scholarly journals A Nonhomogeneous Boundary Value Problem for Steady State Navier-Stokes Equations in a Multiply-Connected Cusp Domain

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2022
Author(s):  
Kristina Kaulakytė ◽  
Konstantinas Pileckas
Keyword(s):  
Boundary Value Problem ◽  
Stokes Equations ◽  
Boundary Value ◽  
Singular Part ◽  
Navier Stokes ◽  
Connected Components ◽  
Dirichlet Integral ◽  
Existence Of A Solution ◽  
Flow Rates ◽  
Multiply Connected

The boundary value problem for the steady Navier–Stokes system is considered in a 2D multiply-connected bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with nonzero flow rates over connected components of the boundary is studied. It is also supposed that there is a source/sink in O. In this case the solution necessarily has an infinite Dirichlet integral. The existence of a solution to this problem is proved assuming that the flow rates are “sufficiently small” . This condition does not require the norm of the boundary data to be small. The solution is constructed as the sum of a function with the finite Dirichlet integral and a singular part coinciding with the asymptotic decomposition near the cusp point.

scholarly journals ON NONHOMOGENEOUS BOUNDARY VALUE PROBLEM FOR THE STATIONARY NAVIER-STOKES EQUATIONS IN A SYMMETRIC CUSP DOMAIN

2021 ◽  
Vol 26 (1) ◽  
pp. 55-71
Author(s):  
Kristina Kaulakytė ◽  
Neringa Klovienė
Keyword(s):  
Boundary Value Problem ◽  
Stokes Equations ◽  
Boundary Value ◽  
Navier Stokes ◽  
Cusp Point ◽  
Multiply Connected Domain ◽  
Navier Stokes Equations ◽  
Multiply Connected ◽  
Nonhomogeneous Boundary ◽  
Stationary Navier Stokes Equations

The nonhomogeneous boundary value problem for the stationary NavierStokes equations in 2D symmetric multiply connected domain with a cusp point on the boundary is studied. It is assumed that there is a source or sink in the cusp point. A symmetric solenoidal extension of the boundary value satisfying the LerayHopf inequality is constructed. Using this extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak symmetric solution is proved. No restrictions are assumed on the size of fluxes of the boundary value.

Fluid Dynamics of the Elliptic Porous Slider

1978 ◽  
Vol 45 (2) ◽  
pp. 435-436 ◽  
Author(s):  
L. T. Watson ◽  
T. Y. Li ◽  
C. Y. Wang
Keyword(s):  
Boundary Value Problem ◽  
Moving Objects ◽  
Stokes Equations ◽  
Boundary Value ◽  
Initial Approximation ◽  
Frictional Resistance ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Transformation Of Variables ◽  
Lift And Drag

Fluid cushioned porous sliders are useful in reducing the frictional resistance of moving objects. This paper studies the elliptic slider. After a transformation of variables, the Navier-Stokes equations reduce to a nonlinear two-point boundary-value problem. This boundary-value problem was solved by a homotopy-type method, which did not require a good initial approximation to the solution. The problem was solved for several Reynolds numbers and ellipse eccentricities. Lift and drag calculations show that an elliptic porous slider should be operated along the minor axis.

scholarly journals ON SINGULAR SOLUTIONS OF THE STATIONARY NAVIER-STOKES SYSTEM IN POWER CUSP DOMAINS

2021 ◽  
Vol 26 (4) ◽  
pp. 651-668
Author(s):  
Konstantinas Pileckas ◽  
Alicija Raciene
Keyword(s):  
Asymptotic Expansion ◽  
Stokes System ◽  
Boundary Value ◽  
Singular Solutions ◽  
Navier Stokes ◽  
Asymptotic Decomposition ◽  
Dirichlet Integral ◽  
Existence Of A Solution ◽  
Source Sink ◽  
Formal Asymptotic Expansion

The boundary value problem for the steady Navier–Stokes system is considered in a 2D bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with a nonzero flow rate is studied. In this case there is a source/sink in O and the solution necessarily has an infinite Dirichlet integral. The formal asymptotic expansion of the solution near the singular point is constructed and the existence of a solution having this asymptotic decomposition is proved.

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