Stokes-like Problem Research Articles

The nonconforming virtual element method for Oseen’s equation using a stream-function formulation

We approximate the solution of the stream function formulation of the Oseen equations on general domains by designing a nonconforming Morley-type virtual element method. Under a suitable assumption on the continuous problem’s coefficients, the discrete scheme is well-posed. By introducing an enriching operator, we derive an a priori estimate of the error in a discrete H2 norm. By post-processing the discrete stream function, we compute the discrete velocity and vorticity fields. Furthermore, we recover an approximate pressure field by solving a Stokes-like problem in a nonconforming Crouzeix–Raviart-type virtual element space that is in a Stokes-complex relation with the Morley-type space of the virtual element approximation. Finally, we confirm our theoretical estimates by solving benchmark problems that include a convex and a nonconvex domain.

A numerical approximation with WLS/SUPG algorithm for solving White–Metzner viscoelastic flows

In this paper, we consider a numerical method for the White–Metzner model of viscoelastic fluid flows by a combination of the weighted least-squares (WLS) method and the streamline upwind/Petrov–Galerkin (SUPG) method. The constitutive equation is decoupled from the momentum and continuity equations, and the approximate solution is computed iteratively by solving the Stokes-like problem and a linearized constitutive equation using WLS and SUPG, respectively. The elastic viscous split stress formulation (EVSS) introduced in [27] is used for the discretization of the constitutive equation. An a priori error estimate for the WLS/SUPG method is derived and numerical results supporting the estimate are presented. We encounter no limit of the Weissenberg number for all values of ε satisfying 0<ε<1, a dimensionless material parameter, in our calculations.

Curvature-induced instability of a Stokes-like problem with non-standard boundary conditions

We present an analysis of a set of parametrized boundary conditions for a Stokes–Brinkman model in two space dimensions, discretized by finite elements. We particularly point out an instability which arises when these boundary conditions are posed on a curved line, which then leads to unphysical oscillations. In contrast to a Navier-slip condition, which is prone to Babuška's paradox, this instability can be traced back to the continuous level. We claim that the stability in these cases depend on the amount of curvature at the boundary, which is shown in a reduced setting in cylinder coordinates. The transition to a two dimensional Cartesian case is then based on numerical studies, which further substantiate the claim. Lastly, stabilization techniques are motivated that enhance the continuous FEM setting and are conveniently able to deal with arising oscillations.

Newton’s algorithm for magnetohydrodynamic equations with the initial guess from Stokes-like problem

The magnetohydrodynamic equations are second order nonlinear partial differential equations which are coupled by fluid velocity and magnetic fields and we consider to apply the Newton’s algorithm to solve them. It is well known that the choice of a proper initial guess is critical to assure the convergence of Newton’s iterations in solving nonlinear partial differential equations. In this paper, we provide a good initial guess for Newton’s algorithm when it is applied for solving magnetohydrodynamic equations.

A Lagrange-Galerkin hp-Finite Element Method for a 3D Nonhydrostatic Ocean Model

We introduce in this paper a Lagrange-Galerkin hp-finite element method to calculate the numerical solution of a nonhydrostatic ocean model. The Lagrange-Galerkin method yields a Stokes-like problem the solution of which is computed by a second-order rotational splitting scheme that separates the calculation of the velocity and pressure, the latter is decomposed into hydrostatic and nonhydrostatic components. We have tested the method in flows where the nonhydrostatic effects are important. The results are very encouraging.

Decoupled algorithm for solving Phan-Thien–Tanner viscoelastic fluid by finite element method

In this paper we present a decoupled algorithm for the viscoelastic fluid flow obeying the Phan-Thien–Tanner (PTT) differential model. The method consists to solve alternatively a Stokes-like problem by weighted least-squares (WLS) finite element method, and the constitutive equation by streamline upwind Petrov–Galerkin (SUPG) method. A priori error estimate for the WLS/SUPG finite element method is derived. The existence and uniqueness of the approximate solution are obtained using fixed point techniques. We show that the mapping of the decoupled algorithm is locally contracting. The planar channel flow problem is used to illustrate our theoretical results. We encounter no limit of the Weissenberg number for reasonable values of ε, a dimensionless material parameter, in our calculations.

Fully Discrete Finite Element Approximations of a Polymer Gel Model

This paper proposes and analyzes some fully discrete (multiphysics) finite element methods for a displacement-pressure model which describes swelling dynamics of polymer gels under mechanical constraints. By introducing an “elastic pressure” we first present a reformulation of the original model. We then propose a time-stepping scheme which decouples the PDE system at each time step into two subproblems, one of which is a Stokes-like problem for the displacement vector field (of the solid network of the gel) and the other is a diffusion problem for a “pseudopressure” field (of the solvent of the gel). To make such a multiphysics approach feasible, it is vital to find admissible constraints to resolve the uniqueness issue for the Stokes-like problem and to construct a good boundary condition for the diffusion equation so that it also becomes uniquely solvable. The key to the first difficulty is to discover certain conserved quantities for the PDE solution, and the solution to the second difficulty is to use the Stokes-like problem to generate a boundary condition for the diffusion problem. The time-stepping scheme allows one to use any convergent Stokes solver (and its code) together with any convergent diffusion equation solver (and its code) to solve the polymer gel model. In the paper, the Taylor–Hood mixed finite element method combined with the continuous linear finite element method are chosen as an example to present the ideas and to demonstrate the viability of the proposed multiphysics approach. It is proved that, under a mesh constraint, both the semidiscrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for both semidiscrete and fully discrete methods. Numerical experiments are also presented to show the performance of the proposed approach and methods.

An Algorithm for Solving Reactive Advection-Dispersion Problems

The physical problem of advection, dispersion, and reaction of a suite of chemicals is formulated mathematically as a set of coupled time-dependent nonlinear partial differential equations. Partially based on the Gear scheme for time discretization, the algorithm developed in this paper decouples equations into separate linearized problems. Then, using finite element approximation, these resulting linear advection-dispersion equations are further transformed, via a re-assembling of variational formulation, into an elliptic Stokes-like problem, which can be solved by an existing Stokes solver previously employed when solving the flow problem to produce a velocity background. The methodology is implemented in the simulation of transport and transformations of an electron donor, an electron acceptor, and active biomass. This application provides insights into biofilm evolution and pore clogging, and demonstrates mesh self-adjustment in the process of biofilm growth.

A Level-Set Method for Computing Solutions to Viscoelastic Two-Phase Flow

A finite-element code based on the level-set method is developed for simulating the motion of viscoelastic two-phase flow problems. This method is a generalization of the finite-difference approach described in [1–4] for computing solutions to two-phase problems of inviscid and viscous fluids. The Marchuk–Yanenko operator-splitting technique is used to decouple the difficulties associated with the nonlinear convection term, the incompressibility constraint, the viscoelastic term, and the interface motion problem. The nonlinear convection problem is solved using a least-squares conjugate gradient algorithm, and the Stokes-like problem is solved using a conjugate gradient algorithm. The constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor, while the convection term in the constitutive equation is discretized using a third-order upwinding scheme. The code is verified by performing a convergence study to show that the results are independent of the mesh and time-step sizes. Using our code we have studied the deformation of drops in simple shear and pressure-driven flows and of bubbles in gravity-driven flows over a wide range of dimensionless capillary (Ca) and Deborah numbers (De). For a Newtonian bubble rising in a quiescent viscoelastic liquid we find that there are limiting values of the parameters De and Ca, above which the bubble assumes a characteristic shape with a cusp-like trailing edge. The front of the bubble, however, remains round, as the local viscoelastic and viscous stresses act to round the bubble. In a pressure-driven flow the drop is stretched so that its front, which is closer to the channel center, remains round, and the trailing edge, which is closer to the channel wall, becomes sharp. These numerical results are in agreement with the experimental observations.

Highly parallel time integration of viscoelastic flows

A highly parallel time integration method is presented for calculating viscoelastic flows with the DEVSS-G/DG finite element discretization. The method is a synthesis of an operator splitting time integration method that decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. Both steps are performed in parallel. The discontinuous finite element discretization of the hyperbolic constitutive equation leads to highly-parallel element-by-element calculation of the stress at each time step. The Stokes-like problem is solved by using the BiCGStab Krylov iterative method implemented with the block complement and additive levels method (BCALM) preconditioner. The solution method is demonstrated for the calculation of two-dimensional (2D) flow of an Oldroyd-B fluid around an isolated cylinder confined between two parallel plates. These calculations use extremely fine finite elements and expose new features of the solution structure.

A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows

A distributed Lagrange multiplier/fictitious domain method (DLM) is developed for simulating the motion of rigid particles suspended in the Oldroyd-B fluid. This method is a generalization of the one described in [R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, Int. J. Multiphase Flows 25 (1998) 755–794] where the motion of particles suspended in a Newtonian fluid was simulated. In our implementation of the DLM method, the fluid–particle system is treated implicitly using a combined weak formulation in which the forces and moments between the particles and fluid cancel. The governing equations for the Oldroyd-B liquid are solved everywhere, including inside the particles. The flow inside the particles is forced to be a rigid body motion by a distribution of Lagrange multipliers. We use the Marchuk–Yanenko operator-splitting technique to decouple the difficulties associated with the incompressibility constraint, nonlinear convection and viscoelastic terms. The constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor, while the convection term in the constitutive equation is discretized using a third-order upwinding scheme. The nonlinear convection problem is solved using a least square conjugate gradient algorithm, and the Stokes-like problem is solved using a conjugate gradient algorithm. The code is verified performing a convergence study to show that the results are independent of the mesh and time step sizes. Our simulations show that, when particles are dropped in a channel, and the viscoelastic Mach number ( M) is less than 1 and the elasticity number ( E) is greater than 1, the particles chain along the flow direction; this agrees with the results presented in [P.Y. Huang, H.H. Hu, D.D. Joseph, J. Fluid Mech. 362 (1998) 297–325]. In our simulations of the fluidization of 102 particles in a two-dimensional bed, we find that the particles near the channel walls form chains that are parallel to the walls, but the distribution of particles away from the walls is relatively random.

A Mixed Spectral/Wavelet Method for the Solution of the Stokes Problem

The paper presents a mixed wavelet/spectral Chebychev method for solving the unsteady 2D Stokes equations in the vorticity-stream function formulation with periodicity condition in one direction. After an appropriate time discretisation of the equations, one has to solve at each time step a stationary Stokes-like problem. A capacitance matrix method is used to eliminate the problem of boundary conditions. This leads to solving a series of Helmholtz problems. The spatial discretisation makes use of the wavelet method in the periodic direction and the spectral collocation Chebychev method in the non-periodic direction. The resolution of the discrete Helmholtz problem is done by means of the diagonalisation technique in the non-periodic direction. The system then splits into a sequence of one dimensionnal periodic Helmholtz problems which are efficiently inverted using FFTs. Numerical tests show both the stability and the accuracy of the method.