Small Data Assumption Research Articles

An unconditional existence result for elastohydrodynamic piezoviscous lubrication problems with Elrod-Adams model of cavitation

An unconditional existence result of a solution for a steady fluid-structure problem is stated. More precisely, we consider an incompressible fluid in a thin film, ruled by the Reynolds equation coupled with a surface deformation modelled by a nonlinear non local Hertz law. The viscosity is supposed to depend nonlinearly on the fluid pressure. Due to the apparition of a mushy region, the two-phase flow satisfies a free boundary problem defined by a pressure-saturation model. Such a problem has been studied with simpler free boundaries models (variational inequality), or with boundary conditions imposing small data assumptions. We show that up to a realistic hypothesis on the asymptotic pressure-viscosity behaviour it is possible to obtain an unconditional solution of the problem.

A Fundamental Limit on the Heat Flux in the Control of Incompressible Channel Flow

This paper proves that there are no zero-net wall-transpiration control strategies that can sustain net heat flux below the laminar level in an incompressible channel flow with constant-temperature walls. The result represents a fundamental limit on the performance of a controlled nonlinear system as measured by a linear cost function over a broad class of admissible initial conditions and control inputs, not a zero-sum tradeoff in the frequency domain or time domain. Both buoyancy effects (via the Boussinesq approximation) and viscous heating effects are accounted for, and phenomenological justification for the result is also given. The boundedness of solutions of the two-way coupled Navier-Stokes/energy equations (when both buoyancy and viscous heating are accounted for) is also discussed, and a new proof of existence under an appropriate small-data assumption is provided.

An unconditional existence result for the quasi-variational elastohydrodynamic free boundary value problem

In this paper a two-dimensional quasi-variational inequality arising in elastohydrodynamic lubrication is studied for non-constant viscosity. So far, existence results for such piezo–viscous problems require an L ∞ property for an auxiliary problem. For the usual pressure–viscosity relations, this property needs small data assumptions which are not observed in experimental conditions. In the present work, such small data assumptions are proved unnecessary for existence results. Besides well-established monotonicity behavior for the viscosity–pressure relation, the only condition used here is on the asymptotic behavior for this law as the pressure tends to infinity. If the procedure used here, namely the introduction of a reduced pressure by Grubin transform followed by a regularization procedure, appears somewhat classical, the way in which an upper bound is obtained is completely new.

Analysis of the Oseen-viscoelastic fluid flow problem

In this article we study the numerical approximation of an Oseen type model for viscoelastic fluid flow. Existence and uniqueness of the continuous and approximate solutions, under a small data assumption, are proved. Error estimates for the numerical approximations are also derived. Numerical experiments are presented which support the error estimates, and which demonstrate the relevance of the small data assumption for the solvability of the continuous and discrete systems.

Approximation of time-dependent, multi-component, viscoelastic fluid flow

In this article we analyse a fully discrete approximation to the time dependent viscoelasticity equations allowing for multicomponent fluid flow. The Oldroyd B constitutive equation is used to model the viscoelastic stress. For the discretization, time derivatives are replaced by backward difference quotients, and the non-linear terms are linearized by lagging appropriate factors. The modeling equations for the individual fluids are combined into a single system of equations using a continuum surface model. The numerical approximation is stabilized by using an SUPG approximation for the constitutive equation. Under a small data assumption on the true solution, existence of the approximate solution is proven. A priori error estimates for the approximation in terms of the mesh parameter h, the time discretization parameter Δ t, and the SUPG coefficient ν are also derived. Numerical simulations of viscoelastic fluid flow involving two immiscible fluids are also presented.

Solvability of the Navier--Stokes System with L 2 Boundary Data

We prove the existence of the very weak solution of the Dirichlet problem for the Navier—Stokes system with L 2 boundary data. Under the small data assumption we also prove the uniqueness. We use the penalization method to study the linearized problem and then apply Banach's fixed point theorem for the nonlinear problem with small boundary data. We extend our result to the case with no small data assumption by splitting the data on a large regular and small irregular part.

Analysis and Numerical Approximation of a Stationary MHD Flow Problem with Nonideal Boundary

We are concerned with the steady flow of a conducting fluid, confined to a bounded region of space and driven by a combination of body forces, externally generated magnetic fields, and currents entering and leaving the fluid through electrodes attached to the surface. The flow is governed by the Navier--Stokes equations (in the fluid region) and Maxwell's equations (in all of space), coupled via Ohm's law and the Lorentz force. By means of the Biot--Savart law, we reduce the problem to a system of integro-differential equations in the fluid region, derive a mixed variational formulation, and prove its well-posedness under a small-data assumption. We then study the finite-element approximation of solutions (in the case of unique solvability) and establish optimal-order error estimates. Finally, an implementation of the method is described and illustrated with the results of some numerical experiments.

Existence theory to the equations in viscoelasticity at finite deformations

In this paper we prove under the assumption of small initial data the global existence of a classical solution to the equations in viscoelasticity, associated with a free damping boundary condition. We also show that if we choose the initial data large enough, blow up will occur in finite time.