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This paper presents a spectral multidomain method for solving the Navier‐Stokes equations in the vorticity‐stream function formulation. The algorithm is based on an extensive use of the influence matrix technique and so leads to a direct method without any iterative process. Numerical results concerning the Czochralski melt configuration are reported and compared with spectral monodomain solutions to show the advantage of the domain decomposition for such a problem which solution presents a singular behaviour.

Free and moving boundary problems require the simultaneous solution of unknown field variables and the boundaries of the domains on which these variables are defined. There are many technologically important processes that lead to moving boundary problems associated with fluid surfaces and solid‐fluid boundaries. These include crystal growth, metal alloy and glass solidification, melting and flame propagation. The directional solidification of semi‐conductor crystals by the Bridgman—Stockbarger method^1,2 is a typical example of such a complex process. A numerical model of this growth method must solve the appropriate heat, mass and momentum transfer equations and determine the location of the melt—solid interface. In this work, a Chebyshev pseudospectral collocation method is adapted to the problem of directional solidification. Implementation involves a solution algorithm that combines domain decomposition, a finite‐difference preconditioned conjugate minimum residual method and a Picard type iterative scheme.

The quality of crystals grown in space can be diversely affected by the melt flows induced by g‐jitter associated with a space vehicle. This paper presents a full three‐dimensional (3D) transient finite element analysis of the complex fluid flow and heat and mass transfer phenomena in a simplified Bridgman crystal growth configuration under the influence of g‐jitter perturbations and magnetic fields.

The model development is based on the Galerkin finite element solution of the magnetohydrodynamic governing equations describing the thermal convection and heat and mass transfer in the melt. A physics‐based re‐numbering algorithm is used to make the formidable 3D simulations computationally feasible. Simulations are made using steady microgravity, synthetic and real g‐jitter data taken during a space flight.

Numerical results show that g‐jitter drives a complex, 3D, time dependent thermal convection and that velocity spikes in response to real g‐jitter disturbances in space flights, resulting in irregular solute concentration distributions. An applied magnetic field provides an effective means to suppress the deleterious convection effects caused by g‐jitter. Based on the simulations with applied magnetic fields of various strengths and orientations, the magnetic field aligned with the thermal gradient provides an optimal damping effect, and the stronger magnetic field is more effective in suppressing the g‐jitter induced convection. While the convective flows and solute transport are complex and truly 3D, those in the symmetry plane parallel to the direction of g‐jitter are essentially two‐dimensional (2D), which may be approximated well by the widely used 2D models.

The physics‐based re‐numbering algorithm has made possible the large scale finite element computations for 3D g‐jitter flows in a magnetic field. The results indicate that an applied magnetic field can be effective in suppressing the g‐jitter driven flows and thus enhance the quality of crystals grown in space.

The low Mach number approximation of the Navier—Stokes equations is of similar nature to the equations for incompressible flow. A major difference, however, is the appearance of a space‐ and time‐varying density that introduces a supplementary non‐linearity. In order to solve these equations with spectral space discretization, an iterative solution method has been constructed and successfully applied in former work to two‐dimensional natural convection and isobaric combustion with one direction of periodicity. For the extension to other geometries efficiency is an important point, and it is therefore desirable to devise a direct method which would have, in the best case, the same stability properties as the iterative method. The present paper discusses in a systematic way different approaches to this aim. It turns out that direct methods avoiding the diffusive time step limit are possible, indeed. Although we focus for discussion and numerical investigation on natural convection flows, the results carry over for other problems such as variable viscosity flows, isobaric combustion, or non‐homogeneous flows.

Deals with the formulation and application of temporal and spatial spectral element approximations for the solution of convection‐diffusion problems. Proposes a new high‐order splitting space‐time spectral element method which exploits space‐time discontinuous Galerkin for the first hyperbolic substep and space continuous‐time discontinuous Galerkin for the second parabolic substep. Analyses this method and presents its characteristics in terms of accuracy and stability. Also investigates a subcycling technique, in which several hyperbolic substeps are taken for each parabolic substep; a technique which enables fast, cost‐effective time integration with little loss of accuracy. Demonstrates, by a numerical comparison with other coupled and splitting space‐time spectral element methods, that the proposed method exhibits significant improvements in accuracy, stability and computational efficiency, which suggests that this method is a potential alternative to existing schemes. Describes several areas for future research.

The multi‐domain generalized differential quadrature method is applied in this paper to simulate the flows in Czochralski crystal growth. The effect of interface treatment on the numerical solution is studied through four types of interface approximations. The performance of those four interface approximations is validated by a benchmark problem suggested by Wheeler. It is demonstrated in this study that the multi‐domain GDQ approach is an efficient method which can obtain accurate numerical solutions by using very few grid points, and the overlapped interface approximation provides the most accurate numerical results.

The purpose of this paper is to introduce a novel approach based on the high-order matrix derivative of the Bernstein basis and collocation method and its employment to solve an interesting and ill-posed model in the heat conduction problems, homogeneous backward heat conduction problem (BHCP).

By using the properties of the Bernstein polynomials the problems are reduced to an ill-conditioned linear system of equations. To overcome the unstability of the standard methods for solving the system of equations an efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-condition system.

The presented numerical results through table and figures demonstrate the validity and applicability and accuracy of the technique.

A novel method based on the high-order matrix derivative of the Bernstein basis and collocation method is developed and well-used to obtain the numerical solutions of an interesting and ill-posed model in heat conduction problems, homogeneous BHCP with high accuracy.

The present study aims to investigate the inverse-thermocapillary effect in an evaporating thin liquid film of self-rewetting fluid, which is a dilute aqueous solution (DAS) of long-chain alcohol.

A long-wave evolution model modified for self-rewetting fluids is used to study the inverse thermocapillary characteristics of an evaporating thin liquid film. The flow attributed to the inverse thermocapillary action is manifested through the streamline plots and the evaporative heat transfer characteristics are quantified and analyzed.

The thermocapillary flow induced by the negative surface tension gradient drives the liquid from a low-surface-tension (high temperature) region to a high-surface-tension (low temperature) region, retarding the liquid circulation and the evaporation strength. The positive surface tension gradients of self-rewetting fluids induce inverse-thermocapillary flow. The results of different working fluids, namely, water, heptanol and DAS of heptanol, are examined and compared. The thermocapillary characteristic of a working fluid is significantly affected by the sign of the surface tension gradient and the inverse effect is profound at a high excess temperature. The inverse thermocapillary effect significantly enhances evaporation rates.

The current investigation on the inverse thermocapillary effect in a self-rewetting evaporating thin film liquid has not been attempted previously. This study provides insights on the hydrodynamic and thermal characteristics of thermocapillary evaporation of self-rewetting liquid, which give rise to significant thermal enhancement of the microscale phase-change heat transfer devices.

The purpose of this paper is to numerically study transient natural convective flow in a square cavity with partially heated and cooled vertical walls, thermally insulated top wall and linearly heated bottom wall.

The governing equations of motion are non‐dimensionalized and reformulated using stream function‐vorticity approach. Alternating direction implicit finite difference scheme is used to solve the coupled equations.

The transient results obtained for different values of Grashof number (Gr) and fixed Prandtl number Pr = 0.733 are presented in the form of isotherms, streamlines, bifurcation diagram and time series. The transition from steady to oscillatory motions is analyzed in detail with respect to Gr. The flow is observed to be steady up to Gr ≈ 2 × 10⁴. A time‐periodic unsteady solution first appears at Gr = 20,900 and the amplitude of the fluctuation grows as Gr is increased.

The study is limited to laminar flow in a square cavity. Further extension of this work could include the influence of various choices of Prandtl number and the effect of aspect ratio. Buoyancy‐driven convection in a sealed cavity with differentially heated walls is a prototype of many industrial applications such as energy‐efficient design of buildings and rooms, convective heat transfer associated with boilers, etc.

The paper presents an original computer program written in FORTRAN to solve the partial differential equations.

The interaction of a global model (GM) and a local (regional) model (LM) of heat flow is considered under the framework of so‐called “one‐way nesting”. In this framework, the GM is constructed in a large domain with coarse discretization in space and time, while the LM is set in a small subdomain with fine discretization.

The GM is solved first, and its results are then used via some boundary transfer operator (BTO) on the GM–LM interface in order to solve the LM. Past experience in various fields of application has shown that one has to be careful in the choice of BTO to be used on the GM–LM interface, since this choice affects both the stability and accuracy of the computational scheme. Here the problem is first theoretically analyzed for the linear heat equation, and stable BTOs are identified. Then numerical experiments are performed with one‐way nesting in a two‐dimensional channel for heat flow with and without radiation emission and linear reaction, using four different BTOs.

Among other conclusions, it is shown that the “negative Robin” BTO is unstable, whereas the Dirichlet, Neumann and “positive Robin” BTO are all stable. It is also shown that in terms of accuracy, the Neumann and “positive Robin” BTOs should be preferred over the Dirichlet BTO.

This study may be the first step in analyzing BTO accuracy and stability for more general atmospheric systems.