Starting solutions for some unsteady unidirectional flows of Oldroyd-B fluids

Analytical expressions for the velocity fields corresponding to the motions of an Oldroyd-B fluid due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient and pressure jumps are determined by means of the Fourier sine transform. The corresponding solutions for a Maxwell and a Newtonian fluid appear as limiting cases of the solutions established here. Relevant physical properties of the flows and their dependence on material and geometry parameters are discussed.     

On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid

This paper deals with exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. The fractional calculus approach is used in the constitutive relationship of fluid model. Analytical expressions for the velocity field and the corresponding shear stress for flows due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined using Fourier sine and Laplace transforms. The obtained solutions are presented under integral and series forms in terms of the Mittag–Leffler functions. For α = β = 1, our solutions tend to the similar solutions for ordinary Oldroyd-B fluid. A comparison between generalized and ordinary Oldroyd-B fluids is shown by means of graphical illustrations.     

On exact solutions for some oscillating motions of a generalized Oldroyd-B fluid

This paper deals with exact solutions for some oscillating motions of a generalized Oldroyd-B fluid. The fractional calculus approach is used in the constitutive relationship of fluid model. Analytical expressions for the velocity field and the corresponding shear stress for flows due to oscillations of an infinite flat plate as well as those induced by an oscillating pressure gradient are determined using Fourier sine and Laplace transforms. The obtained solutions are presented under integral and series forms in terms of the Mittag–Leffler functions. For α = β = 1, our solutions tend to the similar solutions for ordinary Oldroyd-B fluid. A comparison between generalized and ordinary Oldroyd-B fluids is shown by means of graphical illustrations.     

Mathematical modelling and study of the consolidation of an elastic saturated soil with an incompressible fluid by FEM

The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the stability of the method as well as the estimation of the error by using semi-discretization in time. Finally, we then solved this one by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data.     

Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1112–1119, August, 2005.     

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.     

Flow due to a plate that applies an accelerated shear to a second grade fluid between two parallel walls perpendicular to the plate

The velocity field and the shear stresses corresponding to the motion of a second grade fluid between two side walls, induced by an infinite plate that applies an accelerated shear stress to the fluid, are determined by means of the integral transforms. The obtained solutions, presented under integral form in term of the solutions corresponding to the flow due to a constant shear on the boundary, satisfy all imposed initial and boundary conditions. In the absence of the side walls, they reduce to the similar solutions over an infinite plate. The Newtonian solutions are obtained as limiting cases of the general solutions. The influence of the side walls on the fluid motion as well as a comparison between the two models is shown by graphical illustrations.     

New exact solutions for magnetohydrodynamic flows of an Oldroyd-B fluid

This paper presents the new exact analytical solutions for magnetohydrodynamic (MHD) flows of an Oldroyd-B fluid. The explicit expressions for the velocity field and the associated tangential stress are established by using the Laplace transform method. Three characteristic examples: (i) flow due to impulsive motion of plate, (ii) flow due to uniformly accelerated plate, and (iii) flow due to non-uniformly accelerated plate are considered. The solutions for the hydrodynamic flows are special cases of the presented solutions. Moreover, the similar solutions corresponding to Maxwell and Newtonian fluids in the presence as well as absence of a magnetic field appear as the limiting cases of our solutions. The influences of the exerted magnetic field on the flow are also graphically presented and discussed. In particular, graphical results for the Oldroyd-B fluid are compared with those of a Newtonian fluid.     

Exact solutions for the flow of an Oldroyd-B fluid due to an infinite flat plate

The unsteady flow of an Oldroyd-B fluid due to an infinite flat plate, subject to a translation motion of linear time-dependent velocity in its plane, is studied by means of the Laplace transform. The velocity field and the associated tangential stress corresponding to the flow induced by the constantly accelerating plate as well as those produced by the impulsive motion of the plate are obtained as special cases. The solutions that have been determined, in all accordance with the solutions established using the Fourier transform, reduce to those for a Newtonian fluid as a limiting case. The similar solutions for a Maxwell fluid are also obtained.     

Remarks on the minimizing geodesic problem in inviscid incompressible fluid mechanics

We consider L2 minimizing geodesics along the group of volume preserving maps SDiff(D) of a given 3-dimensional domain D. The corresponding curves describe the motion of an ideal incompressible fluid inside D and are (formally) solutions of the Euler equations. It is known that there is a unique possible pressure gradient for these curves whenever their end points are fixed. In addition, this pressure field has a limited but unconditional (internal) regularity. The present paper completes these results by showing: (1) the uniqueness property can be viewed as an infinite dimensional phenomenon (related to the possibility of relaxing the corresponding minimization problem by convex optimization), which is false for finite dimensional configuration spaces such as O(3) for the motion of rigid bodies; (2) the unconditional partial regularity is necessarily limited.     

A note on the flow induced by a constantly accelerating plate in an Oldroyd-B fluid

The velocity field and the adequate tangential stress that is induced by the flow due to a constantly accelerating plate in an Oldroyd-B fluid, are determined by means of Fourier sine transforms. The solutions corresponding to a Maxwell, Second grade and Navier–Stokes fluid appear as limiting cases of the solutions obtained here. However, in marked contrast to the solution for a Navier–Stokes fluid, in the case of an Oldroyd-B fluid oscillations are set up which decay exponentially with time.     

The Rayleigh–Stokes problem for an edge in an Oldroyd-B fluid

The velocity fields corresponding to an incompressible fluid of Oldroyd-B type subject to a linear flow within an infinite edge are determined for all values of the relaxation and retardation times. The well known solution for a Navier–Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appears as a limiting case of our solutions. To cite this article: C. Fetecau, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 979–984.     

Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid

We prove an existence result of weak solutions for an interaction problem between an elastic structure and a compressible fluid in three space dimensions. Solutions are defined as long as there is no collision and as long as conditions of non-interpenetration and of preservation of orientation are satisfied by the displacement field of the structure. To cite this article: M. Boulakia, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Non-linear analysis of the consolidation of an elastic saturated soil with incompressible fluid and variable permeability by FEM

Research interest in the mechanical behaviour of soils is growing as a result of an increasing number of geomechanical problems involving consolidation effects. The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid and variable permeability. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the convergence of the method using a technique based on the proof of solution’s existence. Finally, we then solved this constitutive model by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data. Therefore, the model can be used for quantitative predictions of consolidation behaviour of soils with permeability dependent on the settlement.     

A note on decay of potential vortex in an Oldroyd-B fluid through a porous space

The problem of decay of a potential vortex in an Oldroyd-B fluid filling the porous space is studied. The flow problem is first modeled and then solved by employing the Hankel transform. Analytical expressions of the velocity field and the associated tangential tension are developed. The well known solutions for a Newtonian fluid as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of the present solutions. Finally, some graphical results describing the influence of porous space parameters are sketched and interpreted.     

Analysis of incompressible miscible displacement in porous media by characteristics collocation method

Miscible displacement of one incompressible fluid by another in a porous medium is modelled by a coupled system of two partial differential equations. The pressure equation is elliptic, whereas the concentration equation is parabolic but normally convection‐dominated. In this article, the collocation scheme is used to approximate the pressure equation and another characteristics collocation scheme to treat concentration equation. Existence and uniqueness of solutions of the algorithm are proved. Optimal order error estimate is demonstrated. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On exact solutions of Stokes second problem for a Burgers’ fluid, I. The case γ < λ2/4

In the present work, the exact solutions of Stokes second problem for a Burgers’ fluid are investigated. The expressions for the velocity field and the corresponding tangential stress are obtained when the relaxation times satisfy the condition γ < λ²/4. The solutions have been determined by means of Laplace transform. Only one initial condition is necessary for velocity and these solutions presented in the forms of simple or multiple integrals in terms of Bessel functions. The corresponding solutions for a Newtonian fluid as well as Oldroyd-B fluid appear as the limiting cases of the presented results. The obtained solutions are graphically analyzed for the variations of interesting flow parameters. Moreover, a comparison for velocity is made with Oldroyd-B and Newtonian fluids.     

Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative

This paper presents an analysis for magnetohydrodynamic (MHD) flow of an incompressible generalized Oldroyd-B fluid inducing by an accelerating plate. Where the no-slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is introduced to establish the constitutive relationship of a viscoelastic fluid. Closed form solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete Laplace transform of the sequential fractional derivatives. The solutions for no-slip condition and no magnetic field can be derived as the special cases. Furthermore, the effects of various parameters on the corresponding flow and shear stress characteristics are analyzed and discussed in detail.     

Flow of a generalized Maxwell fluid induced by a constantly accelerating plate between two side walls

The unsteady flow of a viscoelastic fluid with the fractional Maxwell model, induced by a constantly accelerating plate between two side walls perpendicular to the plate, is investigated by means of the integral transforms. Exact solutions for the velocity field are presented under integral and series forms in terms of the derivatives of generalized Mittag–Leffler functions. The corresponding solutions for Maxwell fluids are obtained as limiting cases for β → 1. In the absence of the side walls, all solutions that have been determined reduce to those corresponding to the motion over an infinite plate.