Fractional derivatives of solutions of the Navier-Stokes equations

1971 ◽  
Vol 40 (2) ◽  
pp. 139-154 ◽  
Author(s):  
Marvin Shinbrot
Keyword(s):  
Fractional Derivatives ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Derivatives Of

Navier Stokes Derivative Estimates in Three Dimensions with Boundary Values and Body Forces

1991 ◽  
Vol 43 (6) ◽  
pp. 1161-1212 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Stokes Equations ◽  
Finite Energy ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Boundary Values ◽  
Vector Solution ◽  
Navier Stokes Equations ◽  
Body Forces ◽  
Derivatives Of

AbstractFor a vector solution u(x, t) with finite energy of the Navier Stokes equations with body forces and boundary values on a region Ω ⊆ R3 for t > 0, conditions are established on the L6/5(Ω) and L2(Ω) norms of derivatives of the data that ensure the estimates and max , up to any given integer value of the weighted order 2r+s, where r or s = s1 + s2 + s3 > 0 and 0 < T < ∞.

Parametric Study of a Turbine Blade Cascade Using a New Parametric Flow Solver Turb’Opty

2002 ◽  
Author(s):  
S. Moreau ◽  
S. Aubert ◽  
M. N’Diaye ◽  
P. Ferrand ◽  
J. Tournier ◽  
...  
Keyword(s):  
Turbine Blade ◽  
Stokes Equations ◽  
Taylor Series Expansion ◽  
Navier Stokes ◽  
Reference Solution ◽  
Navier Stokes Equations ◽  
Flow Solver ◽  
Blade Cascade ◽  
Polynomial Reconstruction ◽  
Derivatives Of

A new parameterized CFD solver Turb’Opty™ has been developed based on a Taylor series expansion to high order derivatives of the solutions of the discretized Navier-Stokes equations. The method has been successfully applied to the laminar compressible flow field of the T106 turbine blade cascade. Comparisons with the classical CFD results have validated the accuracy of the parameterized solutions obtained by a simple polynomial reconstruction around a reference solution. The CPU efficiency has been emphasized by quickly computing the performance maps (power and losses) of this blade cascade. Wide industrial perspectives of turbomachinery global optimization are finally demonstrated by coupling this method with a simple genetic algorithm.

Analyticity and Existence of the Keller–Segel–Navier–Stokes Equations in Critical Besov Spaces

2018 ◽  
Vol 18 (3) ◽  
pp. 517-535 ◽  
Author(s):  
Minghua Yang ◽  
Zunwei Fu ◽  
Suying Liu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Gevrey Regularity ◽  
Navier Stokes Equations ◽  
Global Existence Of Solutions ◽  
Regularity Estimates ◽  
Gevrey Estimates ◽  
The Cauchy Problem ◽  
Critical Besov Spaces ◽  
Derivatives Of

Abstract This paper deals with the Cauchy problem to the Keller–Segel model coupled with the incompressible 3-D Navier–Stokes equations. Based on so-called Gevrey regularity estimates, which are motivated by the works of Foias and Temam [20], we prove that the solutions are analytic for a small interval of time with values in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly imply higher-order derivatives of solutions in Besov and Lebesgue spaces. Moreover, we prove that the existence of a positive constant {\tilde{C}} such that the initial data {(u_{0},n_{0},c_{0}):=(u_{0}^{h},u_{0}^{3},n_{0},c_{0})} satisfy \tilde{C}\bigl{(}\lVert(n_{0},c_{0})\rVert_{\dot{B}^{-2+3/q}_{q,1}(\mathbb{R}^% {3})\times\dot{B}^{3/q}_{q,1}(\mathbb{R}^{3})}+\lVert u_{0}^{h}\rVert_{\dot{B}% ^{-1+3/p}_{p,1}(\mathbb{R}^{3})}+\lVert u_{0}^{h}\rVert_{\dot{B}^{-1+3/p}_{p,1% }(\mathbb{R}^{3})}^{\alpha}\lVert u_{0}^{3}\rVert_{\dot{B}^{-1+3/p}_{p,1}(% \mathbb{R}^{3})}^{1-\alpha}\bigr{)}\leq 1 for certain conditions on {p,q} and α implies the global existence of solutions with large initial vertical velocity component.

scholarly journals The Analysis of Fractional-Order Navier-Stokes Model Arising in the Unsteady Flow of a Viscous Fluid via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Pongsakorn Sunthrayuth ◽  
Rasool Shah ◽  
A. M. Zidan ◽  
Shahbaz Khan ◽  
Jeevan Kafle
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Navier Stokes ◽  
Power Series Method ◽  
Navier Stokes Equations ◽  
Residual Power Series ◽  
Residual Power

This paper presents a new method that is constructed by combining the Shehu transform and the residual power series method. Precisely, we provide the application of the proposed technique to investigate fractional-order linear and nonlinear problems. Then, we implemented this new technique to obtain the result of fractional-order Navier-Stokes equations. Finally, we provide three-dimensional figures to help the effect of fractional derivatives on the actions of the achieved profile results on the proposed models.

On Shinbrot’s conjecture for the Navier-Stokes equations

1993 ◽  
Vol 440 (1910) ◽  
pp. 537-540 ◽  
Keyword(s):  
Weak Solution ◽  
Fractional Derivatives ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Maximal Theorem

Marvin Shinbrot conjectured that the weak solution of the Navier-Stokes equations possess fractional derivatives in time of any order less than 1/2. In this paper, using the Hardy-Littlewood maximal theorem we prove that the conjecture is true in the two-dimensional case and it is true conditionally in the three-dimensional case.

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