7,227 research outputs found
Thin Film Motion of an Ideal Fluid on the Rotating Cylinder Surface
The shallow water equations describing the motion of thin liquid film on the
rotating cylinder surface are obtained. These equations are the analog of the
modified Boussinesq equations for shallow water and the Korteweg-de Vries
equation. It is clear that for rotating cylinder the centrifugal force plays
the role of the gravity. For construction the shallow water equations
(amplitude equations) usual depth-averaged and multi-scale asymptotic expansion
methods are used. Preliminary analysis shows that a thin film of an ideal
incompressible fluid precesses around the axis of the cylinder with velocity
which differs from the angular velocity of rotating cylinder. For the
mathematical model of the liquid film motion the analytical solutions are
obtained by the Tanh-Function method. To illustrate the integrability of the
equations the Painleve analysis is used. The truncated expansion method and
symbolic computation allows to present an auto-Backlund transformation. The
results of analysis show that the exact solutions of the model correspond to
the solitary waves of different types.Comment: 10 pages, 9 figures. arXiv admin note: text overlap with
arXiv:nlin/0311028 by other author
Hodograph Method and Numerical Integration of Two Hyperbolic Quasilinear Equations. Part I. The Shallow Water Equations
In paper [S.I. Senashov, A. Yakhno. 2012. SIGMA. Vol.8. 071] the variant of
the hodograph method based on the conservation laws for two hyperbolic
quasilinear equations of the first order is described. Using these results we
propose a method which allows to reduce the Cauchy problem for the two
quasilinear PDE's to the Cauchy problem for ODE's. The proposed method is
actually some similar method of characteristics for a system of two hyperbolic
quasilinear equations. The method can be used effectively in all cases, when
the linear hyperbolic equation in partial derivatives of the second order with
variable coefficients, resulting from the application of the hodograph method,
has an explicit expression for the Riemann-Green function. One of the method's
features is the possibility to construct a multi-valued solutions. In this
paper we present examples of method application for solving the classical
shallow water equations.Comment: 19 pages, 5 figure
Low-order models of 2D fluid flow in annulus
The two-dimensional flow of viscous incompressible fluid in the domain
between two concentric circles is investigated numerically. To solve the
problem, the low-order Galerkin models are used. When the inner circle rotates
fast enough, two axially asymmetric flow regimes are observed. Both regimes are
the stationary flows precessing in azimuthal direction. First flow represents
the region of concentrated vorticity. Another flow is the jet-like structure
similar to one discovered earlier in Vladimirov's experiments.Comment: 12 pages, 15 figure
Hodograph Method and Numerical Solution of the Two Hyperbolic Quasilinear Equations. Part III. Two-Beam Reduction of the Dense Soliton Gas Equations
The paper presents the solutions for the two-beam reduction of the dense
soliton gas equations (or Born-Infeld equation) obtained by analytical and
numerical methods. The method proposed by the authors is used. This method
allows to reduce the Cauchy problem for two hyperbolic quasilinear PDEs to the
Cauchy problem for ODEs. In some respect, this method is analogous to the
method of characteristics for two hyperbolic equations. The method is
effectively applicable in all cases when the explicit expression for the
Riemann-Green function for some linear second order PDE, resulting from the use
of the hodograph method for the original equations, is known. The numerical
results for the two-beam reduction of the dense soliton gas equations, and the
shallow water equations (omitting in the previous papers) are presented. For
computing we use the different initial data (periodic, wave packet).Comment: 22 pages, 11 figures. arXiv admin note: substantial text overlap with
arXiv:1503.0176
Mathematical Model of a pH-gradient Creation at Isoelectrofocusing. Part IV. Theory
The mathematical model describing the non-stationary natural pH-gradient
arising under the action of an electric field in an aqueous solution of
ampholytes (amino acids) is constructed. The model is a part of a more general
model of the isoelectrofocusing (IEF) process. The presented model takes into
account: 1) general Ohm's law (electric current flux includes the diffusive
electric current); 2) dissociation of water; 3) difference between isoelectric
point (IEP) and isoionic point (PZC -- point of zero charge). We also study the
Kohlraush's function evolution and discuss the role of the Poisson-Boltzmann
equation.Comment: 15 pages, 1 figur
Interactions between discontinuities for binary mixture separation problem and hodograph method
The Cauchy problem for first-order PDE with the initial data which have a
piecewise discontinuities localized in different spatial points is completely
solved. The interactions between discontinuities arising after breakup of
initial discontinuities are studied with the help of the hodograph method. The
solution is constructed in analytical implicit form. To recovery the explicit
form of solution we propose the transformation of the PDEs into some ODEs on
the level lines (isochrones) of implicit solution. In particular, this method
allows us to solve the Goursat problem with initial data on characteristics.
The paper describes a specific problem for zone electrophoresis (method of the
mixture separation). However, the method proposed allows to solve any system of
two first-order quasilinear PDEs for which the second order linear PDE, arising
after the hodograph transformation, has the Riemann-Green function in explicit
form.Comment: 19 pages, 11 figure
Anomalous pH-gradient in Ampholyte Solution
A mathematical model describing a steady pH-gradient in the solution of
ampholytes in water has been studied with the use of analytical, asymptotic,
and numerical methods. We show that at the large values of an electric current
a concentration distribution takes the form of a piecewise constant function
that is drastically different from a classical Gaussian form. The correspondent
pH-gradient takes a stepwise form, instead of being a linear function. A
discovered anomalous pH-gradient can crucially affect the understanding of an
isoelectric focusing process.Comment: 5 pages, 2 figure
Mathematical Model of a pH-gradient Creation at Isoelectrofocusing. Part II. Numerical Solution of the Stationary Problem
The mathematical model describing the natural textrm{pH}-gradient arising
under the action of an electric field in an aqueous solution of ampholytes
(amino acids) is constructed and investigated. This paper is the second part of
the series papers \cite{Part1,Part3,Part4} that are devoted to pH-gradient
creation problem. We present the numerical solution of the stationary problem.
The equations system has a small parameter at higher derivatives and the
turning points, so called stiff problem. To solve this problem numerically we
use the shooting method: transformation of the boundary value problem to the
Cauchy problem. At large voltage or electric current density we compare the
numerical solution with weak solution presented in Part 1.Comment: 14 pages, 8 figure
Rotating electrohydrodynamic flow in a suspended liquid film
The mathematical model of a rotating electrohydrodynamic flow in a thin
suspended liquid film is proposed and studied. The motion is driven by the
given difference of potentials in one direction and constant external
electrical field \vE_\text{out} in another direction in the plane of a film.
To derive the model we employ the spatial averaging over the normal coordinate
to a film that leads to the average Reynolds stress that is proportional to
|\vE_\text{out}|^3. This stress generates tangential velocity in the vicinity
of the edges of a film that, in turn, causes the rotational motion of a liquid.
The proposed model is aimed to explain the experimental observations of the
\emph{liquid film motor} (see arXiv:0805.0490v2).Comment: 12 pages, 9 figures. (Submitted to Phys. Rev. E
Modeling of zonal electrophoresis in plane channel of complex shape
The zonal electrophoresis in the channels of complex forms is considered
mathematically with the use of computations. We show that for plane S-type
rectangular channels stagnation regions can appear that cause the strong
variations of the spatial distribution of an admixture. Besides, the shape of
an admixture zone is strongly influenced by the effects of electromigration and
by a convective mixing. Taking into account the zone spreading caused by
electromigration, the influence of vertex points of cannel walls, and
convection would explain the results of electrophoretic experiments, which are
difficult to understand otherwise.Comment: 13 pages, 10 figure
- …