Three-dimensional motion of a viscous incompressible fluid in a narrow tube

An approximate solution of the problem of unsteady motion of a viscous incompressible fluid in a long narrow deformable tube at low Reynolds numbers is obtained. Pressure oscillations and tube deformation are shown to be related by an integrodifferential equation. The solution obtained extends the Poiseuille solution in elliptic tubes to the case of comparatively arbitrary small deformations in terms of the tube length and angle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 28–32, July–August, 2009.     

Some properties of three-dimensional Biltrami flows

The investigation of Beltrami flows is important for the research on the mechanism of turbulent structure. In this paper the general solutions of the Beltrami flows are given, which depend explicitly on the solutions of three independent Helmholtz equations with scalar unknowns. Velocity fields of Beltrami flows can then be obtained explicitly after the application of some curl operations on the solutions of Helmholtz equations. On the basis of the exact solutions of Euler equations given above, we obtain one kind of exact solutions of non-steady Navier-Stokes equations which are also the Beltrami flows. Some interesting examples of Beltrami flows other than “ABC flows”, “Kolmogolov flows”, “Rayleigh-Bernard flows”, “Q-flows” are given. The detailed analytic results of these examples will be published in the near future.     

Asymptotic Solution for a Kind of Boundary Layer Problem

This paper studies the partial differential equation with a small coefficient in the highest-order item. This kind of equation is also named as boundary layer problem. The Burgers equation and modified Burgers equation are analyzed in this approach. First, these equations are transferred into the strong nonlinear ones, and then the corresponding strong nonlinear equations are solved based on the perturbation method. The results from the asymptotic method are comparable with those obtained from numerical computation. An erratum to this article is available at .     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

Ｔｈｅａｕｔｈｏｒｓｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1 ] -[7] .Ｎｏｗｗｅｒａｉｓｅａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｏｎａｐａｒｔｏｆｄｏｍａｉｎ .Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ　　ｕｔ-λε(ｘ) ( μ(ｕ)ｕｘ) ｘ Ｋｘ(ｕ) ｆ(ｘ ,ｔ,ｕ) =0 ,(ｔ,ｘ) ∈ ( 0 ,Ｔ)× ( ( 0 ,α) ∪ (…     

Blow-up of solution for a generalized Boussinesq equation

This paper studies the initial boundary value problem for a generalized Boussinesq equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method.Moreover,it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method.     

To construct a vector field with given curl function and divergence function

Two conclusions have been achieved in this paper. Firstly, a formal solution of the equations , has been derived with different point of view from commonly known classical method developed by Helmholtz(1), (2), (3).Secondly, a method to construct a vector field with given curl function and divergence function has been given in terms of the above solution.     

Solvability of a class of second-order quasilinear boundary value problems

The second-order quasilinear boundary value problems are considered when the nonlinear term is singular and the limit growth function at the infinite exists. With the introduction of the height function of the nonlinear term on a bounded set and the consideration of the integration of the height function, the existence of the solution is proven. The existence theorem shows that the problem has a solution if the integration of the limit growth function has an appropriate value.     

On an Exact Analytical Solution of the Boussinesq Equation

A useful exact analytical solution of the Boussinesq equation is discussed and is the most general solution presently available, and in particular yields a solution for a finite aquifer. It provides insight into the physical processes arising during the exchange of water between an aquifer and a free body of water of varying height as an application and extension of Barenblatt's solution. We also illustrate the value of such a solution to check numerical and approximate schemes.     

A new method for obtaining the exact solutions of the Navier-Stokes (NS) equations and its applications

In this paper we propose a new method for obtaining the exact solutions of the Mavier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author^[1,2], Using the present method we can find some new exact solutions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a variety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates. The project supported by National Natural Science Foundation of China.     

Optimum shape of a cavity for the collection of a soil-saturating viscous fluid

The process of inflow of an oil-bearing fluid from an infinite region of porous soil into a cavity (reservoir) and its subsequent pumping into a well located at the center of the reservoir is investigated. The cavity is chosen from the class of flattened ellipsoids of revolution. Thus, a combined problem of the optimization of the shape of a cavity of given volume for which the maximum seepage flow is achieved and the approximate determination of the additional constraints on the cavity dimensions necessary to ensure the outflow of the oil-bearing fluid is considered.     

Similarity solutions for non-Newtonian power-law fluid flow

The problem of the boundary layer flow of power law non-Newtonian fluids with a novel boundary condition is studied.The existence and uniqueness of the solutions are examined,which are found to depend on the curvature of the solutions for different values of the power law index n.It is established with the aid of the Picard-Lindel¨of theorem that the nonlinear boundary value problem has a unique solution in the global domain for all values of the power law index n but with certain conditions on the curvature of the solutions.This is done after a suitable transformation of the dependent and independent variables.For 0 n 1,the solution has a positive curvature,while for n 1,the solution has a negative or zero curvature on some part of the global domain.Some solutions are presented graphically to illustrate the results and the behaviors of the solutions.     

A new exact solution for the flow of a Maxwell fluid past an infinite plate

A new exact solution corresponding to the flow of a Maxwell fluid over a suddenly moved flat plate is determined. This solution is in all accordance with a previous one and for λ→0 it goes to the well-known solution for Navier-Stokes fluids.     

Oscillation of solutions for a class of nonlinear neutral parabolic differential equations boundary value problem

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｅｔｏｔｈｅａｃｃｕｒｅｎｃｅｏｆａｇｒｅａｔｄｅａｌｏｆｍａｔｈｅｍａｔｉｃｍｏｄｅｌｌｉｎｇｓｄｅｓｃｒｉｂｅｄｉｎｐａｒｔｉａｌｆｕｎｃｔｉｏｎａｌｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｉｎｔｈｅａｐｐｌｉｃａｔｉｏｎｆｉｅｌｄｓｏｆｍｅｃｈａｎｉｃｓａｎｄｃｏｎｔｒｏｌｔｈｅｏｒｙ[1],ｇｒｅａｔｉｎｔｅｒｅｓｔｗａｓａｒｏｕｓｅｄａｓｔｏｔｈｅｓｔｕｄｉｅｓｏｎｔｈｅｐｒｏｐｅｒｔｙｏｆｔｈｅｓｏｌｕｔｉｏｎｓｏｆｔｈｉｓｔｙｐｅｏｆｅｑｕａｔｉｏｎｓ.Ｇ…     

WELL-POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INVISCID COMPRESSIBLE ADIABATIC FLUID ＊

Introduction Itisreasonabletoconsiderarealfluidasanidealoneinfluidmechanicsundermany conditions.Forinstance,ingeneral,forthedistributionofthefluidfieldaroundtheaerocraft,mostpartofthefluidfieldmayberegardedasidealfluidexceptforasmallpartwhereeffects ofviscousandheatconductioninthethinlayernearthesurfacemustbeconsidered.Evenif thefluidiscompletelysupposedasidealonethroughoutthefluidfield,thequitereasonable resultsarealsogained,thereforestudyingidealfluidhasnotonlytheoreticalsignificancebut also…     

The generation of three-dimensional internal waves and attendant boundary layers in a viscous continuously stratified fluid. Construction of an analytical solution

An analytical solution of a linearized problem of the emission of periodic internal waves by part of a plane which oscillates with a small amplitude in an arbitrary direction in a viscous exponentially stratified fluid is constructed. Solutions of the dispersion equation are given for all positions of the emitting surface (arbitrary, vertical, horizontal, and critical when one of the beam propagation directions is collinear with the emitting surface). The possibility of transition to the case of a uniform fluid, which is important for applications, is analyzed.     

Existence of boundary value problems for TFD equation in quantum mechanics

ＩｎｔｒｏｄｕｃｔｉｏｎＴｏｓｔｕｄｙｐｏｔｅｎｔｉａｌｓａｎｄｃｈａｒｇｅｄｅｎｓｉｔｉｅｓｏｆａｔｏｍｓｉｎｑｕａｎｔｕｍｍｅｃｈａｎｉｃｓ,ｉｔｉｓｓｕｍｍｅｄｕｐａｓｓｏｌｖｉｎｇＳｃｈｒｄｉｎｇｅｒｅｑｕａｔｉｏｎｗｉｔｈｗａｖｅｆｕｎｃｔｉｏｎψ(ｒ):-122 Ｖ(ｒ)ψ(ｒ)=Ｅψ(ｒ),(1)｜ψ(0)｜<Ｍ,｜ψ(ｒ)｜｜ｒ｜→∞=0(2)ａｎｄｔｈｅｃｒｕｘｏｆｔｈｅｍａｔｔｅｒｉｓｇｏｉｎｇｔｏｄｅｔｅｒｍｉｎｅＶ(ｒ).Ｔｈｕｓ,ｔｈｅｋｎｏｗｎＴｈｏｍａｓ＿Ｆｅｒｍｉｅｑｕａｔｉｏｎ,ｏｂｔａｉｎｅｄｉ…     

EXISTENCE THEOREMS OF SOLUTIONS FOR TWO－POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS IN BANACH SPACES

ＥＸＩＳＴＥＮＣＥＴＨＥＯＲＥＭＳＯＦＳＯＬＵＴＩＯＮＳＦＯＲＴＷＯ－ＰＯＩＮＴＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＳＥＣＯＮＤＯＲＤＥＲＯＲＤＩＮＡＲＹＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＩＮＢＡＮＡＣＨＳＰＡＣＥＳＺｈａｎｇＳｈｉｓ...     

A necessary and sufficient set of boundary integral equations with indirect unknowns for plate bending problem

A boundary integral representation of plane biharmonic function is established rigorously by the method of unanalytical continuation in the present paper. In this representation there are two boundary functions and four constants which bear a one to one correspondence to biharmonic functions. Therefore the set of boundary integral equations with indirect unknowns based on this representation is equivalent to the original differential equation formulation.     

Kinematically Excited Stokes Flow in a Right-Angle Wedge (Plane Problem)

An exact solution is obtained for a Stokes flow in a right-angle wedge. The flow is kinematically excited at the bottom edge and discontinues at the corner point __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 60–69, September 2005.     

Transient flow towards a well in an aquifer including the effect of fluid inertia

Transient propagation of weak pressure perturbations in a homogeneous, isotropic, fluid saturated aquifer has been studied. A damped wave equation for the pressure in the aquifer is derived using the macroscopic, volume averaged, mass conservation and momentum equations. The equation is applied to the case of a well in a closed aquifer and analytical solutions are obtained to two different flow cases. It is shown that the radius of influence propagates with a finite velocity. The results show that the effect of fluid inertia could be of importance where transient flow in porous media is studied.List of symbols b Thickness of the aquifer, m - c ₀ Wave velocity, m/s - k Permeability of the porous medium, m² - n Porosity of the porous medium - p( ,t) Pressure, N/m² - Q Volume flux, m³/s - r Radial coordinate, m - r _w Radius of the well, m - s Transform variable - S Storativity of the aquifer - S _d(r, t) Drawdown, m - t Time, s - T Transmissivity of the aquifer, m²/s - ( ,t) Velocity of the fluid, m/s - Coordinate vector, m - z Vertical coordinate, m - Coefficient of compressibility, m²/N - Coefficient of fluid compressibility, m²/N - Relaxation time, s - (r, t) Hydraulic potential, m - Dynamic viscosity of the fluid, Ns/m² - Dimensionless radius - Density of the fluid, Ns²/m⁴ - (, ) Dimensionless drawdown - Dimensionless time - , x Dummy variables - ₀, ₁ Auxilary functions