Equation (1) is just Newton’s law f = ma for a fluid element subject to the external force f = (fi(x, t))1 i n and to the forces arising from pressure and friction. Equation (2) just says that the fluid is incompressible. For physically reasonable solutions, we want to make sure u(x, t) does now grow large as |x| → ∞. Hence, we will restrict attention to forces f and initial conditions u◦ that ...

We consider a flow of incompressible Newtonian fluid through a pipe with helical shape. We suppose that the flow is governed by the prescribed pressure drop between pipe’s ends. Such model has relevance to some important engineering applications. Under small data assumption, we prove the existence and uniqueness of the weak solution to the corresponding Navier-Stokes system with pressure bounda...

The particular example given in §6 of Crowdy and Tanveer [1] to illustrate the general theory is in error because it is not consistent with the original assumptions underlying the theory. The theory developed in earlier sections assumes that a single-valued analytic function F(ζ, t) can be found in the annulus ρ(t) < |ζ | < 1 when the integration constants AO(t) and AI (t) in equations (16) and...

We study in this paper the movement of a rigid solid inside an incompressible Navier-Stokes flow, within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface, through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no-slip conditions lead to unrealistic collisional behavior between the solid and t...

In this paper, we introduce the Navier-Stokes equations with a new boundary condition. In this context, we show the existence and uniqueness of the solution of the weak formulation associated with the proposed problem. To solve this latter, we use the discretization by mixed finite element method. In addition, two types of a posteriori error indicator are introduced and are shown to give global...

We consider incompressible Navier-Stokes equations in a bounded 3D regular domain, coupled with the so-called dynamic boundary condition. rigorously establish principle of linearized stability. Namely, given smooth stationary state, we prove that equation has complete basis generalized eigenfunctions, and non-linear stability depends on supremum real part spectrum usual way. work class weak sol...