Incompressibility Condition Research Articles

On the use of multibody dynamics techniques to simulate fluid dynamics and fluid–solid interaction problems

A multibody dynamics-based solution to the fluid dynamics problem is compared herein to two established Lagrangian-based techniques used by the computational fluid dynamics (CFD) community. The multibody dynamics-based solution has two salient attributes: it enforces the incompressibility condition through bilateral kinematic constraints, and it treats the coupling with the solid phase via unilateral kinematic constraints. The multibody dynamics-based solution, called herein the Kinematically Constrained Smoothed Particle Hydrodynamics (KCSPH) method, is a Lagrangian approach to solving the CFD problem. It relies on the Smoothed Particle Hydrodynamics (SPH) method to discretize the spatial differential operators in the Navier–Stokes equations, and on the Newton–Euler equations of multibody dynamics to convect the SPH particles forward in time. We show that the multibody dynamics-based approach is efficient and accurate by comparing its performance with the two most commonly used SPH algorithms in the CFD community: the weakly compressible SPH (WCSPH), and the implicit SPH (ISPH) methods. The comparison is carried out in conjunction with four tests: an incompressibility benchmark test, dam break, floating cylinder, and sloshing tank. We conclude that KCSPH is a robust alternative to conventional CFD approaches for fluid–solid interaction (FSI) problems with complex/moving boundaries. The solvers and models used herein are publicly available in an open-source software called Chrono; the implementations use GPU (for WCSPH and ISPH), and multicore CPU (for KCSPH) parallel computing.

Numerical investigation of solitary wave interaction with a flapper wave energy converter using incompressible SPH method

In this research, mesh-free incompressible smoothed particle hydrodynamics (ISPH) was used to study the oscillatory behavior of a flapper wave energy converter located at the end of a numerical tank under the impact of solitary waves. A two-step projection approach including prediction and correction steps was employed to conduct temporal discretization of the governing equations for inviscid fluid flows. At different times, the pressure fields were implicitly calculated by the ISPH method to solve a pressure Poisson equation (PPE) derived upon the enforcement of the incompressibility conditions. To the authors’ knowledge, not many mesh-less numerical studies have been accomplished addressing different aspects of absorption of wave power using oscillating flappers. In this study, effects of different parameters, including solitary wave height, flapper spring stiffness, moment of inertia of the wall withstanding wave loads, initial angle of the flapper and initial longitudinal wave crest–flapper distance for a pair of solitary waves, were examined to analyze the wave energy absorption performance of the flapper. The results showed the necessity of adopting optimum spring stiffness to maximize the wave power absorption. Otherwise, wave overtopping may occur for sub-optimal spring stiffness values, while the absorbed wave power was lower than maximal for too rigid flappers. It was further found that the flapper oscillation period decreased with increasing the spring stiffness, with the other parameters imposing no significant effect on the period. As another finding, it was figured out that the moment of inertia of the downstream wall must be optimal to capture maximum wave power. The initial angle of the flapper was also found to be an important factor contributing to wave power absorption. The maximal absorbed wave power was achieved when a pair of solitary waves was flowing toward the flapper in such a way that the second wave hit the flapper at the end of the flapper period, causing some resonance phenomenon. Damping process of the oscillatory motion due to the flow viscosity and flapper friction was also considered. The results of the present work can be used to maximize the wave power absorption by adjusting the wave converter design considering the wave height, spring stiffness, moment of inertia and longitudinal distance of wave crest between two successive solitary waves.

Unsteady Aerodynamic Characteristics of Pitching Flat Plates at Low Reynolds Numbers

A computational study is conducted on thin flat plates to simulate flows of Reynolds numbers at to provide understanding and guidance for micro air vehicles and other low-Reynolds-number airfoil designs. A synergistic effort between experiments and validated and computational fluid dynamics (CFD) tools were used as part of this study. The CFD tool used in this study is an established Reynolds-averaged Navier–Stokes (RANS) solver with a Spalart–Allmaras turbulence model and a correlation-based laminar–turbulent boundary-layer transition model. The computational method was validated against experimental data for flat plates under steady and unsteady kinematic conditions. The objective of the study was to understand unsteady characteristics of a thin flat plate undergoing harmonic pitching (no plunging) around the quarter chord under incompressible flow conditions. Pitching amplitudes were limited to 10 deg to ensure there was no effect of dynamic stall. The focus of this study is to characterize unsteady aerodynamics based on reduced frequency, which was varied between 0.005, 0.05, and 0.5. The unsteady condition of 0.05 was compared against experiments, 2-D RANS, and 3-D hybrid RANS/large-eddy simulation formulations. It was observed that the lift characteristics were reasonably well predicted by the CFD tools when compared to experimental observations. At a reduced frequency of , the pitching motion causes an apparent stabilization of vortices resulting in higher oscillatory lift amplitude than the static value. The 3-D RANS better predicted the pitching-moment characteristics compared to the 2-D RANS, attributed to the breakdown of the leading-edge vortex. It was observed that, although thinner flat-plate airfoils have a higher maximum lift coefficient compared to the thicker NACA 0012, they also produce higher instantaneous pitching moment.

Investigation of the Tribological Performance of Heterogeneous Slip/No-Slip Journal Bearing Considering Thermo-Hydrodynamic Effects

The slip boundary has an important influence on hydrodynamic journal bearing. However, less attention has been paid to the positive effect of slip on thermal behaviour. In this study, a computational fluid dynamics (CFD) analysis investigating the thermo-hydrodynamic (THD) characteristics of heterogeneous slip/no-slip bearings running under steady, incompressible, and turbulent conditions is presented. A comprehensive analysis is made to investigate the THD behaviours of heterogeneous slip/no-slip bearings in terms of lubricant pressure, temperature distribution, volume fraction of vapor, and load-carrying capacity when they are running under different shaft rotational speeds. The multiphase cavitation model is adopted to represent the real operational condition of the journal bearing. Numerical results show that the load-carrying capacity of the heterogeneous slip/no-slip bearing can be significantly increased by up to 100% depending on the rotational speed. It is also observed that there is an optimal journal rotational speed for maximizing the load-carrying capacity. An insightful new finding is revealed in a numerical framework, wherein it is found that by introducing the heterogeneous slip/no-slip pattern, the maximum temperature can be reduced by up to 25% in comparison with a conventional bearing.

A Weakly Non-hydrostatic Shallow Model for Dry Granular Flows

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $\mu(I)$ rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope.

Estimating the moments of a random forcing field of 2D fluid from image sequences using energy minimisation method

In this paper, we consider a version of energy minimisation technique applied to images of a 2D fluid flow. The two Navier–Stokes equations describe the static flow of a 2D fluid in terms of velocity field, (u, v), pressure field, p and forcing field, f. Apart from these two Navier–Stokes equations, we have the incompressibility condition to evaluate the three parameters. While implementing this system, random noise (usually non-Gaussian) creeps into the random force field underline{f}(x,y). We denote this random field by delta underline{f}(x,y) having zero mean and non-trivial second and third moments. We assume that these two moments are known except for some unknown parameters underline{theta } (like mean, variance, co-variance, skewness, etc.) which we wish to estimate. In the proposed technique, we first calculate the approximate shift in the average fluid energy defined as a quadratic function of the velocity field. The energy method then requires that underline{theta } should be such that this average increases in the energy due to the random forcing component be minimised. We should, however, note that the standard statistical approach to force field estimation is to calculate the velocity field as a function of the force field and then adopt the statistical moment matching technique. Such an approach assumes spatial ergodicity of the velocity field. This approach to force field estimation is more accurate from the statistical moment matching view point but works only if velocity measurements are made. The former technique of energy minimisation does not require any velocity measurements. Both of these techniques are discussed in this paper and MATLAB simulations presented.

КОНЕЧНЫЕ ДЕФОРМАЦИИ ТОРОИДАЛЬНОЙ ОБОЛОЧКИ

The stress-strain state of a nonlinear elastic shell exposed to the internal pressure is considered. A surface of the shell is toroidal in shape in the initial state. The Lagrangian coordinates of the shell are assigned to a cylindrical system. The kinematic characteristics of the process are shown: a law of the motion of points, vectors of a material basis, a strain affinor and its polar decomposition, the Cauchy-Green strain measure and tensor, the Finger measure, and the “left” and the“right” Hencky strain tensors. Neglecting the shear components of the stress tensor, a constitutive relation is obtained as a quasilinear relation between true stress tensor and the Hencky corotation tensor. A system of equilibrium equations is presented in terms of physical components of the true stress tensor in the Lagrangian coordinates. Using the equilibrium equations and the incompressibility condition, a closed system of nonlinear ordinary differential equations is obtained to determine six unknown functions, depending on the angle indicating a position of the points along the cross-section in the initial state. The method of successive approximations is applied to estimate stress tensor components and to derive logarithms of the elongations of material fibers.

Calculation of rubber-metal silent-blocks under quasi-static loading

Abstract. In this paper, an algorithm for calculating rubber-metal silent blocks (hinges) under the action of a lateral quasi-static load is presented. Silent blocks of a welded type made of new brands of rubbers, which are widely used in vibration machines of various types as elastic links, are considered. A calculation is given for a very long hinge, for which the length is large compared to its outer diameter. In the calculation, it was assumed that there are no axial displacements, and the angular and radial displacements can be represented as a product of arbitrary functions of the radial coordinate and the sine and cosine of the angular coordinate, respectively. The relationship between these functions is obtained from the condition of rubber incompressibility. From the condition of the minimum total energy of the system, we have a linear inhomogeneous differential equation of the third order for one of these functions. By solving it under known boundary conditions, we obtain final expressions for the radial and angular displacement, and, consequently, for the displacement of the inner cage. With taking these expressions into account, a solution was also obtained for the hinge, the length of which cannot be considered infinite in comparison with its diameter. In this case, axial displacements should also be considered. Besides, it is assumed that the functions of the radial coordinate for the radial and angular displacement can be represented as a linear combination of the corresponding functions for the long hinge. The corresponding function for axial displacement can be found from the condition of volume constancy. The linear combination coefficients are obtained from a system of two linear algebraic equations, to which the minimum condition for the total energy of the system leads. The exact expression for the movement for the short hinge is rather cumbersome. But for the most common sizes of rubber-metal hinges, you can use a series expansion of the expression for displacement and thus get a fairly simple formula. By comparing the resulting expression with the expression for displacement of the long hinge, you can see that the formula for the infinitely long hinge can only be used if a certain condition is met that binds the dimensions of the hinge. At the end of the paper, an example of calculating a rubber-metal element ШРМ-102, which is under the action of a radial load, is given. The rubber layer in it is made of a new medium-filled rubber made of natural rubber. The obtained value of the displacement of the inner cage is in good agreement with the experimental data.

Stress distribution in a rubber-metal block under compression by static load

Abstract. The problem of determining the stress-strain state of a prismatic rubber-metal block when it is compressed by a static load is considered. A mathematical model is proposed, which assumes that when a block is compressed by a vertical load, its horizontal sections remain horizontal, the metal components of the block are rigid, non-deformable, points located on a vertical line during loading remain on a parabola, and rubber is considered incompressible. Accepted assumptions provided an opportunity to obtain expressions for movements along the axes of symmetry, which provide the expected parabolicity during compression of the rubber-metal block and satisfy the condition of incompressibility. The formulas for normal and tangential stresses arising during block compression are derived, which satisfy the differential equation of equilibrium of the elementary volume, Hooke’s law, and take into account the boundary conditions on the sides of the rubber body. Given the geometric parameters of the block and the mechanical characteristics of the rubber, it was possible to calculate its drawdown depending on the degree of static load, as well as to analyze the distribution of normal and tangential stresses that occur in the block. The developed mathematical model of the compression process allowed, as an example, to analyze the stress-strain state of a rubber-metal block of the BRM‑102 type, for the manufacture of which rubber of the brand 51‑1562 was used. As a result of solving the problem, distribution diagrams of normal, tangent, and total stresses arising in dangerous sections (on the upper or lower sides of the rubber body) under loading are constructed. It was found that the maximum normal stresses and maximum total arise on the vertical axis of symmetry of the block; maximum tangential stresses are realized at corner points. The ability to receive this type of illustration allows you to visually determine which is more rational to use: one block with a horizontal section of 200(100 mm or two blocks with a size of 100(100 mm. The proposed mathematical model allows us to study the stress-strain state of a rubber-metal block under compression, and can also be used in the design of a block with predetermined characteristics. Keywords: rubber-metal block, stress-strain state, displacements, stresses, relative deformation, mathematical model.

MATHEMATICAL MODELING OF COLD PRESSING THE SHEET COMPOSITE

Statement of the problem. The article examines the problem of cold pressing, which is the most important technological component in the production of sheet composite, which is widely studied in the repair and construction works in the interior decoration of residential and industrial premises. The solution to this problem is carried out on the basis of a physical and mathematical model under the assumption that the rheological properties of the deformable medium correspond to the principles of ideal plasticity and a flat deformable state. Within the framework of the problem, in two dimensions of quasistatic compression between absolutely rigid parallel-approaching plates of a thin ideally plastic layer, the stress-strain state of a composite medium is studied. It is believed that in the absence of volumetric loads, the condition of incompressibility of the medium and the associated flow law is fulfilled. Based on the hypothesis of the linear distribution of tangential stresses over the thickness of the deformable layer, analytical expressions for the statistical and kinematic characteristics of the deformation are obtained, and the condition at the edges of the rough plates makes it possible to determine the coefficient of slip thorns, which makes it possible to control the pressing process.Results and conclusions. It was established that the components of the strain rate are directly proportional to the plate approach speed, and the normal stresses acting in the pressing direction are independent of the loading speed, decreasing in magnitude from the center to the periphery.Keywords: yield strength, pressing, plasticity condition, mathematical model.

Numerical simulation of flashing using a pressure-based compressible multiphase approach and a thermodynamic cavitation model

A recently published pressure-based compressible multiphase flow model Labois and Narayanan (2017) is validated for applications relevant to safety of pressurized systems, such as flashing of high-pressure water through valves and nozzles. The pressure-based compressible multiphase flow solver is based on non-conservative discretization of the mixture continuity equation and has the advantage of being applicable to interface tracking and n-phase mixture formulations. For simulation of flashing, two well-known cavitation models Singhal et al.(2002), Yuan et al.(2001) have been implemented along with thermodynamic effects such as latent heat of phase change, variable saturation pressure, and heat capacities. The limitations of the well-established cavitation models in terms of their applicability to compressible flows have been clarified. The model was applied to the Super Moby Dick experiment Rousseau (1987). Good flow rates, and pressure and void fraction variations were obtained for three chosen conditions by tuning the cavitation model under incompressible conditions. It was found that the void fraction stops increasing beyond the throat due to thermodynamic effects where the latent heat and heat capacity are functions of pressure and temperature. The reduction of latent heat and rapid changes in liquid and vapour heat capacities close to the saturation line at high pressures was shown to be important. The accuracy, robustness, and efficiency of the pressure-based method has been proven for flashing under strong depressurization conditions. The cavitation model predictions are strongly sensitive to model parameters such as the nucleate density making the models not general enough to be applied to different problems without calibration. It was shown that there is significant room for development of improved cavitation models especially focussing on the constant number density constraint due to the strong sensitivity of the current models to this input parameter. The assumption of incompressibility and homogeneity of the phasic velocities is also identified as serious limitations requiring further study.

Undrained cylindrical and spherical cavity expansion in elastic–viscoplastic soils

Various undrained cavity expansion solutions for elastic–plastic soil have been proposed previously. However, no solution has been presented for elastic–viscoplastic (EVP) soil until now. This paper presents a general solution method for solving the classical one-dimensional (1D) boundary value problem (BVP) for undrained cylindrical or spherical cavity expansion in EVP soil with an emphasis on the rate effect of soil. The solution method is summarized as three standard procedures: (i) obtaining the soil displacement and strain under incompressible conditions, (ii) calculating the effective stress of soil through a suitable constitutive law, and (iii) obtaining the pore pressure by numerically solving the stress equilibrium equation through the finite difference method (FDM) or other numerical solution techniques. The numerical algorithms for calculating the effective stress and pore pressure are very simple, without any complex iteration processes, and they require little calculation time but provide high computational accuracy. In addition, some numerical results are given to investigate the influence of the cavity expansion velocity on the cavity expansion response. The proposed solution procedure is general and can be applied not only for the EVP model but also for other plasticity models, and the given EVP solution can be applied to interpret the rate effect of the cone penetration test and pile penetration.

Vortex-ring-induced elliptical drop deformation process in ambient liquid under an impact: The effects of the drop shape and other parameters

In this paper, the elliptical drop deformation process in ambient liquid under an impact was systematically studied based on an experimental method combined with computational fluid dynamics (CFD). The effects of nondimensional parameters (i.e., shape parameter e, viscosity ratio η, and density ratio ε) on the vortex dynamics during the drop deformation process in ambient liquids were systematically investigated. The severity of the elliptical drop deformation process is apparently promoted by e, i.e., with larger e, the vortex dynamics during the drop deformation process is strengthened, and a more severe deformation process occurs. Under a relatively low η, the vortex dynamics can be further strengthened by increasing e, while the vortex dynamics shows an opposite trend by decreasing e under a relatively high η. With increasing ε, the vortex dynamics is apparently strengthened. Under the incompressible condition, the vortex dynamics is approximately unaffected by increasing e.

Effect of Rotor Spacing and Duct Diffusion Angle on the Aerodynamic Performances of a Counter-Rotating Ducted Fan in Hover Mode

The aerodynamic performance of a counter-rotating ducted fan in hover mode is numerically analyzed for different rotor spacings and duct diffusion angles. The design of the counter-rotating fan is inspired by a custom-designed single rotor ducted fan used in a previous study. The numerical model to predict the aerodynamic performance of the counter-rotating ducted fan is developed by adopting the frozen rotor approach for steady-state incompressible flow conditions. The relative angle between the front and the rear rotor is examined due to the usage of the frozen rotor model. The results show that the variation of thrust for the different relative angles is extremely low. The aerodynamic performances are evaluated by comparing the thrust, thrust coefficient, power coefficient, and figure of merit (FOM). The thrust, thrust coefficient, and FOM slightly increase with increasing rotor spacing up to 200 mm, regardless of the duct diffusion angle, and reduce on further increase in the rotor spacing. The duct diffusion angle of 0° generates about 9% higher thrust and increases the FOM by 6.7%, compared with the 6° duct diffusion angle. The duct diffusion angle is highly effective in improving the thrust and FOM of the counter-rotating ducted fan, rather than the rotor spacing.

Lattice-Boltzmann simulation of creeping generalized Newtonian flows: Theory and guidelines

The accuracy of the lattice-Boltzmann (LB) method is related to the relaxation time controlling the flow viscosity. In particular, it is often recommended to avoid large fluid viscosities in order to satisfy the low-Knudsen-number assumption that is essential to recover hydrodynamic behavior at the macroscopic scale, which may in principle limit the possibility of simulating creeping flows and non-Newtonian flows involving important viscosity variations. Here it is shown, based on the continuous Boltzmann equations, that a two-relaxation-time (TRT) model can however recover the steady Navier-Stokes equations without any restriction on the fluid viscosity, provided that the Knudsen number is redefined as a function of both relaxation times. This effective Knudsen number is closely related to the previously-described parameter controlling numerical errors of the TRT model, providing a consistent theory at both the discrete and continuous levels. To simulate incompressible flows, the viscous incompressibility condition Ma2/Re≪1 also needs to be satisfied, where Ma and Re are the Mach and Reynolds numbers. This concept is extended by defining a local incompressibility factor, allowing one to locally control the accuracy of the simulation for flows involving varying viscosities. These theoretical arguments are illustrated based on numerical simulations of the two-dimensional flow past a square cylinder. In the case of a Newtonian flow, the viscosity independence is confirmed for relaxation times up to 104, and the ratio Ma2/Re=0.1 is small enough to ensure reliable incompressible simulations. The Herschel-Bulkley model is employed to introduce shear-dependent viscosities in the flow. The proposed numerical strategy allows to achieve major viscosity variations, avoiding the implementation of artificial viscosity cut-off in high-viscosity regions. Highly non-linear flows are simulated over ranges of the Bingham number Bn∈[1,1000] and flow index n∈[0.2,1.8], and successfully compared to prior numerical works based on Navier-Stokes solvers. This work provides a general framework to simulate complex creeping flows, as encountered in many biological and industrial systems, using the lattice-Boltzmann method.

Upper Bound Shakedown Analysis of Plates Utilizing the C$$^{1}$$ Natural Element Method

This paper proposes a numerical solution method for upper bound shakedown analysis of perfectly elasto-plastic thin plates by employing the C $$^{1}$$ natural element method. Based on the Koiter’s theorem and von Mises yield criterion, the nonlinear mathematical programming formulation for upper bound shakedown analysis of thin plates is established. In this formulation, the trail function of residual displacement increment is approximated by using the C $$^{1}$$ shape functions, the plastic incompressibility condition is satisfied by introducing a constant matrix in the objective function, and the time integration is resolved by using the Konig’s technique. Meanwhile, the objective function is linearized by distinguishing the non-plastic integral points from the plastic integral points and revising the objective function and associated equality constraints at each iteration. Finally, the upper bound shakedown load multipliers of thin plates are obtained by direct iterative and monotone convergence processes. Several benchmark examples verify the good precision and fast convergence of this proposed method.

Effect of fiber exclusion in uniaxial tensile tests of soft biological tissues

The effect of the exclusion of the compressed fibers in the identification of material parameters from uniaxial tensile tests on two orthogonal strips is investigated. The micro-structurally based constitutive model with two dispersion parameters developed by Holzapfel and his colleagues is utilized in the study. A new exclusion method, based on the coefficient reflecting the percentage of stretched fibers, is proposed. The material parameters are identified by using experimental data from 30 uniaxial tensile tests (5 donors, 6 strips per donor) and a genetic algorithm code that is capable to find the optimal set of parameters. The contraction of the strip width computed by using the hyperelastic model with the identified material parameters is compared to the experimental data for two human aortas (one from literature and one experiment, specific for this study), in order to show the accuracy of the identified model. The complex behavior of the thickness deformation of the strip is also obtained and compared to the experimental data derived from in-plane measurements and the incompressibility condition. Results show that the in-plane fiber exclusion is appropriate for aortic material characterization with uniaxial tensile tests, reducing very significantly the computational cost. At the same time, thickness growth of strips during uniaxial tests is possible, depending on fiber dispersion and orientation.

Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

AbstractIn this article, we propose to investigate numerically the steady bifurcation points and bifurcated branches in fluid mechanics by employing high‐order mesh‐free geometric progression algorithms. These algorithms are based on the use of the geometric progression (GP) in a high‐order mesh‐free approach. The first proposed algorithm is applied on a strong formulation using the moving least squares (MLS) approximation coupled with GP (HO‐MLS‐GPM). While the second proposed algorithm is applied on a weak formulation using the element‐free Galerkin (EFG) coupled also with GP (HO‐EFG‐GPM). The incompressibility condition is taken by introducing the penalty technique to transform the stationary Navier–Stokes equations verified by the pressure and velocity into ones verified by only the velocity. The high‐order mesh‐free algorithm permits to transform this nonlinear equations into a succession of linear ones. The GP allows to detect with precision the bifurcation points and the Lyapunov–Schmidt reduction is coupled with HO‐MLS‐GPM and HO‐EFG‐GPM as a continuation procedure to follow the many bifurcated branches. The aim of this resolution strategy concerns the treatment of the bifurcation phenomena for a fluid flow through an expansion in several geometries, where the steady flow becomes unstable after a critical Reynolds value. The obtained results are compared with those presented in literature and with those computed using the high‐order finite element algorithm coupled with GP.

A five-field augmented fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem

In this work we introduce and analyze a new augmented fully-mixed formulation for the stationary Navier–Stokes/Darcy coupled problem. Our approach employs, on the free-fluid region, a technique previously applied to the stationary Navier–Stokes equations, which consists of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, together with the pressure. In addition, by using the incompressibility condition, the pressure is eliminated, and since the convective term forces the free-fluid velocity to live in a smaller space than usual, we augment the resulting formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations. On the other hand, in the Darcy region we apply the usual dual-mixed formulation, which yields the introduction of the trace of the porous media pressure as an associated Lagrange multiplier. The latter is connected with the fact that one of the transmission conditions involving mass conservation becomes essential and must be imposed weakly. In this way, we obtain a five-field formulation where the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium, and the aforementioned Lagrange multiplier, are the corresponding unknowns. The well-posedness analysis is carried out by combining the classical Babuška–Brezzi theory and the Banach fixed-point theorem. A proper adaptation of the arguments exploited in the continuous analysis allows us to state suitable hypotheses on the finite element subspaces ensuring that the associated Galerkin scheme is well-posed and convergent. In particular, Raviart–Thomas elements of lowest order for the pseudostress and the Darcy velocity, continuous piecewise linear polynomials for the free-fluid velocity, piecewise constants for the Darcy pressure, together with continuous piecewise linear elements for the Lagrange multiplier, constitute feasible choices. Finally, we provide several numerical results illustrating the performance of the Galerkin method and confirming the theoretical rates of convergence.

Density Functional Approach to the Molecular Theory of Rod-Coil Diblock Copolymers

The general density functional approach is used to develop a molecular-statistical theory of rod-coil diblock copolymers which is valid in the case of both weak and strong segregation. The free energy of the rod-coil copolymer is expressed as a functional of the number densities of rod and coil monomers which depend on the translational and orientational degrees of freedom. The equilibrium densities are determined by minimization of the free energy functional and depend on the orientational and translational order parameters of the monomers. The order parameters are calculated numerically by minimization of the free energy taking into account the incompressibility condition within the formalism of Lagrange multipliers. Phase diagrams are obtained and the profiles of orientational and translational order parameters are presented as functions of temperature and the fraction of coil fragments. It is shown that the lamellar phase possesses strong orientational order and the stability of the phase is increasing with the increasing fraction of rod monomers.