On MHD flow of an incompressible viscous fluid

In this paper, we apply Homotopy Perturbation Method (HPM) to find the analytical solutions of nonlinear MHD flow of an incompressible viscous fluid through convergent or divergent channels in presence of a high magnetic field. The flow of an incompressible electrically conducting viscous fluid in convergent or divergent channels under the influence of an externally applied homogeneous magnetic field is studied both analytically and numerically. The graphs are presented to reveal the physical characteristics of flow by changing angles of the channel, Hartmann and Reynolds numbers.     

Numerical solution of laminar incompressible generalized Newtonian fluids flow

This paper deals with the numerical solution of laminar viscous incompressible flows for generalized Newtonian fluids in the branching channel. The generalized Newtonian fluids contain Newtonian fluids, shear thickening and shear thinning non-Newtonian fluids. The mathematical model is the generalized system of Navier-Stokes equations. The finite volume method combined with an artificial compressibility method is used for spatial discretization. For time discretization the explicit multistage Runge-Kutta numerical scheme is considered. Steady state solution is achieved for t → ∞ using steady boundary conditions and followed by steady residual behavior. For unsteady solution a dual-time stepping method is considered. Numerical results for flows in two dimensional and three dimensional branching channel are presented.     

Numerical modelling of linear and nonlinear diffusion equations by compact finite difference method

In this work, accurate solutions to linear and nonlinear diffusion equations were introduced. A combination of a sixth-order compact finite difference scheme in space and a low-storage third-order total variation diminishing Runge-Kutta scheme in time have been used for treatment of these equations. The computed results with the use of this technique have been compared with the exact solution to show the accuracy of it. Here, the approximate solution to the diffusion equations has been obtained easily and elegantly with neither transforming nor linearizing the equation. The present method is seen to be a very good alternative method to some existing techniques for realistic problems.     

A finite element model for nonlinear behaviour of piezoceramics under weak electric fields

It has been experimentally observed that piezoceramic materials exhibit different types of nonlinearities under different combinations of electric and mechanical fields. When excited near resonance in the presence of weak e to a Duffinor such as jump phenomena and presence of superharmonics in the response spectra. There has not been much work in the litrature to model these types of nonlinearities. Some authors have developed one-dimensional models for the above phenomenon and derived closed-form solutions for the displacement response of piezo-actuators. However, the generalized three-dimensional (3-D) formulation of electric enthalpy, the variational formulation and the FEM implementation have not yet been addressed, which are the focus of this paper. In this work, these nonlinearities have been modelled in a 3-D piezoelectric continuum using higher order quadratic and cubic terms in the generalized electric enthalpy density function. The coupled nonlinear finite element equations have been derived using variational formulation. A special linearization technique for assembling the nonlinear matrices and solution of the resulting nonlinear equations has been developed. The method has been used for simulating the nonlinear frequency response of a lead zirconate titanate plate excited near its first in-plane vibration resonance frequency with sinusoidal excitations of different electric field strengths. The results have been compared with those of the experiment.     

Optimal control of non-isothermal viscous fluid flow

For flow inside a four-to-one contraction domain, we minimize the vortex that occurs in the corner region by controlling the heat flux along the corner boundary. The problem of matching a desired temperature along the outflow boundary is also considered. The energy equation is coupled with the mass, momentum, and constitutive equations through the assumption that viscosity depends on temperature. The latter three equations are a non-isothermal version of the three-field Stokes–Oldroyd model, formulated to have the same dependent variable set as the equations governing viscoelastic flow. The state and adjoint equations are solved using the finite element method. Previous efforts in optimal control of fluid flows assume a temperature-dependent Newtonian viscosity when describing the model equations, but make the simplifying assumption of a constant Newtonian viscosity when carrying out computations. This assumption is not made in the current work.     

An effective computer modelling approach to the study of aeroelastic characteristics of an aircraft composite wing with high aspect ratio

Static aeroelastic and flutter characteristics of an aircraft composite wing with high aspect ratio were analysed by an effective Computational Fluid Dynamics and Computational Structure Dynamics coupled method. Effects of stiffness distribution on aeroelastic characteristics were considered. Honeycomb core sandwich composite was considered to be equivalent to an orthotropic material by stiffness and inertance equivalent method to allow highly efficient numerical simulation, which was used for analysis of bending and torsional stiffness distribution. The results showed that the redistributed aerodynamic load leads to a decrease of pressure difference between the upper and lower airfoils. The flutter speed of the composite wing is near 0.64 Ma. Both bending and torsional stiffness increases with a small increase of beam size. Stiffness of the wing root has a major influence generally on the static aeroelastic characteristics. Both the lift coefficient and the loss percent decrease with a small increase of beam size. Effects of stiffness distribution on frequency are not obvious. Flutter speed remains close to the initial value when the beam size is changed.     

Steady magnetohydrodynamic flow of an incompressible viscous fluid involving an ignorable co-ordinate

Résumé On considère l'écoulement visqueux régulier et incompressible ainsi que la distribution de température d'un fluide conducteur d'électricité contenu dans un conduit. En utilisant une forme spéciale du potentiel vectoriel magnétique et en changeant les variables des coordonnées, on démontre la possibilité de réduire le problème de l'écoulement à la résolution de l'équation de Poisson et de deux équations indépendantes linéaires différentielles et partielles du deuxième ordre. Les résultats obtenus sont employés pour déterminer l'écoulement et la distribution de la température d'une forme peu commune de l'écoulement de Couette.     

On the accuracy of the viscous form in simulations of incompressible flow problems

We present a numerical study of drag/lift and flux estimates using two forms of Navier‐Stokes equations (NSE) that are equivalent in the continuum formulation but not in the discrete finite element formulation. The two investigated forms of the NSE differ in the viscous term that is represented in one form by νΔ u with ν being the viscosity and 2ν?·?^S u in the other form where ?^S represents the deformation tensor. The study consists of numerical analysis of the two forms and computations of drag/lift, pressure drop on the cylinder problem and computations of flux for the Poiseuille flow. The main objective is to provide a clear comparison of the reference values for the maximal drag and lift coefficient at the cylinder and for the pressure difference between the front and the back of the cylinder at the final time for the two forms of NSEs. Our computational results of the reference values do not differ significantly between the two forms, but the differences are there. For the Poiseuille flow, the differences in the flux computations were much smaller, and this agreed with the computationally obtained results of the divergence of the velocity field. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 523–541, 2012     

High-order approximations for an incompressible viscous flow on a rough boundary

In this paper, we propose approximations of fluid flow that could be used for obtaining wall laws of higher order. We consider the two-dimensional laminar fluid flow, modeled by the incompressible Stokes system in a straight channel with a rough side. The roughness is periodic and the ratio of the amplitude of the rough part and the size of the flow domain is denoted by ?, being a small number. We impose periodic boundary conditions on the flow. We generalize the boundary layers needed for the construction of flow approximations of higher order with respect to ?. The existence of the layers and their features are discussed. Finally we give the error estimates for the approximations and establish an explicit wall law.     

Short-time dynamics of an elliptic cylinder moving in a viscous incompressible free-surface flow

The joint motion of a viscous incompressible fluid and a completely submerged elliptic cylinder is analyzed at short times. The cylinder is assumed to start from rest and move horizontally at a constant acceleration. A feature of the problem is that, at high accelerations, the fluid becomes detached from the cylinder surface and a cavity is formed. The problem is generalized to an elliptic cylinder floating on the surface of a viscous fluid.     

Homogenization of an incompressible viscous flow in a porous medium with double porosity

We study the homogenization of an incompressible viscous flow in a porous medium with double porosity. We derive a macroscopic model with local Navier–Stokes system in the large cavities, Darcy law in the thinner porous rock, and a contact law between the two. We use Γ-convergence methods associated with multi-scale convergence notions in order to get this limit law. We exhibit a critical ratio between the two scales of the pores.     

Numerical simulation of the sedimentation of a tripole-like body in an incompressible viscous fluid

In this note, we discuss the application of a methodology combining distributed Lagrange multiplier based fictitious domain techniques, finite-element approximations and operator splitting, to the numerical simulation of the motion of a tripole-like rigid body falling in a Newtonian incompressible viscous fluid. The motion of the body is driven by the hydrodynamical forces and gravity. The numerical simulation shows that the distribution of mass of this rigid body and added moment of inertia compared to a simple cylinder (circular or elliptic) plays a significant role on the particle-fluid interaction. Apparently, for the parameters examined, the action of the moving rigid body on the fluid is stronger than the hydrodynamic forces acting on the rigid body.     

MIXED FINITE ELEMENT METHODS FOR A STRONGLY NONLINEAR PARABOLIC PROBLEM

1.IntroductionFOrsecondorderellipticproblems,themixedmethodwasdescribedandanalyzedbymanyauthors[1--3]inthecaseoflinearequationsindivergenceform,aswellasin[4,5]forquasilinearornonlinearproblemsindivergenceform.JohnsonandThorn6e[6]consideredalternativeproofSofthepreviouslyknownerrorestimatesforsuchmethodsintheellipticcase.Theyalsoanalyzedthemixedfiniteelementmethodforthepaxabolicequationgivenbypt--ho~f.Garcia[7]studiedtheconvergenceofmixedfiniteelementapproximationstoquasilinearparabolicequati…     

On non‐stationary viscous incompressible flow through a cascade of profiles

The paper deals with theoretical analysis of non‐stationary incompressible flow through a cascade of profiles. The initial‐boundary value problem for the Navier–Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd.     

Geometric depletion of vortex stretch in 3D viscous incompressible flow

New geometric constraints on vorticity are obtained which suppress possible development of finite-time singularity from the nonlinear vortex stretching mechanism. We find a new condition on the smoothness of the direction of vorticity in the vortical region which yields regularity. We also detect a regularity condition of isotropy type on vorticity in the intensive vorticity region via a new cancellation principle. This is in contrast with the one of isotropy type on the curl of vorticity obtained recently by A. Ruzmaikina and Z. Gruji? [A. Ruzmaikina, Z. Gruji?, On depletion of the vortex-stretching term in the 3D Navier-Stokes equations, Comm. Math. Phys. 247 (2004) 601-611]. We improve as well all of their results by eliminating their assumption that the initial vorticity ω₀ is required to be in L¹.     

Formulation of boundary conditions for vorticity in viscous incompressible flow problems

For the stream function-vorticity formulation of the Navier-Stokes equations, vorticity boundary conditions are required on the body surface and the far-field boundary. A two-parameter approximating formula is derived that relates the velocity and vorticity on the outer boundary of the computational domain. The formula is used to construct an algorithm for correcting the conventional far-field boundary conditions. Specifically, a soft boundary condition is set for the vorticity and a uniform flux is specified for the transversal velocity. A third-order accurate three-parameter formula for the vorticity on the wall is derived. The use of the formula does not degrade the convergence of the iterative process of finding the vorticity as compared with a previously derived and tested two-parameter formula.     

Existence and numerical modelling of vortex rings with elliptic boundaries

We revisit in this paper the theory of axisymmetric vortex rings in an ideal fluid. The boundary separating the vortex ring from the external (potential) flow is assumed of elliptic shape. For a given distribution of vorticity in the vortex core, we theoretically put into evidence the critical parameter for the existence of non-trivial solutions, thus confirming the numerical observation of Durst et al. [ZAMP 32 (1981) 156]. A sharp estimation of the critical threshold is analytically derived. Theoretical predictions are confirmed by numerical simulations using finite elements. A new numerical algorithm is presented and shown to display better performances compared to previous published algorithms using finite differences. The convergence of the iterative algorithm is proved using the theory of elliptic partial differential equations with discontinuous nonlinearities.