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Theodore D Drivas - One of the best experts on this subject based on the ideXlab platform.
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boundary conditions and polymeric drag reduction for the navier stokes equations
Archive for Rational Mechanics and Analysis, 2021Co-Authors: Theodore D DrivasAbstract:Reducing wall drag in turbulent pipe and channel flows is an issue of great practical importance. In engineering applications, end-functionalized polymer chains are often employed as agents to reduce drag. These are polymers which are floating in the solvent and attach (either by adsorption or through irreversible chemical binding) at one of their chain ends to the substrate (wall). We propose a PDE model to study this setup in the simple setting where the solvent is a viscous incompressible Navier–Stokes fluid occupying the bulk of a smooth domain $$\Omega \subset {\mathbb {R}}^d$$ , and the wall-grafted polymer is in the so-called mushroom regime (inter-polymer spacing on the order of the typical polymer length). The microscopic description of the polymer enters into the macroscopic description of the fluid motion through a dynamical boundary condition on the wall-tangential stress of the fluid, something akin to (but distinct from) a history-dependent slip-length. We establish the global well-posedness of strong Solutions in two-spatial dimensions and prove that the inviscid limit to the strong Euler Solution holds with a rate. Moreover, the wall-friction factor $$\langle f\rangle $$ and the global energy dissipation $$\langle \varepsilon \rangle $$ vanish inversely proportional to the Reynolds number $$\mathbf{Re } $$ . This scaling corresponds to Poiseuille’s law for the friction factor $$\langle f\rangle \sim 1/\mathbf{Re } $$ for laminar flow and thereby quantifies drag reduction in our setting. These results are in stark contrast to those available for physical boundaries without polymer additives modeled by, for example, no-slip conditions, where no such results are generally known even in two-dimensions.
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boundary conditions and polymeric drag reduction for the navier stokes equations
arXiv: Analysis of PDEs, 2019Co-Authors: Theodore D DrivasAbstract:Reducing wall drag in turbulent pipe and channel flows is an issue of great practical importance. In engineering applications, end-functionalized polymer chains are often employed as agents to reduce drag. These are polymers which are floating in the solvent and attach (either by adsorption or through irreversible chemical binding) at one of their chain ends to the substrate (wall). We propose a PDE model to study this setup in the simple setting where the solvent is a viscous incompressible Navier-Stokes fluid occupying the bulk of a smooth domain $\Omega\subset \mathbb{R}^d$, and the wall-grafted polymer is in the so-called mushroom regime (inter-polymer spacing on the order of the typical polymer length). The microscopic description of the polymer enters into the macroscopic description of the fluid motion through a dynamical boundary condition on the wall-tangential stress of the fluid, something akin to (but distinct from) a history-dependent slip-length. We establish global well-posedness of strong Solutions in two-spatial dimensions and prove that the inviscid limit to the strong Euler Solution holds with a rate. Moreover, the wall-friction factor $\langle f\rangle$ and the global energy dissipation $\langle \varepsilon\rangle$ vanish inversely proportional to the Reynolds number $Re$. This scaling corresponds to Poiseuille's law for the friction factor $\langle f\rangle \sim1/ Re$ for laminar flow and thereby quantifies drag reduction in our setting. These results are in stark contrast to those available for physical boundaries without polymer additives modeled by, e.g., no-slip conditions, where no such results are generally known even in two-dimensions.
Russel E Caflisch - One of the best experts on this subject based on the ideXlab platform.
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zero viscosity limit for analytic Solutions of the navier stokes equation on a half space ii construction of the navier stokes Solution
Communications in Mathematical Physics, 1998Co-Authors: M Sammartino, Russel E CaflischAbstract:This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct Solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes Solution is constructed through a composite asymptotic expansion involving the Solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes Solution goes to an Euler Solution outside a boundary layer and to a Solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the Solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.
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navier stokes equations on an exterior circular domain construction of the Solution and the zero viscosity limit
Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 1997Co-Authors: Russel E Caflisch, M SammartinoAbstract:Abstract In this Note, we consider the limit of Navier-Stokes equations on a circular domain. By an explicit construction of the Solution, it is proved that, when viscosity goes to zero, Solution converges to the Euler Solution outside the boundary layer and to the Prandtl Solution inside the boundary layer.
M Sammartino - One of the best experts on this subject based on the ideXlab platform.
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zero viscosity limit for analytic Solutions of the navier stokes equation on a half space ii construction of the navier stokes Solution
Communications in Mathematical Physics, 1998Co-Authors: M Sammartino, Russel E CaflischAbstract:This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct Solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes Solution is constructed through a composite asymptotic expansion involving the Solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes Solution goes to an Euler Solution outside a boundary layer and to a Solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the Solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.
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navier stokes equations on an exterior circular domain construction of the Solution and the zero viscosity limit
Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 1997Co-Authors: Russel E Caflisch, M SammartinoAbstract:Abstract In this Note, we consider the limit of Navier-Stokes equations on a circular domain. By an explicit construction of the Solution, it is proved that, when viscosity goes to zero, Solution converges to the Euler Solution outside the boundary layer and to the Prandtl Solution inside the boundary layer.
James P Kelliher - One of the best experts on this subject based on the ideXlab platform.
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boundary layer analysis of the navier stokes equations with generalized navier boundary conditions
Journal of Differential Equations, 2012Co-Authors: Gungmin Gie, James P KelliherAbstract:We study the weak boundary layer phenomenon of the Navier–Stokes equations with generalized Navier friction boundary conditions, u⋅n=0, [S(u)n]tan+Au=0, in a bounded domain in R3 when the viscosity, e>0, is small. Here, S(u) is the symmetric gradient of the velocity, u, and A is a type (1,1) tensor on the boundary. When A=αI we obtain Navier boundary conditions, and when A is the shape operator we obtain the conditions, u⋅n=(curlu)×n=0. By constructing an explicit corrector, we prove the convergence, as e tends to zero, of the Navier–Stokes Solutions to the Euler Solution both in the natural energy norm and uniformly in time and space.
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boundary layer analysis of the navier stokes equations with generalized navier boundary conditions
arXiv: Analysis of PDEs, 2011Co-Authors: Gungmin Gie, James P KelliherAbstract:We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $\epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1, 1) tensor on the boundary. When the tensor is a multiple of the identity we obtain Navier boundary conditions, and when the tensor is the shape operator we obtain conditions in which the vorticity vanishes on the boundary. By constructing an explicit corrector, we prove the convergence of the Navier-Stokes Solutions to the Euler Solution as the viscosity vanishes. We do this both in the natural energy norm with a rate of order $\epsilon^{3/4}$ as well as uniformly in time and space with a rate of order $\epsilon^{3/8 - \delta}$ near the boundary and $\epsilon^{3/4 - \delta'}$ in the interior, where $\delta, \delta'$ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.
Harold S. Park - One of the best experts on this subject based on the ideXlab platform.
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Surface stress effects on the critical buckling strains of silicon nanowires
Computational Materials Science, 2012Co-Authors: Harold S. ParkAbstract:Abstract The objective of this paper is to quantify how nanoscale surface stresses impact the critical buckling strains of silicon nanowires. These insights are gained by using nonlinear finite element calculations based upon a multiscale, finite deformation constitutive model that incorporates nanoscale surface stress and surface elastic effects to study the buckling behavior of silicon nanowires that have cross sectional dimensions between 10 and 25 nm under axial compressive loading. The key finding is that, in contrast to existing surface elasticity Solutions, the critical buckling strains are found to show little deviation from the classical bulk Euler Solution. The present results suggest that accounting for axial strain relaxation due to surface stresses may be necessary to improve the accuracy and predictive capability of analytic linear surface elastic theories.