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The purpose of this paper is to investigate the stability of a suspension containing both gyrotactic and oxytactic microorganisms for the case when the suspension occupies a horizontal layer of finite depth. The lower boundary of the layer is assumed rigid while at the upper boundary both situations of rigid and stress‐free boundary conditions are considered.

Linear instability analysis is utilized, and the obtained eigenvalue problem is solved analytically using a one‐term Galerkin method.

The obtained eigenvalue equation relates three Rayleigh numbers, the traditional thermal Rayleigh number and two bioconvection Rayleigh numbers, for gyrotactic and oxytactic microorganisms.

Only the case of non‐oscillatory instability (which always occurs when heating from the bottom is considered) is analyzed. Further experimental research is needed to elucidate possible interaction between gyrotactic and oxytactic microorganisms. The developed theory is applicable only for dilute suspensions.

This paper extends the traditional theory of bio‐thermal convection to the case when the suspension contains two types of motile microorganisms exhibiting different behaviors.

The purpose of this paper is to determine both analytically and numerically the existence of smooth, cusped and sharp shock wave solutions to a one-dimensional model of microfluidic droplet ensembles, water flow in unsaturated flows, infiltration, etc., as functions of the powers of the convection and diffusion fluxes and upstream boundary condition; to study numerically the evolution of the wave for two different initial conditions; and to assess the accuracy of several finite difference methods for the solution of the degenerate, nonlinear, advection--diffusion equation that governs the model.

The theory of ordinary differential equations and several explicit, finite difference methods that use first- and second-order, accurate upwind, central and compact discretizations for the convection terms are used to determine the analytical solution for steadily propagating waves and the evolution of the wave fronts from hyperbolic tangent and piecewise linear initial conditions to steadily propagating waves, respectively. The amplitude and phase errors of the semi-discrete schemes are determined analytically and the accuracy of the discrete methods is assessed.

For non-zero upstream boundary conditions, it has been found both analytically and numerically that the shock wave is smooth and its steepness increases as the power of the diffusion term is increased and as the upstream boundary value is decreased. For zero upstream boundary conditions, smooth, cusped and sharp shock waves may be encountered depending on the powers of the convection and diffusion terms. For a linear diffusion flux, the shock wave is smooth, whereas, for a quadratic diffusion flux, the wave exhibits a cusped front whose left spatial derivative decreases as the power of the convection term is increased. For higher nonlinear diffusion fluxes, a sharp shock wave is observed. The wave speed decreases as the powers of both the convection and the diffusion terms are increased. The evolution of the solution from hyperbolic tangent and piecewise linear initial conditions shows that the wave back adapts rapidly to its final steady value, whereas the wave front takes much longer, especially for piecewise linear initial conditions, but the steady wave profile and speed are independent of the initial conditions. It is also shown that discretization of the nonlinear diffusion flux plays a more important role in the accuracy of first- and second-order upwind discretizations of the convection term than either a conservative or a non-conservative discretization of the latter. Second-order upwind and compact discretizations of the convection terms are shown to exhibit oscillations at the foot of the wave’s front where the solution is nil but its left spatial derivative is largest. The results obtained with a conservative, centered second--order accurate finite difference method are found to be in good agreement with those of the second-order accurate, central-upwind Kurganov--Tadmor method which is a non-oscillatory high-resolution shock-capturing procedure, but differ greatly from those obtained with a non-conservative, centered, second-order accurate scheme, where the gradients are largest.

A new, one-dimensional model for microfluidic droplet transport, water flow in unsaturated flows, infiltration, etc., that includes high-order convection fluxes and degenerate diffusion, is proposed and studied both analytically and numerically. Its smooth, cusped and sharp shock wave solutions have been determined analytically as functions of the powers of the nonlinear convection and diffusion fluxes and the boundary conditions. These solutions are used to assess the accuracy of several finite difference methods that use different orders of accuracy in space, and different discretizations of the convection and diffusion fluxes, and can be used to assess the accuracy of other numerical procedures for one-dimensional, degenerate, convection--diffusion equations.

In 1982, Smith and Hutton published comparative results of several different convection‐diffusion schemes applied to a specially devised test problem involving near‐discontinuities and strong streamline curvature. First‐order methods showed significant artificial diffusion, whereas higher‐order methods gave less smearing but had a tendency to overshoot and oscillate. Perhaps because unphysical oscillations are more obvious than unphysical smearing, the intervening period has seen a rise in popularity of low‐order artificially diffusive schemes, especially in the numerical heat‐transfer industry. This paper presents an alternative strategy of using non‐artificially diffusive higher‐order methods, while maintaining strictly monotonic transitions through the use of simple flux‐limiter constraints. Limited third‐order upwinding is usually found to be the most cost‐effective basic convection scheme. Tighter resolution of discontinuities can be obtained at little additional cost by using automatic adaptive stencil expansion to higher order in local regions, as needed.

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.

In the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.

Numerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.

Accurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.

The present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.

The paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

The purpose of this paper is to present an efficient improved version of Implicit Pressure-Explicit Saturation (IMPES) method for the solution of incompressible two-phase flow model based on the discontinuous Galerkin (DG) numerical scheme.

The governing equations, based on the wetting-phase pressure-saturation formulation, are discretized using various primal DG schemes. The authors use H(div) velocity reconstruction in Raviart-Thomas space (RT_0 and RT_1), the weighted average formulation, and the scaled penalties to improve the spatial discretization. It uses a new improved IMPES approach, by using the second-order explicit Total Variation Diminishing Runge-Kutta (TVD-RK) as temporal discretization of the saturation equation. The main purpose of this time stepping technique is to speed up computation without losing accuracy, thus to increase the efficiency of the method.

Utilizing pressure internal interpolation technique in the improved IMPES scheme can reduce CPU time. Combining the TVD property with a strong multi-dimensional slope limiter namely, modified Chavent-Jaffre leads to a non-oscillatory scheme even in coarse grids and highly heterogeneous porous media.

The presented locally conservative scheme can be applied only in 2D incompressible two-phase flow modeling in non-deformable porous media. In addition, the capillary pressure discontinuity between two adjacent rock types assumed to be negligible.

The proposed numerical scheme can be efficiently used to model the incompressible two-phase flow in secondary recovery of petroleum reservoirs and tracing immiscible contamination in aquifers.

The paper describes a novel version of the DG two-phase flow which illustrates the effects of improvements in special discretization. Also the new improved IMPES approach used reduces the computation time. The non-oscillatory scheme is an efficient algorithm as it maintains accuracy and saves computation time.

High order schemes, which are widely used in DNS and LES, received increasing attention in recent years with a number of variants being developed. However most of these schemes have difficulties in achieving high order accuracy near the boundary points. In order to solve this problem, the analytical discrete method (ADM) is proposed and presented in this paper. In addition, this method is convenient to construct the higher order WENO (weighted essentially non‐oscillatory) scheme. Application of the ADM‐WENO scheme to shock‐tube problems and compressible mixing flows has shown it is robust and accurate in both shock‐capturing and complex flow structures detection.

The use of explicit finite difference schemes for low Stefan number problems with moving interface was largely abandoned because they require small time intervals (large CPU time) to obtain accurate non‐oscillatory solutions. This paper uses these type of schemes for better estimations of the dynamics of the solid—liquid interface. The scheme in which time and radial intervals are constant, uses a local, continuous, time‐dependent radial coordinate to define the instantaneous location of the interface. Taylor series expansions which result in a polynomial fit are used for forward and backward interpolation of temperatures of nodal points in the vicinity of the interface. A distinction is made between the left and right position of the interface relative to the closest nodal point. The algorithm handles accurately and effectively the non‐linearities near the interface thus producing accurate stable solutions with relatively low CPU time. The scheme which obviously may be applied to large Stefan number problems, is also suitable for time dependent boundary conditions as well as temperature dependent physical properties. The results obtained by the scheme were in excellent agreement with ones derived from an approximate analytical solution which is applicable in the low Stefan number range.

In this paper new cubic v‐splines monotonic one‐dimensional profiles are presented, for the finite volume solution of convection‐diffusion problems. By studying the profile in normalized variables, some weight functions have been determined for the profile. Being free of the requirement that the volumes be equal, the volume size can be reduced where needed. Numerical properties of the proposed method were formally analysed and are confirmed by numerical examples included here.

A new finite volume (FV) method is proposed for the solution of convection‐diffusion equations defined on 2D convex domains of general shape. The domain is approximated by a polygonal region; a structured non‐uniform mesh is defined; the domain is partitioned in control volumes. The conservative form of the problem is solved by imposing the law to be verified on each control volume. The dependent variable is approximated to the second order by means of a quadratic profile. When, for the hyperbolic equation, discontinuities are present, or when the gradient of the solution is very high, a cubic profile is defined in such a way that it enjoys unidirectional monotonicity. Numerical results are given.

This paper aims to study thermocapillarity‐induced flow of thin liquid films covering heated horizontal walls with 2D topography.

A numerical model based on the 2D solution of heat and fluid flow within the liquid film, the gas above the film and the structured wall is developed. The full Navier‐Stokes equations are solved and coupled with the energy equation by a finite difference algorithm. The movable gas‐liquid interface is tracked by means of the volume‐of‐fluid method. The model is validated by comparison with theoretical and experimental data showing a good agreement.

It is demonstrated that convective motion within a film on a structured wall exists at any nonzero Marangoni number. The motion is caused by surface tension gradients induced by temperature differences at the gas‐liquid interface due to the spatial structure of the heated wall. These simulations predict that the maximal flow velocity is practically independent from the film thickness, and increases with increasing temperature difference between the wall and the surrounding gas. It is found that an abrupt change in wall temperature causes rupture of the liquid film. The thermocapillary convection notably enhances heat transfer in liquid films on heated structured walls.

Our solutions are restricted to the case of periodic wall structure, and the flow is enforced to be periodic with a period equal to that of the wall.

The reported results are useful for design of the heat transfer equipment.

New effects in thermocapillary convection are presented and studied using a developed numerical model.