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George Gabriel Stokes : life, science and faith
This edited collection of essays brings together experts in mathematics, physics and the history of science to cover the many facets of Stokes's life in a scholarly but accessible way to mark the bicentenary of his birth.
Book
The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations
In this work we prove that the original (Bassi and Rebay in J Comput Phys 131:267–279,
1997
) scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier–Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. (SIAM J Sci Comput 36:B835–B867,
2014
) for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu.
Journal Article
On the steady motion of a coupled system solid-liquid
The authors study the unconstrained (free) motion of an elastic solid $\\mathcal B$ in a Navier-Stokes liquid $\\mathcal L$ occupying the whole space outside $\\mathcal B$, under the assumption that a constant body force $\\mathfrak b$ is acting on $\\mathcal B$. More specifically, the authors are interested in the steady motion of the coupled system $\\{\\mathcal B,\\mathcal L\\}$, which means that there exists a frame with respect to which the relevant governing equations possess a time-independent solution. The authors prove the existence of such a frame, provided some smallness restrictions are imposed on the physical parameters, and the reference configuration of $\\mathcal B$ satisfies suitable geometric properties.
eBook
Adaptive servo-ventilation therapy improves cardiac sympathetic nerve activity, cardiac function, exercise capacity, and symptom in patients with chronic heart failure and Cheyne-Stokes respiration
Adaptive servo-ventilation (ASV) therapy has been reported to be effective for improving central sleep apnea (CSA) and chronic heart failure (CHF). The purpose of this study was to clarify whether ASV is effective for CSA, cardiac sympathetic nerve activity (CSNA), cardiac symptoms/function, and exercise capacity in CHF patients with CSA and Cheyne-Stokes respiration (CSR-CSA).
In this study, 31 CHF patients with CSR-CSA and a left ventricular ejection fraction (LVEF) ≤ 40% were randomized into an ASV group and a conservative therapy (non-ASV) group for 6 month. Nuclear imagings with 123I-Metaiodobenzylguanidine (MIBG) and 99mTc-Sestamibi were performed. Exercise capacity using a specific activity scale (SAS) and the New York Heart Association (NYHA) class were evaluated. CSNA was evaluated by 123I-MIBG imaging, with the delayed heart/mediastinum activity ratio (H/M), delayed total defect score (TDS), and washout rate (WR).
The ASV group had significantly better (P < .05) results than the non-ASV group with respect to the changes of AHI (−20.8 ± 14.6 vs −0.5 ± 8.1), TDS (−7.9 ± 4.3 vs 1.4 ± 6.0), and H/M(0.16 ± 0.16 vs −0.04 ± 0.10) on 123I-MIBG imaging, as well as the changes of LVEF (5.3 ± 3.9% vs 0.7 ± 32.6%), SAS (1.6 ± 1.4 vs 0.3 ± 0.7), and NYHA class (2.2 ± 0.4 vs 2.7 ± 0.5) after 6-month therapy.
Performing ASV for 6 months achieved improvement of CSR-CSA, CSNA, cardiac symptoms/function, and exercise capacity in CHF patients with CSR-CSA.
Journal Article
From bead to rod: Comparison of theories by measuring translational drag coefficients of micron-sized magnetic bead-chains in Stokes flow
Frictional drag force on an object in Stokes flow follows a linear relationship with the velocity of translation and a translational drag coefficient. This drag coefficient is related to the size, shape, and orientation of the object. For rod-like objects, analytical solutions of the drag coefficients have been proposed based on three rough approximations of the rod geometry, namely the bead model, ellipsoid model, and cylinder model. These theories all agree that translational drag coefficients of rod-like objects are functions of the rod length and aspect ratio, but differ among one another on the correction factor terms in the equations. By tracking the displacement of the particles through stationary fluids of calibrated viscosity in magnetic tweezers setup, we experimentally measured the drag coefficients of micron-sized beads and their bead-chain formations with chain length of 2 to 27. We verified our methodology with analytical solutions of dimers of two touching beads, and compared our measured drag coefficient values of rod-like objects with theoretical calculations. Our comparison reveals several analytical solutions that used more appropriate approximation and derived formulae that agree with our measurement better.
Journal Article
On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows
The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, H(div)-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using nonrobust discretizations and the improvements obtained by utilizing pressure-robust discretizations.
Journal Article
Acetazolamide Attenuates Hunter-Cheyne-Stokes Breathing but Augments the Hypercapnic Ventilatory Response in Patients with Heart Failure
Acetazolamide has been used to attenuate Hunter-Cheyne-Stokes breathing with central sleep apnea (CSA) associated with heart failure. However, the mechanisms underlying this improvement remain to be fully elucidated.
We hypothesized that acetazolamide stabilizes CSA by attenuating the ventilatory sensitivity to CO2, which is increased in patients with heart failure and is thought to be the major mechanism mediating CSA.
Six consecutive male patients with stable systolic heart failure and CSA (apnea-hypopnea index [AHI] ≥ 15 episodes/h) were randomized to a double-blind crossover protocol with acetazolamide or placebo received 1 hour before bedtime for six nights with 2 weeks of wash-out. Under both conditions, we measured the hypercapnic ventilatory response (HCVR), arterial blood Pco2, steady-state metabolic CO2 production, overnight attended polysomnography, and also assessed cardiac and pulmonary function.
Compared with placebo, acetazolamide significantly decreased the AHI (65 ± 32 vs. 31 ± 19 events/h, mean ± SD). Acetazolamide increased the HCVR slope by 55% (3.3 ± 1.7 vs. 5.1 ± 2.4 L/min/mm Hg; P = 0.03), an increase that far exceeded the 12% fall in arterial Pco2 (P = 0.02). The acetazolamide-induced change in the balance of these effects (ΔHCVR × Pco2) was inversely associated with the reduction in AHI (r = 0.8; P = 0.045).
This placebo-controlled study indicates that acetazolamide improves CSA in patients with heart failure despite an increase in the slope of the HCVR. However, because the degree of HCVR elevation inhibits the improvement in unstable breathing, an increased CO2 chemosensitivity may be a key mechanism underlying an incomplete resolution of CSA with acetazolamide.
Journal Article