19 research outputs found

    Abstract kinetic equations with positive collision operators

    Full text link
    We consider "forward-backward" parabolic equations in the abstract form Jdψ/dx+Lψ=0Jd \psi / d x + L \psi = 0, 0<x<τ 0< x < \tau \leq \infty, where JJ and LL are operators in a Hilbert space HH such that J=J=J1J=J^*=J^{-1}, L=L0L=L^* \geq 0, and kerL=0\ker L = 0. The following theorem is proved: if the operator B=JLB=JL is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation μψx(x,μ)=b(μ)2ψμ2(x,μ) \mu \frac {\partial \psi}{\partial x} (x,\mu) = b(\mu) \frac {\partial^2 \psi}{\partial \mu^2} (x, \mu), 0<x<τ 0<x<\tau, μR \mu \in \R, as well as to other parabolic equations of the "forward-backward" type. The abstract kinetic equation Tdψ/dx=Aψ(x)+f(x) T d \psi/dx = - A \psi (x) + f(x), where T=TT=T^* is injective and AA satisfies a certain positivity assumption, is considered also.Comment: 20 pages, LaTeX2e, version 2, references have been added, changes in the introductio

    Interphase mass transfer between liquid-liquid counter-current flows. I. Velocity distribution

    No full text
    A theoretical analysis of liquid–liquid counter-current flow in laminar boundary layers with a flat interphase based on the similarity-variables method has been made. The numerical results for the velocity distribution in both phases are obtained. The dissipation energy in a boundary layer is found and the results corresponding to counter-current and co-current flows are compared. The comparison shows significant differences in the dissipation energy values in the cases of co-current and counter-current flows
    corecore