original problem. The classical integral transform technique is employed, and the problem solution is written as an expansion in terms of eigenfunctions obtained from an eigenvalue problem with spatially variable coefficients. To solve this eigenvalue problem, the finite difference method with a second-order accuracy scheme was employed. In order to illustrate the application of the methodology, the problem of heat conduction within a multilayered pipe was considered, including the identification of a defect in the junction region of a laminated composite, in which the adhesion of two different materials is imperfect. The failure identification was carried out by estimating the thermal conductivity throughout the region containing adhesive using the formulation and the solution of an inverse problem within the Bayesian framework.

Integral transform analysis of convective heat transfer within wavy walls channels

The generalized integral transform technique (GITT) is employed in the hybrid numerical-analytical solution of the two-dimensional Navier–Stokes and energy equations in wavy walls channels. The flow is considered laminar and incompressible for a Newtonian fluid with temperature-independent physical properties, while the walls temperatures are kept uniform along the channel length. The streamfunction-only formulation is adopted, which eliminates the pressure field and automatically satisfies the continuity equation. A thorough convergence analysis is performed for the streamfunction field, temperature field, friction factor, and local Nusselt number to illustrate the method robustness. The verification of the present GITT results is also performed by comparing the centerline velocity, friction factor, average temperature, and local Nusselt number with equivalent results from the COMSOL Multiphysics simulation platform, with overall very good agreement. The influence of the governing parameters such as Reynolds number and wavy-wall amplitude on the velocity and temperature fields is also analyzed, demonstrating their importance in the convective heat transfer behavior.

A hybrid analytical-numerical solution for a circular pile under lateral load in multilayered soil

A hybrid analytical-numerical solution is proposed to solve the problem of a laterally loaded pile with a circular cross-section in multilayered soils. In the pile-soil model, the lateral load is located at the pile head including both lateral force and bending moment. The single pile is considered as a beam on elastic foundation while shear beams model the soil column below the pile toe. The differential equations governing pile deflections are derived based on the energy principles and variational approaches. The differential equations are solved iteratively by using the finite element method that provides results of pile deflection, rotation angle, shear force, and bending moment along the pile and equivalent stiffness of the pile-soil system. The modulus reduction equation is also developed to match the proposed results well to the three-dimensional finite element analyses. Several examples are conducted to validate the proposed method by comparing the analysis results with those of existing analytical solutions, the three-dimensional finite element solutions.
 Keywords:
 beam on elastic foundation; finite element method; pile; energy principle; lateral load.

Single domain integral transform analysis of natural convection in cavities partially filled with heat generating porous medium

An analysis of natural convection within a rectangular cavity partially filled with a heat-generating porous medium is carried out through the Generalized Integral Transform Technique (GITT), in which the laminar flow and energy equations are solved with automatic error control. A single domain reformulation strategy is adopted to rewrite the governing equations within the fluid and the porous medium as a single heterogeneous medium formulation, with spatially variable physical properties and source terms that account for the abrupt transition of the two regions. This fundamental study is motivated by the analysis of wet storage of spent nuclear fuel elements with passive cooling of the pool and physical conclusions are drawn from the hybrid numerical-analytical solution. Increases in the Rayleigh number with constant internal heat generation are found to lower the maximum temperature within the cavity. Moreover, decreasing the aspect ratio also has positive effects on the cooling of the cavity.

On the mass transport in membraneless flow batteries with flow-by configuration

Progress on the analysis of the mass transport phenomena in a membraneless redox flow battery with porous electrodes in flow-by configuration is here reported. A species transport model for a typical redox reactant interacting with a porous electrode is proposed. A hybrid numerical-analytical solution is obtained through the Generalized Integral Transform Technique (GITT), adopting a single domain formulation that includes both the porous and pure fluid regions. The influence of the Reynolds number and the thickness of the electrode on the parameters of interest is theoretically examined. The importance of considering the simultaneous development of flow and mass transport is analyzed, and the presence of the transversal convective flux of species proves to have a significant role on the generation of current inside the electrode. A scaling of the limiting current density with ∼Re0.41 is demonstrated and some physical conclusions are drawn. Guidelines for the prevention of crossover are also offered, with increasing Reynolds number and decreasing relative thickness of the electrode having a positive effect, as far as avoiding the mixed potentials effects is concerned. The physical insights attained through the present analysis should add to the efforts in achieving membraneless redox flow batteries with performances comparable to membrane-based devices of similar size.

Hybrid integral transforms for flow development in ducts partially filled with porous media.

A hybrid numerical-analytical solution is developed for laminar flow development in a parallel plate duct partially filled with porous media. The integral transform method is employed in combination with a single domain reformulation strategy for representing the heterogeneous media within the channel. A novel eigenfunction expansion basis is proposed, including abrupt spatial variations of physical properties due to the domain transitions. The introduction of the new basis allows for a solution with similar convergence rates as in previous applications with simpler formulations, as demonstrated through a careful convergence analysis of the expansions. The inherent automatic error control characteristic of the integral transforms approach then provides benchmark results for the developing velocity profile. Moreover, a physical analysis further verifies the consistency of both the proposed expansion and the mixed symbolic-numerical code developed. A detailed verification with a finite-element commercial code is also performed.

Integral transform solution for the analysis of hydrodynamic lubrication of thrust bearings

PurposeThe aim of this paper is related to an analysis of hydrodynamic lubrication of circular thrust bearings.Design/methodology/approachThe modified Reynolds equation was treated to obtain a hybrid numerical-analytical solution through the generalized integral transform technique (GITT) for the problem.FindingsNumerical results for the engineering parameters such as pressure field, load capacity and power consumption were thus produced as functions of the radial and circumferential directions. These parameters depend on the geometry of sector-shaped used: Rayleigh pad with 4, 8 and 16 steps. Comparing among them, on the numerical point of view, the Rayleigh pad geometry with N = 16 steps has a better satisfactory performance because it has a lower power consumption.Originality/valueThe present GITT results and those obtained by the finite volume method (FVM) from previous works in the literature were confronted to verify whether the results are consistent and to demonstrate the capacity of the GITT approach in handling thrust bearing problems.

Analysis of magnetohydrodynamic natural convection in closed cavities through integral transforms

A hybrid numerical-analytical solution is proposed to analyze MHD (magnetohydrodynamic) natural convection of an electrically-conducting fluid within a square cavity, differentially heated at the sidewalls and subjected to an inclined external magnetic field. The first goal is to expand the spectrum of application of the so called Generalized Integral Transform Technique (GITT), dealing with a multiphysics formulation, while further demonstrating the relative merits of the proposed eigenfunction expansion approach in handling highly nonlinear and coupled systems of partial differential equations. The second goal is to provide a set of benchmark results in this important application for quantities of practical interest in determining the heat transfer rates, such as the average Nusselt number. The two-dimensional steady state equations are written in dimensionless form using the streamfunction-only formulation and are subsequently solved with the GITT approach, under automatic relative error control. Critical comparisons are performed against previous work reported in the literature, both computational and experimental, together with the corresponding physical interpretations, for different values of the governing parameters, such as Grashof number, Hartmann number, Prandtl number, and magnetic field inclination angle.

Size-Dependent Vibration of Axially Moving Viscoelastic Micro-Plates Based on Sinusoidal Shear Deformation Theory

In the present research, vibration and instability of axially moving viscoelastic micro-plate is investigated. Sinusoidal shear deformation theory (SSDT) is utilized due to its accuracy of polynomial functions than other plate theories. Based on Kelvin’s model, the viscoelastic structural properties of micro-plate are taken into consideration. The modified couple stress theory (MCST) is employed because of its capability to interpret the size effect. Using Hamilton’s principle, equations of motion are obtained and solved by hybrid analytical–numerical solution at different boundary conditions. Influences of various parameters such as size effect, axially moving speed, viscoelastic structural damping coefficient, thickness and aspect ratio on the vibration characteristics of moving viscoelastic micro-plate are discussed in detail. The results indicated that the critical speed of moving micro-plate is strongly dependent on the aspect ratio, therefore, the low aspect ratio should be considered for optimum design of this kind of moving micro-devices. The results of this investigation can be used in design and manufacturing of axially moving systems at the micro-scale such as micro-magnetic tapes.

Inverse problem analysis for identification of reaction kinetics constants in microreactors for biodiesel synthesis

The theoretical analysis for the design of microreactors in biodiesel production is a complicated task due to the complex liquid-liquid flow and mass transfer processes, and the transesterification reaction that takes place within these microsystems. Thus, computational simulation is an important tool that aids in understanding the physical-chemical phenomenon and, consequently, in determining the suitable conditions that maximize the conversion of triglycerides during the biodiesel synthesis. A diffusive-convective-reactive coupled nonlinear mathematical model, that governs the mass transfer process during the transesterification reaction in parallel plates microreactors, under isothermal conditions, is here described. A hybrid numerical-analytical solution via the Generalized Integral Transform Technique (GITT) for this partial differential system is developed and the eigenfunction expansions convergence rates are extensively analyzed and illustrated. The heuristic method of Particle Swarm Optimization (PSO) is applied in the inverse analysis of the proposed direct problem, to estimate the reaction kinetics constants, which is a critical step in the design of such microsystems. The results present a good agreement with the limited experimental data in the literature, but indicate that the GITT methodology combined with the PSO approach provide a reliable computational algorithm for direct-inverse analysis in such reactive mass transfer problems.

Escoamento laminar em dutos de setor circular

The fluid flow inside ducts is of great importance in engineering, because it is present in most domestic, commercial, and industrial application equipments. In this work was presented a hybrid analytical-numerical solution to hydrodynamic problem of fully developed laminar flow inside circular sector ducts. In order to facilitate the analytical treatment and the application of the boundary conditions, a Conformal Transform was used. Thereafter, the Generalized Integral Transform Technique was applied on the momentum equation to obtain the velocity field. Numerical results were obtained for quantities of practical interest, such as maximum and minimum velocity, Fanning friction factor, Poiseuille number, Hagenbach factor, and hydrodynamic entry length. The results were compared, as much as possible, with the parameter values available in the literature and they presented a great agreement, with a difference less than 1.5%. Keywords: Integral Transform, Conformal Transform, Laminar Flow, Newtonian Fluid, Circular Sector.