scholarly journals Weak–strong uniqueness for a class of generalized dissipative weak solutions for non-homogeneous, non-Newtonian and incompressible fluids

2022 ◽  
Vol 64 ◽  
pp. 103426
Author(s):  
Jakub Woźnicki
Keyword(s):  
Weak Solutions ◽  
Incompressible Fluids ◽  
Strong Uniqueness

scholarly journals Uniqueness of Weak Solutions to an Electrohydrodynamics Model

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yong Zhou ◽  
Jishan Fan
Keyword(s):  
Besov Space ◽  
Weak Solutions ◽  
Uniqueness Result ◽  
Strong Uniqueness ◽  
Homogeneous Besov Space

This paper studies uniqueness of weak solutions to an electrohydrodynamics model inℝd(d=2,3). Whend=2, we prove a uniqueness without any condition on the velocity. Ford=3, we prove a weak-strong uniqueness result with a condition on the vorticity in the homogeneous Besov space.

On steady flows of fluids with pressure– and shear–dependent viscosities

2005 ◽  
Vol 461 (2055) ◽  
pp. 651-670 ◽  
Author(s):  
M. Franta ◽  
J. Málek ◽  
K. R. Rajagopal
Keyword(s):  
Shear Rate ◽  
Weak Solutions ◽  
Velocity Gradient ◽  
Dirichlet Boundary ◽  
Incompressible Fluids ◽  
Dirichlet Boundary Conditions ◽  
Steady Flows ◽  
Existence Of Weak Solutions ◽  
Wide Range ◽  
Homogeneous Dirichlet Boundary Conditions

There are many technologically important problems such as elastohydrodynamics which involve the flows of a fluid over a wide range of pressures. While the density of the fluid remains essentially constant during these flows whereby the fluid can be approximated as being incompressible, the viscosity varies significantly by several orders of magnitude. It is also possible that the viscosity of such fluids depends on the shear rate. Here we consider the flows of a class of incompressible fluids with viscosity that depends on the pressure and shear rate. We establish the existence of weak solutions for the steady flows of such fluids subjected to homogeneous Dirichlet boundary conditions and to specific body forces that are not necessarily assumed to be small. A novel aspect of the study is the manner in which we treat the pressure that allows us to establish its compactness, as well as that of the velocity gradient. The method draws upon the physics of the problem, namely that the notion of incompressibility is an idealization that is attained by letting the compressibility of the fluid to tend to zero.

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