Nonlinear stability of boundary layer solution to the Boltzmann equation with diffusive effect at the boundary

2009 ◽  
Vol 50 (10) ◽  
pp. 103303 ◽  
Author(s):  
Qianzhu Tian ◽  
Jie Sun
Keyword(s):  
Boundary Layer ◽  
Boltzmann Equation ◽  
Nonlinear Stability ◽  
Boundary Layer Solution ◽  
The Boltzmann Equation ◽  
Layer Solution

Convective diffusion to a slowly rotating spherical electrode; Basic model for Re → O, Pe → ∞

1983 ◽  
Vol 48 (6) ◽  
pp. 1571-1578 ◽  
Author(s):  
Ondřej Wein
Keyword(s):  
Boundary Layer ◽  
Flow Regime ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Creeping Flow ◽  
Péclet Number ◽  
Spherical Electrode ◽  
Basic Model ◽  
Boundary Layer Solution ◽  
Layer Solution

Theory has been formulated of a convective rotating spherical electrode in the creeping flow regime (Re → 0). The currently available boundary layer solution for Pe → ∞ has been confronted with an improved similarity description applicable in the whole range of the Peclet number.

The solution of heat-transfer problems by the Wiener—Hopf technique II. Trailing edge of a hot film

1974 ◽  
Vol 337 (1610) ◽  
pp. 395-412 ◽  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Half Plane ◽  
Dimensionless Temperature ◽  
Fluid Temperature ◽  
Trailing Edge ◽  
Boundary Layer Solution ◽  
Leading Term ◽  
Hot Film ◽  
Layer Solution

An incompressible fluid of constant thermal diffusivity k , flows with velocity u = Sy in the x -direction, where S is a scaling factor for the velocity gradient at the wall y = 0. The region — L ≤ x ≤ 0 is occupied by a heated film of temperature T 1 , the rest of the wall being insulated. Far from the film the fluid temperature is T 0 < T 1 . The finite heated film is approximated by a semi-infinite half-plane x < 0 by assuming that the boundary-layer solution is valid somewhere on the finite region upstream of the trailing edge. Exact solutions in terms of Fourier inverse integrals are obtained by using the Wiener-Hopf technique for the dimensionless temperature distribution on the half-plane x > 0 and the heat transfer from the heated film. An asymptotic expansion is made in inverse powers of x and the coefficient of the leading term is used to calculate the exact value of the total heat-transfer as a function of the length L . It is shown that the boundary layer solution differs from the exact solution by a term of order L -1/3 for large L . An expansion in powers of x for the heat transfer upstream of the trailing edge is also found. Application of the theory, together with that of Springer & Pedley (1973), to hot films used in experiments are discussed for the range of values of L(S/K) ½ , up to 20.

EXISTENCE OF BOUNDARY LAYER SOLUTIONS TO THE BOLTZMANN EQUATION

2004 ◽  
Vol 02 (04) ◽  
pp. 337-363 ◽  
Author(s):  
CHIUN-CHUAN CHEN ◽  
TAI-PING LIU ◽  
TONG YANG
Keyword(s):  
Boundary Layer ◽  
Boltzmann Equation ◽  
Boundary Data ◽  
Hard Sphere Model ◽  
Far Field ◽  
Sphere Model ◽  
Existence Of A Solution ◽  
Solvability Conditions ◽  
Potential Models ◽  
The Boltzmann Equation

In this paper, we consider the existence of boundary layer solutions to the Boltzmann equation for a hard potential with angular cut-off. The boundary condition is imposed for incoming particles of Dirichlet type and the solution tends to a global Maxwellian in the far field. Similar to the problem on the hard sphere model studied in [17], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian, and there is an implicit solvability conditions yielding the co-dimensions of the boundary data. For hard potential models, the shape of the boundary layers is richer. This compels the introduction of a new norm which is a function of both position and velocity.

Mixing at the interface between two fluids in porous media: a boundary-layer solution

2007 ◽  
Vol 584 ◽  
pp. 455-472 ◽  
Author(s):  
AMIR PASTER ◽  
GEDEON DAGAN
Keyword(s):  
Boundary Layer ◽  
Sea Water ◽  
Mixing Layer ◽  
Sharp Interface ◽  
Sea Water Intrusion ◽  
Water Intrusion ◽  
Two Fluids ◽  
The Past ◽  
Boundary Layer Solution ◽  
Layer Solution

A lighter fluid (fresh water) flows steadily above a body of a standing heavier one (sea water) in a porous medium. If mixing by transverse pore-scale dispersion is neglected, a sharp interface separates the two fluids. Solutions for interface problems have been derived in the past, particularly for the case of interest here: sea-water intrusion in coastal aquifers. The Péclet number characterizing mixing, Pe = b′/αT where b′ is the aquifer thickness and αT is transverse dispersivity, is generally much larger than unity. Mixing is nevertheless important in a few applications, particularly in the development of a transition layer near the interface and in entrainment of sea water within this layer. The equations of flow and transport in the mixing zone comprise the unknown flux, pressure and concentration fields, which cannot be separated owing to the presence of density in the gravity term. They are nonlinear because of the advective term and the dependence of the dispersion coefficients on flux, the latter making the problem different from that of mixing between streams in laminar viscous flow.The aim of the study is to solve the mixing-layer problem for sea-water intrusion by using a boundary-layer approximation, which was used in the past for the case of uniform flow of the upper fluid, whereas here the two-dimensional flux field is non-uniform. The boundary-layer solution is obtained in a few steps: (i) analytical potential flow solution of the upper fluid above a sharp interface is adopted; (ii) the equations are reformulated with the potential and streamfunction of this flow serving as independent variables; (iii) boundary-layer approximate equations are formulated in terms of these variables; and (iv) simple analytical solutions are obtained by the von Káarmán integral method. The agreement with an existing boundary-layer solution for uniform flow is excellent, and similarly for a solution of a particular case of sea-water intrusion with a variable-density code. The present solution may serve for estimating the thickness of the mixing layer and the rate of sea-water entrainment in applications, as well as a benchmark for more complex problems.

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