Linearized solution of a flow over a nonuniform bottom

A linearized theory is presented for determining the shape of the free surface of a running stream which is disturbed by some irregularities lying on the bottom. The bottom is represented in integral form using Fourier's double-integral theorem. Then following Lamb [3], a linear free-surface profile is obtained for the supercritical and subcritical cases.The results are plotted for the two cases of the flow for different shapes of the bottom, and different values of the Froude number. The effect of the Froude number, the bottom height and the shape of the bottom are discussed.     

Influence of thermal load on the characteristics of a flow with evaporation

The two-layer flows of a liquid and a gas in a horizontal channel are investigated under condition of given gas flow rate. Evaporation on the thermocapillary interface is taken into account. An exact solution is constructed of the Navier–Stokes equations in the Boussinesq approximation, taking into account the Dufour effect in the gas-vapor layer. Within the framework of linear theory, the stability of the obtained solutions and the characteristics of the arising perturbations are studied. The influence is considered of the thickness of the liquid layer and the magnitude of the longitudinal temperature gradient on the structure of the basic flow and perturbations.     

Slow viscous flow through a membrane built up from porous cylindrical particles with an impermeable core

This paper concerns the slow viscous flow through an aggregate of concentric clusters of porous cylindrical particles with Happel boundary condition. An aggregate of clusters of porous cylindrical particles is considered as a hydro-dynamically equivalent to solid cylindrical core with concentric porous cylindrical shell. The Brinkman equation inside the porous cylindrical shell and the Stokes equation outside the porous cylindrical shell in their stream function formulations are used. The drag force acting on each porous cylindrical particle in a cell is evaluated. In certain limiting cases, drag force converges to pre-existing analytical results, such as, the drag on a porous circular cylinder and the drag on a solid cylinder in a Happel unit cell. Representative results are then discussed and presented in graphical forms. The hydrodynamic permeability of the membrane built up from porous particles is evaluated. The variation of hydrodynamic permeability with different parameters is graphically presented. Some new results are reported for flow pattern in the porous region. Being in resemblance with the model of colloid particles with a coating of porous layers due to adsorption phenomenon, results obtained through this model can be useful to study the membrane filtration process.     

Mathematical description of a charged-particle system near a porous membrane

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 1, pp. 114–117, January–February, 1988.     

A model for gravity driven flow of a thin liquid-solid solution with evaporation effects

A vertical substrate is coated with a thin film of a solution consisting of a volatile viscous liquid and a solid solute. The liquid film thins under gravity while the volatile component simultaneously evaporates. We develop a model to predict the evolving film thickness and in so doing we develop an approximation for the later stages of the well-known dip-coating process.Received: October 10, 2003; revised: May 4 and July 19, 2004     

A model for gravity driven flow of a thin liquid-solid solution with evaporation effects

A vertical substrate is coated with a thin film of a solution consisting of a volatile viscous liquid and a solid solute. The liquid film thins under gravity while the volatile component simultaneously evaporates. We develop a model to predict the evolving film thickness and in so doing we develop an approximation for the later stages of the well-known dip-coating process.     

Modeling filmwise condensation of vapor on curvilinear fins with condensate suction from interfin grooves

The nonstationary process of filmwise condensation of vapor on curvilinear fins with condensate suction from interfin grooves is numerically simulated with account taken of surface tension and gravity. The problem is reduced to solving a nonlinear evolution equation for the thickness of the condensate film. We performed calculations of the ethanol vapor condensation at atmospheric pressure on the fins of constant curvature for various temperature differences between the fin surface and the vapor saturation and for various values of the rate of condensate suction from the interfin space. Numerical calculations show that the condensation process is stable in the device (i.e., in the condenser) with condensate suction. Filling the interfin space leads to diminishing the zone of intense condensation and reducing the condensate inflow; therefore, this yields stable equilibrium between the condensation and condensate suction. The changes of the condenser temperature at a constant condensate suction entail variation of the filling level of the interfin groove and establishment of a stationary process, provided that the fin temperature becomes constant.     

Non-isothermal fluid flow through a thin pipe with cooling

The stationary flow of a Boussinesquian fluid with temperature-dependent viscosity through a thin straight pipe is considered. The fluid in the pipe is cooled by the exterior medium. The asymptotic approximation of the solution is built and rigorously justified by proving the error estimate in terms of domain thickness. The boundary layers for the temperature at the ends of the pipe are studied.     

Optimization of the flow through networks with gains

The optimal flow problem in networks with gains is presented through the simplex method. Out of simple theorical conditions, a method is built which needs only a relatively small number memory and quite a short calculation time by computer. Large examples are given; e.g., one test-example of 1000 nodes and 3000 arcs, and one real problem leading to a linear program of 3000 constraints and 8000 arcs.     

Analysis of blood flow through a modelled artery with an aneurysm

The intention of the present work is to carry out a systematic analysis of flow features in a tube, modelled as artery, having a local aneurysm in presence of haematocrit. The arterial model is treated to be axi-symmetric and rigid. The blood, flowing through the modelled artery, is treated to be Newtonian and non-homogeneous. For a thorough quantitative analysis of the flow characteristics such as wall pressure, flow velocity, wall shear stress, the unsteady incompressible Navier-Stokes equations in cylindrical polar co-ordinates under the laminar flow conditions are solved by using the finite-difference method. Finally, the numerical illustrations presented in this paper provide an effective measure to estimate the combined influence of haematocrit and aneurysm on flow characteristics. It is found that the magnitude of wall shear stress and also the length of separation increase with increasing values of the haematocrit parameter. The length of flow separation increases but the peak value of wall shear stress decreases with the increasing length of aneurysm. The peak value of wall shear stress as well as the length of separation increases with the increasing height of the aneurysm.     

Transformation of coordinates with grid condensation toward the curvilinear boundary of the exterior flow region and boundary rectification

We consider a transformation of coordinates that maps the infinite region above the curve y=f(x) (f(x) 0) onto the entire upper halfplane. The original region is interpreted as an exterior flow region, and the curve described by the function y=f(x) as the flow boundary. The coordinate transformation rectifies the curvilinear boundary and can be applied for numerical solution of the equations of motion of a viscous fluid in a region with a curvilinear boundary of arbitrary shape.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 42–47, 1992.     

On the flow of a cosserat fluid through a curved pipe

Summary A theoretical analysis is made of the flow of a Cosserat fluid in a curved pipe under a pressure gradient. It is assumed that the curvature of the pipe is small, that is the radius of the circle in which the central line of the pipe is coiled is large in comparison with the radius of the cross-section. Following Dean [2] a solution is developed by the method of successive approximation. The paths of the particles in the central plane and the projection of the streamlines on the cross-section of the pipe are compared with those of a Newtonian fluid. It is observed that in the theory of Cosserat fluids the curvature of the streamlines in the central plane increases and the motion is slower in the cross-section of the pipe. It is also shown that the rate of flow of a Cosserat fluid through a curved pipe is decreased due to the curvature of the pipe.

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Résumé On fait une analyse théorique de l'écoulement d'un fluide de Cosserat dans un tube sous un gradient de pression. On suppose que la courbature du tube est faible, c'est-à-dire que le radius du cercle qui fait la ligne du centre du tube est fort par rapport au radius de la coupe transversale.D'après Dean [2], on développe une résolution par approximations successives. On fait la comparaison des trajectoires des particules dans le plan central et la projection des lignes d'écoulement sur la coupe transversale du tube avec celles d'un fluide de Newton.On note que dans la théorie des fluides de Cosserat, la courbature des lignes d'écoulement dans le plan central augmente, et que la motion est ralentie dans la coupe transversale du tube. On démontre ensuite que le taux d'écoulement d'un fluide de Cosserat dans un tube courbe se diminue à raison de la courbature.

     

Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is also decoupled in computation, whichmeans that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in implementation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L^∞- norm, and for N in the discrete L^2-norm, respectively, where U and N are the numerical solutions of the KGZ equation. Numerical results verify the theoretical analysis.     

On the stability and uniqueness of the flow of a fluid through a porous medium

In this short note, we study the stability of flows of a fluid through porous media that satisfies a generalization of Brinkman’s equation to include inertial effects. Such flows could have relevance to enhanced oil recovery and also to the flow of dense liquids through porous media. In any event, one cannot ignore the fact that flows through porous media are inherently unsteady, and thus, at least a part of the inertial term needs to be retained in many situations. We study the stability of the rest state and find it to be asymptotically stable. Next, we study the stability of a base flow and find that the flow is asymptotically stable, provided the base flow is sufficiently slow. Finally, we establish results concerning the uniqueness of the flow under appropriate conditions, and present some corresponding numerical results.     

The Point Spectrum of the Linearized Boltzmann Operator with the External-force Term in a Bounded Domain

We will investigate the point spectrum on the imaginary axis of the linearized Boltzmann operator with an external-force potential in a bounded domain whose boundary is sufficiently smooth. The boundary condition considered is the perfectly reflective boundary condition. The point spectrum on the imaginary axis is only equal to {0}. However, the null space varies with the common axes of symmetry of the domain and the external-force potential.     

Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel

The paper presents the transportation of viscoelastic fluid with fractional Maxwell model by peristalsis through a channel under long wavelength and low Reynolds number approximations. The propagation of wall of channel is taken as sinusoidal wave propagation (contraction and relaxation). Homotopy perturbation method (HPM) and Adomian decomposition method (ADM) are used to obtain the analytical approximate solutions of the problem. The expressions of axial velocity, volume flow rate and pressure gradient are obtained. The effects of fractional parameters (α), relaxation time (λ₁) and amplitude (?) on the pressure difference and friction force across one wavelength are calculated numerically for different particular cases and depicted through graphs.