Unilateral problem for the Stokes equations: The well-posedness and finite element approximation

Saito, Norikazu; Sugitani, Yoshiki; Zhou, Guanyu

doi:10.1016/j.apnum.2016.03.002

Cited by 12 publications

(19 citation statements)

References 25 publications

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“…The nonlinear condition consists of the impermeability (no leak) condition

u_{ν} = 0

supplemented with the nonmonotone slip boundary condition of frictional type

- {boldτ}_{τ} false(boldu false) \in k false(u_{τ} false) \partial j false(u_{τ} false)

, where k is a prescribed nonnegative function, and

\partial j

denotes the generalized gradient of a given function

j : {normalΓ}_{1} \times {double-struckR}^{d} \to R

. Condition has been recently proposed and studied in []. It is an analogy to the Signorini‐type condition which models the unilateral contact in the theory of elasticity.…”

Section: Weak Formulation Of the Oseen Problemmentioning

confidence: 99%

“…Example The outflow unilateral boundary condition has been recently suggested and investigated in []. Its introduction is motivated by blood flow simulations which are highly dependent on the choice of artificial boundary conditions posed on the outflow boundary Γ 2 .…”

Section: Weak Solution To the Oseen Problemmentioning

confidence: 99%

“…Next, it serves as a generalization of the free‐traction condition or the so‐called do‐nothing condition of the form

τ false(boldu, π false) = 0 on {normalΓ}_{2} .

In ,

g > 0

is a prescibed constant and h is a given nonnegative function. As explained in [], the physical interpretation and full justification for applying to the fluid‐flow problems are open at present. Note also that the free‐traction condition has no physical justification and it is used because of its simplicity.…”

Section: Weak Solution To the Oseen Problemmentioning

confidence: 99%

“…In all aformentioned papers various types of boundary conditions were associated with the subdifferential of convex functions which lead to variational inequality formulation. On the other hand, well‐posedness of the Navier‐Stokes and Stokes equations with unilateral boundary conditions have been studied only recently in []. Nonmonotone boundary conditions described by the Clarke generalized gradient were studied for Navier‐Stokes equations in [], and for non‐Newtonian fluids and generalized Newtonian fluids in [].…”

Section: Introductionmentioning

confidence: 99%

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A new class of variational‐hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions

Migórski

Dudek

2019

Z Angew Math Mech

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We study a new class of elliptic variational-hemivariational inequalities in a reflexive Banach space. Based on a surjectivity result for an operator inclusion of Clarke's subdifferential type, we prove existence of solution. Then, we apply this result to a mathematical analysis of the steady Oseen model for a generalized Newtonian incompressible fluid. A variational-hemivariational inequality for the flow problem is derived and sufficient conditions for existence of weak solutions are obtained. The mixed boundary conditions involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier-Fujita slip condition, and the threshold slip and leak condition of frictional type. K E Y W O R D Sfrictional type boundary condition, generalized Newtonian fluid, leak, operator inclusion, Oseen model, slip, unilateral boundary condition 2 0 1 0 M A In this paper we introduce and analyse a new class of operator inclusions of subdifferential type and a corresponding class of elliptic variational-hemivariational inequalities in a reflexive Banach space. Our main goal is to provide existence results to these classes of problems, and apply them to examine the stationary flow Oseen model of generalized Newtonian fluid described by equations − Div + ( ⋅ ∇) + ∇ = in Ω, div = 0 in Ω

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“…The nonlinear condition consists of the impermeability (no leak) condition

u_{ν} = 0

supplemented with the nonmonotone slip boundary condition of frictional type

- {boldτ}_{τ} false(boldu false) \in k false(u_{τ} false) \partial j false(u_{τ} false)

, where k is a prescribed nonnegative function, and

\partial j

denotes the generalized gradient of a given function

j : {normalΓ}_{1} \times {double-struckR}^{d} \to R

. Condition has been recently proposed and studied in []. It is an analogy to the Signorini‐type condition which models the unilateral contact in the theory of elasticity.…”

Section: Weak Formulation Of the Oseen Problemmentioning

confidence: 99%

Section: Weak Solution To the Oseen Problemmentioning

confidence: 99%

“…Next, it serves as a generalization of the free‐traction condition or the so‐called do‐nothing condition of the form

τ false(boldu, π false) = 0 on {normalΓ}_{2} .

In ,

g > 0

Section: Weak Solution To the Oseen Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new class of variational‐hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions

Migórski

Dudek

2019

Z Angew Math Mech

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“…The particular purpose of this paper is to establish the well-posedness of these problems. We postpone a study on time discretizations and the finite element approximation to be addressed in future works; a partial result on the finite element approximation for a model (stationary) Stokes problem will be reported in Saito et al [20]. The motivation, advantage and results (without any proofs) are already announced in Saito et al [19].…”

Section: Motivationmentioning

confidence: 99%

The Navier–Stokes Equations Under a Unilateral Boundary Condition of Signorini’s Type

Zhou

Saito

2016

J. Math. Fluid Mech.

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We propose a new outflow boundary condition, a unilateral condition of Signorini's type, for the incompressible Navier-Stokes equations. The condition is a generalization of the standard free-traction condition. Its variational formulation is given by a variational inequality. We also consider a penalty approximation, a kind of the Robin condition, to deduce a suitable formulation for numerical computations. Under those conditions, we can obtain energy inequalities that are key features for numerical computations. The main contribution of this paper is to establish the well-posedness of the Navier-Stokes equations under those boundary conditions. Particularly, we prove the unique existence of strong solutions of Ladyzhenskaya's class using the standard Galerkin's method. For the proof of the existence of pressures, however, we offer a new method of analysis.Mathematics Subject Classification. 35K85, 35Q30, 76D05.

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Energy Inequalities and Outflow Boundary Conditions for the Navier–Stokes Equations

Saito

Sugitani

Zhou

2016

Advances in Computational Fluid-Structure Interaction and Flow Simulation

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Unilateral problem for the Stokes equations: The well-posedness and finite element approximation

Cited by 12 publications

References 25 publications

A new class of variational‐hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions

A new class of variational‐hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions

The Navier–Stokes Equations Under a Unilateral Boundary Condition of Signorini’s Type

Energy Inequalities and Outflow Boundary Conditions for the Navier–Stokes Equations

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