scholarly journals Regularized Stokes Immersed Boundary Problems in Two Dimensions: Well‐Posedness , Singular Limit, and Error Estimates

2020 ◽  
Vol 74 (2) ◽  
pp. 366-449
Author(s):  
Jiajun Tong
Keyword(s):  
Error Estimates ◽  
Two Dimensions ◽  
Singular Limit ◽  
Immersed Boundary ◽  
Well Posedness ◽  
Boundary Problems

scholarly journals Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow

2020 ◽  
Vol 30 (05) ◽  
pp. 847-865
Author(s):  
Gabriel Barrenechea ◽  
Erik Burman ◽  
Johnny Guzmán
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Model Problem ◽  
Finite Element Approximation ◽  
Inviscid Flow ◽  
Element Approximation ◽  
Well Posedness ◽  
Hodge Laplacian ◽  
Conforming Finite Element ◽  
Velocity Approximation

We consider a linearised model of incompressible inviscid flow. Using a regularisation based on the Hodge Laplacian we prove existence and uniqueness of weak solutions for smooth domains. The model problem is then discretised using [Formula: see text](div)-conforming finite element methods, for which we prove error estimates for the velocity approximation in the [Formula: see text]-norm of order [Formula: see text]. We also prove error estimates for the pressure error in the [Formula: see text]-norm.

scholarly journals Fluid Dynamics of Ballistic Strategies in Nematocyst Firing

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 20 ◽  
Author(s):  
Christina Hamlet ◽  
Wanda Strychalski ◽  
Laura Miller
Keyword(s):  
Immersed Boundary Method ◽  
Short Distance ◽  
Reynolds Numbers ◽  
Two Dimensions ◽  
Immersed Boundary ◽  
Small Scale ◽  
Viscous Damping ◽  
Final Velocity ◽  
Inertial Regime ◽  
Surrounding Fluid

Nematocysts are stinging organelles used by members of the phylum Cnidaria (e.g., jellyfish, anemones, hydrozoans) for a variety of important functions including capturing prey and defense. Nematocysts are the fastest-known accelerating structures in the animal world. The small scale (microns) coupled with rapid acceleration (in excess of 5 million g) present significant challenges in imaging that prevent detailed descriptions of their kinematics. The immersed boundary method was used to numerically simulate the dynamics of a barb-like structure accelerating a short distance across Reynolds numbers ranging from 0.9–900 towards a passive elastic target in two dimensions. Results indicate that acceleration followed by coasting at lower Reynolds numbers is not sufficient for a nematocyst to reach its target. The nematocyst’s barb-like projectile requires high accelerations in order to transition to the inertial regime and overcome the viscous damping effects normally encountered at small cellular scales. The longer the barb is in the inertial regime, the higher the final velocity of the projectile when it touches its target. We find the size of the target prey does not dramatically affect the barb’s approach for large enough values of the Reynolds number, however longer barbs are able to accelerate a larger amount of surrounding fluid, which in turn allows the barb to remain in the inertial regime for a longer period of time. Since the final velocity is proportional to the force available for piercing the membrane of the prey, high accelerations that allow the system to persist in the inertial regime have implications for the nematocyst’s ability to puncture surfaces such as cellular membranes or even crustacean cuticle.

Feedback theory to the well-posedness of evolution equations

2020 ◽  
Vol 378 (2185) ◽  
pp. 20190616
Author(s):  
S. Boulite ◽  
S. Hadd ◽  
L. Maniar
Keyword(s):  
Evolution Equations ◽  
Semigroup Theory ◽  
Well Posedness ◽  
Boundary Problems ◽  
Neutral Equations ◽  
Infinite Dimensional ◽  
Research Activities ◽  
Semigroup Approach ◽  
History Of ◽  
And Control

In this paper, we cross the boundary between semigroup theory and general infinite-dimensional systems to bridge the isolated research activities in the two areas. Indeed, we first give a chronological history of the development of the semigroup approach for control theory. Second, we use the feedback theory to prove the well-posedness of a class of dynamic boundary problems. Third, the obtained results are applied to the well-posedness of neutral equations with non-autonomous past. We will also see that the strong connection between semigroup and control theories lies in feedback theory, where different kinds of perturbations appear. This article is part of the theme issue ‘Semigroup applications everywhere’.

Global well-posedness for the averaged Euler equations in two dimensions

2000 ◽  
Vol 138 (3-4) ◽  
pp. 197-209 ◽  
Author(s):  
Shinar Kouranbaeva ◽  
Marcel Oliver
Keyword(s):  
Euler Equations ◽  
Two Dimensions ◽  
Well Posedness

scholarly journals Inverse Boundary Problems for Systems in Two Dimensions

2013 ◽  
Vol 14 (6) ◽  
pp. 1551-1571 ◽  
Author(s):  
Pierre Albin ◽  
Colin Guillarmou ◽  
Leo Tzou ◽  
Gunther Uhlmann
Keyword(s):  
Two Dimensions ◽  
Boundary Problems ◽  
Inverse Boundary

WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE CHERN–SIMONS–DIRAC SYSTEM IN TWO DIMENSIONS

2013 ◽  
Vol 10 (04) ◽  
pp. 735-771 ◽  
Author(s):  
MAMORU OKAMOTO
Keyword(s):  
Cauchy Problem ◽  
Dirac Equation ◽  
Sobolev Space ◽  
Gauge Invariance ◽  
Two Dimensions ◽  
Well Posedness ◽  
Dirac System ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

We consider the Cauchy problem associated with the Chern–Simons–Dirac system in ℝ1+2. Using gauge invariance, we reduce the Chern–Simons–Dirac system to a Dirac equation and we uncover the null structure of this Dirac equation. Next, relying on null structure estimates, we establish that the Cauchy problem associated with this Dirac equation is locally-in-time well-posed in the Sobolev space Hs for all s > 1/4. Our proof uses modified L4-type estimates.

Flows produced by the combined oscillatory rotation and translation of a circular cylinder in a quiescent fluid

2014 ◽  
Vol 764 ◽  
pp. 148-170 ◽  
Author(s):  
Christopher Koehler ◽  
Philip Beran ◽  
Marcos Vanella ◽  
Elias Balaras
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Bluff Body ◽  
Stokes Equations ◽  
Time Windows ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Immersed Boundary ◽  
Dimensionless Time ◽  
Quiescent Fluid

AbstractFlows produced by a circular cylinder undergoing oscillatory rotation and translation in a quiescent fluid have been studied via direct numerical simulations. The incompressible Navier–Stokes equations were solved for large dimensionless time windows using an immersed boundary method with adaptive Cartesian grid refinement. Parametric studies were conducted in two dimensions on the Reynolds number, Keulegan–Carpenter number and phase shift. In addition to the previously reported net thrust case (Blackburn et al., Phys. Fluids, vol. 11, 1999, pp. 4–6), the study catalogued the appearance of several streaming jet regimes with varying deflection angles, deflected and horizontal vortex shedding regimes, and a double mirrored jet regime with varying inter-jet angles, as well as several chaotic cases. Visualizations are presented to clarify each observed flow regime and to illustrate the parameter space. Connections are drawn between these canonical bluff-body deflected wakes and a similar phenomenon observed in aerofoils oscillating at high reduced frequencies in a cross-flow. Also, the discovery of the streaming jet regimes with varying deflection angles opens the door for using these flows as a low-Reynolds-number propulsive mechanism requiring only a two-degree-of-freedom actuator. Simulation results suggest that the flow phenomena observed in two dimensions persist in three dimensions, despite spanwise fluctuations.

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