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The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation.

The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme.

A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification.

A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.

The two‐dimensional steady boundary layer equations, for simultaneous heat and fluid flow within ducts, are handled through the generalized integral transform technique. The momentum and energy equations are integral transformed by eliminating the transversal coordinate and reducing the PDE’s into an infinite system of coupled non‐linear ordinary differential equations for the transformed potentials. An adaptively truncated version of this ODE system is numerically handled through well known initial value problem solvers, with automatic precision control procedures. The explicit inversion formulae are then recalled to provide analytic expressions for velocity and temperature fields and related quantities of practical interest. Typical examples are presented in order to illustrate the hybrid numerical analytical approach and its convergence behaviour.

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis.

The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials.

An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented.

This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

A hybrid numerical‐analytical approach, based on recent developments in the generalized integral transform technique, is presented for the solution of a class of non‐linear transient convection‐diffusion problems. The original partial differential equation is integral transformed into a denumerable system of coupled non‐linear ordinary differential equations, which is numerically solved for the transformed potentials. The hybrid analysis convergence is illustrated by considering the one‐dimensional non‐linear Burgers equation and numerical results are presented for increasing truncation orders of the infinite ODE system.

This paper seeks to analyze transient convection‐diffusion by employing the generalized integral transform technique (GITT) combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behavior of the associated eigenfunction expansions. The idea is to consider analytical approximations of the original problem as filtering solutions, defined within specific ranges of the time variable, which act diminishing the importance of the source terms in the original formulation and yielding a filtered problem for which the integral transformation procedure results in faster converging eigenfunction expansions. An analytical local instantaneous filtering is then more closely considered to offer a hybrid numerical‐analytical solution scheme for linear or nonlinear convection‐diffusion problems.

The approach is illustrated for a test‐case related to transient laminar convection within a parallel‐plates channel with axial diffusion effects.

The developing thermal problem is solved for the fully developed flow situation and a step change in inlet temperature. An analysis is performed on the variation of Peclet number, so as to investigate the importance of the axial heat or mass diffusion on convergence rates.

This paper succeeds in analyzing transient convection‐diffusion via GITT, combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behaviour of the associated eigenfunction expansions.

The integral transform method is employed for the hybrid numerical‐analytical solution of the boundary layer formulation in developing turbulent flow inside channels. An algebraic turbulence model due to Cebeci and Smith is adopted, in light of its popularity demonstrated in the recent numerical simulation literature. Numerical results for the velocity components in a parallel‐plates channel are obtained, and the automatic error control feature of the present approach is demonstrated. Critical comparisons with experimental works, as well as with previous simulations that employed differential turbulence models, are also performed.

Different possibilities for the enhancement of convergence rates in eigenfunction expansions are investigated in the realm of integral transform solutions for partial differential equations. A representative parabolic problem is chosen to illustrate two schemes and their combinations; a filtering technique and an integral balance approach. Numerical results are presented to confirm the relative merits in each proposed procedure.

The aim of this paper is related to an analysis of hydrodynamic lubrication of circular thrust bearings.

The modified Reynolds equation was treated to obtain a hybrid numerical-analytical solution through the generalized integral transform technique (GITT) for the problem.

Numerical 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.

The 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.

The purpose of this paper is to provide an analysis of two‐dimensional laminar flow in the entrance region of wavy wall ducts as obtained from the solution of the steady Navier‐Stokes equations for incompressible flow.

The study is undertaken by application of the generalized integral transform technique in the solution of the steady Navier‐Stokes equations for incompressible flow. The streamfunction‐only formulation is adopted, and a general filtering solution that adapts to the irregular contour is proposed to enhance the convergence behavior of the eigenfunction expansion.

A few representative cases are considered more closely in order to report some numerical results illustrating the eigenfunction expansions convergence behavior. The product friction factor‐Reynolds number is also computed and compared against results from discrete methods available in the literature for different Reynolds numbers and amplitudes of the wavy channel.

The proposed methodology is fairly general in the analysis of different channel profiles, though the reported results are limited to the wavy channel configuration. Future work should also extend the analysis to geometries represented in the cylindrical coordinates with longitudinally variable radius.

The error‐controlled converged results provide reliable benchmark results for the validation of numerical results from computational codes that address the solution of the Navier‐Stokes equations in irregular geometries.

Although the hybrid methodology is already known in the literature, the results here presented are original and further challenges application of the integral transform method in the solution of the Navier‐Stokes equations.

The integral transform method is employed for the hybrid numerical‐analytical solution of two‐dimensional, steady‐state heat conduction within extended surfaces of variable longitudinal profile and temperature dependent thermal conductivity. Numerical results are then obtainable with automatic accuracy, allowing for the establishment of benchmark results and for the validation of approximate solutions. Convergence rates are illustrated for longitudinal fins with trapezoidal and parabolic profiles, and for different values of the governing parameters, Biot number and aspect ratio. In addition, the classical one‐dimensional approximate solutions are critically examined for these typical non‐straight profiles, and the applicability limits are investigated.