91 research outputs found
New exact solutions for the discrete fourth Painlev\'e equation
In this paper we derive a number of exact solutions of the discrete equation
x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})=
{-2z_nx_n^3+(\eta-3\delta^{-2}-z_n^2)x_n^2+\mu^2\over
(x_n+z_n+\gamma)(x_n+z_n-\gamma)},\eqno(1) where and ,
, and are constants. In an appropriate limit (1) reduces
to the fourth \p\ (PIV) equation {\d^2w\over\d z^2} = {1\over2w}\left({\d
w\over\d z}\right)^2+\tfr32w^3 + 4zw^2 + 2(z^2-\alpha)w +{\beta\over
w},\eqno(2) where and are constants and (1) is commonly
referred to as the discretised fourth Painlev\'e equation. A suitable
factorisation of (1) facilitates the identification of a number of solutions
which take the form of ratios of two polynomials in the variable . Limits
of these solutions yield rational solutions of PIV (2). It is also known that
there exist exact solutions of PIV (2) that are expressible in terms of the
complementary error function and in this article we show that a discrete
analogue of this function can be obtained by analysis of (1).Comment: Tex file 14 page
A Bilinear Approach to Discrete Miura Transformations
We present a systematic approach to the construction of Miura transformations
for discrete Painlev\'e equations. Our method is based on the bilinear
formalism and we start with the expression of the nonlinear discrete equation
in terms of -functions. Elimination of -functions from the
resulting system leads to another nonlinear equation, which is a ``modified''
version of the original equation. The procedure therefore yields Miura
transformations. In this letter, we illustrate this approach by reproducing
previously known Miura transformations and constructing new ones.Comment: 7 pages in TeX, to appear in Phys. Letts.
Unsteady thermal boundary layer flows of a Bingham fluid in a porous medium.
In this paper we consider some unsteady free convection flows of a Bingham fluid when it saturates a porous medium. These flows are induced by suddenly raising the constant temperature of a vertical bounding surface from that of the uniform ambient value to a new constant level. As time progresses heatconducts inwards and this induces flow. We consider both a semi-infinite domain and a vertical channel of finite width. Of interest here are (i) how the presence of yield surfaces alters the classical results for Newtonian flows and (ii) the manner in which the locations of the yield surfaces change as time progresses
Baxter T-Q Equation for Shape Invariant Potentials. The Finite-Gap Potentials Case
The Darboux transformation applied recurrently on a Schroedinger operator
generates what is called a {\em dressing chain}, or from a different point of
view, a set of supersymmetric shape invariant potentials. The finite-gap
potential theory is a special case of the chain. For the finite-gap case, the
equations of the chain can be expressed as a time evolution of a Hamiltonian
system. We apply Sklyanin's method of separation of variables to the chain. We
show that the classical equation of the separation of variables is the Baxter
T-Q relation after quantization.Comment: 25 pages, no figures Extended section 10, one reference added.
Version accepted for publication in Jurnal of Mathematical Physic
The linear stability of a Stokes layer with an imposed axial magnetic field
The effects of a uniform axial magnetic field directed towards an oscillating wall in a semi-infinite viscous fluid (or Stokes layer) is investigated. The linear stability and disturbance characteristics are determined using both Floquet theory and via direct numerical simulations. Neutral stability curves and critical parameters for instability are presented for a range of magnetic field strengths. Results indicate that a magnetic field directed towards the boundary wall is stabilizing, which is consistent with that found in many steady flows
The Effect of Internal and External Heating on the Free Convective Flow of a Bingham Fluid in a Vertical Porous Channel
We study the steady free convective flow of a Bingham fluid in a porous channel where heat is supplied by both differential heating of the sidewalls and by means of a uniform internal heat generation. The detailed temperature profile is governing by an external and an internal Darcy-Rayleigh number. The presence of the Bingham fluid is characterised by means of a body force threshold as given by the Rees-Bingham number. The resulting flow field may then exhibit between two and four yield surfaces depending on the balance of magnitudes of the three nondimensional parameters. Some indication is given of how the locations of the yield surfaces evolve with the relative strength of the Darcy-Rayleigh numbers and the Rees-Bingham number. Finally, parameter space is delimited into those regions within which the different types of flow and stagnation patterns arise
Stability of the boundary layer on a rotating disk for power-law fluids
The stability of the flow due to a rotating disk is considered for non-Newtonian fluids, specifically shear-thinning fluids that satisfy the power-law (Ostwald-de Waele) relationship. In this case the basic flow is not an exact solution of the Navier–Stokes equations, however, in the limit of large Reynolds number the flow inside the three-dimensional boundary layer can be determined via a similarity solution. An asymptotic analysis is presented in the limit of large Reynolds number. It is shown that the stationary spiral instabilities observed experimentally in the Newtonian case can be described for shear-thinning fluids by a linear stability analysis. Predictions for the wavenumber and wave angle of the disturbances suggest that shear-thinning fluids may have a stabilising effect on the flow
Screw dynamo in a time-dependent pipe flow
The kinematic dynamo problem is investigated for the flow of a conducting
fluid in a cylindrical, periodic tube with conducting walls. The methods used
are an eigenvalue analysis of the steady regime, and the three-dimensional
solution of the time-dependent induction equation. The configuration and
parameters considered here are close to those of a dynamo experiment planned in
Perm, which will use a torus-shaped channel. We find growth of an initial
magnetic field by more than 3 orders of magnitude. Marked field growth can be
obtained if the braking time is less than 0.2 s and only one diverter is used
in the channel. The structure of the seed field has a strong impact on the
field amplification factor. The generation properties can be improved by adding
ferromagnetic particles to the fluid in order to increase its relative
permeability,but this will not be necessary for the success of the dynamo
experiment. For higher magnetic Reynolds numbers, the nontrivial evolution of
different magnetic modes limits the value of simple `optimistic' and
`pessimistic' estimates.Comment: 10 pages, 12 figure
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