Brief Review of Exact Solutions for Slip-Flow in Ducts and Channels

2012 ◽  
Vol 134 (9) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Exact Solutions ◽  
Slip Flow ◽  
Microchannel Flow ◽  
Approximate Methods

Slip flow in ducts is important in numerous contemporary applications, especially microchannel flow. This Note reviews the existing exact solutions for slip flow. These solutions serve as accuracy standards for approximate methods including numerical or semi-numerical means. Some new solutions are also introduced.

scholarly journals XXIII.—On Systems of Solutions of Homogeneous and Central Equations of the nth Degree and of two or more Variables; with a Discussion of the Loci of such Equations

1890 ◽  
Vol 35 (4) ◽  
pp. 1043-1098
Author(s):  
M'Laren
Keyword(s):  
Exact Solutions ◽  
Plane Curves ◽  
Algebraic Equations ◽  
Linear Variation ◽  
Approximate Methods ◽  
Contour Lines

The purpose of the present paper is to ascertain how far it is possible to find exact solutions or values of x, y, &c., in equations between variables, so that the forms of plane curves and contour-lines of surfaces may be exactly determined. No approximate methods have been admitted, and only those methods have been used which are applicable to algebraic equations of every degree and any number of variables. In the examples given I have generally selected equations of even degree and even powers of the variables. But every such solution evidently includes the solution of the non-central equation of half the degree having corresponding terms and equal coefficients. The methods of solution employed are founded on the following introductory theorem or principle, which may be described as that of Homogeneous or Linear Variation of the quantities.

scholarly journals Symmetry reduction a promising method for heat conduction equations

2019 ◽  
Vol 23 (4) ◽  
pp. 2219-2227
Author(s):  
Yi Tian
Keyword(s):  
Heat Conduction ◽  
Exact Solutions ◽  
Wave Equations ◽  
Reduction Method ◽  
Variational Iteration Method ◽  
Mathematical Tool ◽  
Approximate Methods ◽  
Similarity Reduction ◽  
Homotopy Perturbation ◽  
Powerful Mathematical Tool

Though there are many approximate methods, e. g., the variational iteration method and the homotopy perturbation, for non-linear heat conduction equations, exact solutions are needed in optimizing the heat problems. Here we show that the Lie symmetry and the similarity reduction provide a powerful mathematical tool to searching for the needed exact solutions. Lie algorithm is used to obtain the symmetry of the heat conduction equations and wave equations, then the studied equations are reduced by the similarity reduction method.

scholarly journals Approximate and Exact Solutions to the Singular Nonlinear Heat Equation with a Common Type of Nonlinearity

2020 ◽  
Vol 34 ◽  
pp. 18-34
Author(s):  
A. L. Kazakov ◽  
◽  
L. F. Spevak ◽  
Keyword(s):  
Heat Equation ◽  
Exact Solutions ◽  
Wave Front ◽  
Heat Wave ◽  
Small Neighborhood ◽  
Parabolic Type ◽  
Nonlinear Heat Equation ◽  
Time Interval ◽  
Qualitative Properties ◽  
Approximate Methods

The paper deals with the problem of the motion of a heat wave with a specified front for a general nonlinear parabolic heat equation. An unknown function depends on two variables. Along the heat wave front, the coefficient of thermal conductivity and the source function vanish, which leads to a degeneration of the parabolic type of the equation. This circumstance is the mathematical reason for the appearance of the considered solutions, which describe perturbations propagating along the zero background with a finite velocity. Such effects are generally atypical for parabolic equations. Previously, we proved the existence and uniqueness theorem for the problem considered in this paper. Still, it is local and does not allow us to study the properties of the solution beyond the small neighborhood of the heat wave front. To overcome this problem, the article proposes an iterative method for constructing an approximate solution for a given time interval, based on the boundary element approach. Since it is usually not possible to prove strict convergence theorems of approximate methods for nonlinear equations of mathematical physics with a singularity, verification of the calculation results is relevant. One of the traditional ways is to compare them with exact solutions. In this article, we obtain and study an exact solution of the required type, the construction of which is reduced to integrating the Cauchy problem for an ODE. We obtained some qualitative properties, including an interval estimation of the wave amplitude in one particular case. The performed calculations show the effectiveness of the developed computational algorithm, as well as the compliance of the results of calculations with qualitative analysis.

Torsion and Flexure of Slender Solid Sections

1958 ◽  
Vol 25 (1) ◽  
pp. 115-121
Author(s):  
W. J. Carter
Keyword(s):  
Exact Solutions ◽  
Turbine Blades ◽  
Torsion Problem ◽  
Rectangular Section ◽  
Approximate Methods ◽  
Elastic Plastic ◽  
Shear Problem

Abstract The solution of the torsion problem for a slender rectangular section has been made previously by approximate methods based on the Prandtl membrane analogy. In this paper approximate methods are employed in the solution of both the torsion and flexural shear problem for slender sections having a variety of shapes, most of them being doubly symmetric. Solutions obtained in this manner are compared with exact solutions, when these are available, and otherwise with solutions obtained by relaxation. It is shown that approximate methods provide an adequate solution for elements such as compressor-turbine blades when pretwist and taper can be neglected. Some attention is given to the problem of elastic-plastic torsion and elastic-plastic flexural shear of slender sections.

Effects of Axial Corrugated Roughness on Low Reynolds Number Slip Flow and Continuum Flow in Microtubes

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Zhipeng Duan ◽  
Y. S. Muzychka
Keyword(s):  
Pressure Drop ◽  
Slip Flow ◽  
Velocity Slip ◽  
Analytical Models ◽  
Axial Distance ◽  
Microchannel Flow ◽  
Compressibility Effect ◽  
Continuum Flow ◽  
Experimental Pressure ◽  
The Stokes Equation

The effect of axial corrugated surface roughness on fully developed laminar flow in microtubes is investigated. The radius of a microtube varies with the axial distance due to corrugated roughness. The Stokes equation is solved using a perturbation method with slip at the boundary. Analytical models are developed to predict friction factor and pressure drop in corrugated rough microtubes for continuum flow and slip flow. The developed model proposes an explanation on the observed phenomenon that some experimental pressure drop results for microchannel flow have shown a significant increase due to roughness. The developed model for slip flow illustrates the coupled effects between velocity slip and small corrugated roughness. Compressibility effect has also been examined and simple models are proposed to predict the pressure distribution and mass flow rate for slip flow in corrugated rough microtubes.

scholarly journals Exact solutions for hydrodynamic interactions of two squirming spheres

2017 ◽  
Vol 813 ◽  
pp. 618-646 ◽  
Author(s):  
Dario Papavassiliou ◽  
Gareth P. Alexander
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Slip Surface ◽  
Rotational Velocity ◽  
Near Field ◽  
Three Dimensions ◽  
Hydrodynamic Interactions ◽  
Reciprocal Theorem ◽  
Approximate Methods ◽  
Stokes Drag

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or with each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point-singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head–tail reversal of the swimmer, in the near field, microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of approximately two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.

Exact Solutions for Starting and Oscillatory Flows in an Equilateral Triangular Duct

2016 ◽  
Vol 138 (8) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Helmholtz Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Oscillatory Flow ◽  
Series Solutions ◽  
Eigenvalues And Eigenfunctions ◽  
Approximate Methods ◽  
Oscillatory Flows ◽  
Complete Set ◽  
Starting Flow

Exact series solutions, some in closed-form, for starting flow and oscillatory flow in an equilateral triangular duct are presented. The complete set of eigenvalues and eigenfunctions of the Helmholtz equation is derived, and the method of eigenfunction superposition is used. Exact solutions are rare, fundamental, and serve as accuracy standards for approximate methods.

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