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M.m.r. Williams - One of the best experts on this subject based on the ideXlab platform.

  • Probabilistic methods in heterogeneous neutronic systems
    Annals of Nuclear Energy, 2000
    Co-Authors: M.m.r. Williams
    Abstract:

    Abstract We consider a slab of moderating material in which are embedded absorbing plates at fixed positions. A fast neutron source is incident on the left hand face and we calculate the transmitted fraction, T, from the far face and the reflected fraction, R, from the source face. The effect of the absorbing plates on the thermal neutron flux is characterised by a plate Sink of Strength γ, i.e. the Feinberg-Galanin-Horning representation is used. However, instead of the values of γ being prescribed, they are sampled from a probability distribution in the range (γ l , γ u ). Given the probability distribution of the γ's, we then show how the corresponding probability distribution functions P ( T ) and P ( R ) can be obtained. The problem is solved by two methods; in the first, for each realisation of the γ's we calculate the associated values of T and R. Then, after many realisations (10 5 in this case), we may construct P ( T ) and P ( R ). The second method uses a completely analytical approach which does not involve numerical sampling with random numbers. To proceed we use well-established procedures in the theory of random processes, but because of the algebraic complexity, only two absorbing plates are considered.

Barbara Abraham-shrauner - One of the best experts on this subject based on the ideXlab platform.

  • A ‘simplest’ steady-state Munch-like model of phloem translocation, with source and pathway and Sink
    Functional Plant Biology, 2009
    Co-Authors: William F. Pickard, Barbara Abraham-shrauner
    Abstract:

    In the 80 years since its introduction by Munch, the pressure-driven mass-flow model of phloem translocation has become hegemonic, and has been mathematically modelled in many different fashions but not, to our knowledge, by one that incorporated the equations of hydrodynamics with those of osmosis and slice-source and slice-Sink boundary conditions to yield a system that admits of an analytical steady-state solution for the sap velocity in a single sieve tube. To overcome this situation, we drastically simplified the problem by: (i) justifying a low Peclet number idealisation in which transverse variations could be neglected; (ii) justifying a low viscosity idealisation in which axial pressure drops could be neglected; and (iii) assuming a Sink of Strength sufficient to lower the photosynthate concentration at the extreme distal end of the sieve tube to levels at which it became unimportant. The resulting ordinary nonlinear second-order differential equation in sap velocity and axial position was of a generalised Lienard form with a single forcing parameter; and this is reason enough for the lack of a known analytic solution. However, since the forcing parameter was very large, it was possible to deduce approximate second-order solutions for behavior in the source, Sink and transport regions: the sap velocity is zero at the slice-source, climbs with exponential rapidity to a plateau, maintains this plateau over most of the sieve tube, and then drops with exponential rapidity to zero at the slice-Sink.

Herbert Steinrück - One of the best experts on this subject based on the ideXlab platform.

  • Mixed convection over a horizontal plate : connecting boundary layer flows
    Journal of Applied Mathematics and Mechanics, 1996
    Co-Authors: Herbert Steinrück
    Abstract:

    Consider a flat horizontal plate aligned parallel to a uniform flow with velocity U∞ and temperature T∞. It is assumed that a heat source (or Sink) of Strength Q is at the leading edge of the plate and that the plate is adiabatic everywhere else. In the limit of large Reynolds number Re (boundary layer approximation) a similarity solution exists. Besides the Prandtl number Pr a second dimensionless parameter K appears in the dimensionless equations which describes the Strength of the buoyancy effects. In case of the flow above a plate cooled at its leading edge (K < 0) two solution branches exist for buoyancy parameters K between some critical value K c < 0 and K = 0. Both solution branches are connected at K c . For K small and negative, one solution can be viewed as a perturbation of the Blasisus solution, where the buoyancy effects do not influence the velocity profile significantly, while the other solution branch has a large region with reverse flow. The aim of the present work is to discuss which of the two solutions for K < 0 is of physical relevance and stable with respect to small perturbations. After presenting the basic equations steady flows are presented which connect both similarity solutions.

William F. Pickard - One of the best experts on this subject based on the ideXlab platform.

  • A ‘simplest’ steady-state Munch-like model of phloem translocation, with source and pathway and Sink
    Functional Plant Biology, 2009
    Co-Authors: William F. Pickard, Barbara Abraham-shrauner
    Abstract:

    In the 80 years since its introduction by Munch, the pressure-driven mass-flow model of phloem translocation has become hegemonic, and has been mathematically modelled in many different fashions but not, to our knowledge, by one that incorporated the equations of hydrodynamics with those of osmosis and slice-source and slice-Sink boundary conditions to yield a system that admits of an analytical steady-state solution for the sap velocity in a single sieve tube. To overcome this situation, we drastically simplified the problem by: (i) justifying a low Peclet number idealisation in which transverse variations could be neglected; (ii) justifying a low viscosity idealisation in which axial pressure drops could be neglected; and (iii) assuming a Sink of Strength sufficient to lower the photosynthate concentration at the extreme distal end of the sieve tube to levels at which it became unimportant. The resulting ordinary nonlinear second-order differential equation in sap velocity and axial position was of a generalised Lienard form with a single forcing parameter; and this is reason enough for the lack of a known analytic solution. However, since the forcing parameter was very large, it was possible to deduce approximate second-order solutions for behavior in the source, Sink and transport regions: the sap velocity is zero at the slice-source, climbs with exponential rapidity to a plateau, maintains this plateau over most of the sieve tube, and then drops with exponential rapidity to zero at the slice-Sink.