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.

Ward’s Solitons II: Exact Solutions

1998 ◽  
Vol 50 (6) ◽  
pp. 1119-1137 ◽  
Author(s):  
Christopher Kumar Anand
Keyword(s):  
Algebraic Geometry ◽  
Exact Solutions ◽  
Closed Form ◽  
Solution Space ◽  
Compact Surface ◽  
Closed Form Expression ◽  
Chiral Model ◽  
Form Expression ◽  
Holomorphic Bundles

AbstractIn a previous paper, we gave a correspondence between certain exact solutions to a (2 + 1)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions.

scholarly journals New Conditions for Obtaining the Exact Solutions of the General Riccati Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Lazhar Bougoffa
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Riccati Equation ◽  
Direct Method ◽  
Equivalent Equation ◽  
A Direct Method

We propose a direct method for solving the general Riccati equationy′=f(x)+g(x)y+h(x)y2. We first reduce it into an equivalent equation, and then we formulate the relations between the coefficients functionsf(x),g(x), andh(x)of the equation to obtain an equivalent separable equation from which the previous equation can be solved in closed form. Several examples are presented to demonstrate the efficiency of this method.

Exact Analysis of Dynamic Sliding Indentation at any Constant Speed on an Orthotropic or Transversely Isotropic Half-Space

2002 ◽  
Vol 69 (3) ◽  
pp. 340-345 ◽  
Author(s):  
L. M. Brock
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Plane Strain ◽  
Half Space ◽  
Contact Zone ◽  
Transversely Isotropic ◽  
Exact Formula ◽  
Constant Speed ◽  
Displacement Fields ◽  
Rigid Indentor

A plane-strain study of steady sliding by a smooth rigid indentor at any constant speed on a class of orthotropic or transversely isotropic half-spaces is performed. Exact solutions for the full displacement fields are constructed, and applied to the case of the generic parabolic indentor. The closed-form results obtained confirm previous observations that physically acceptable solutions arise for sliding speeds below the Rayleigh speed, for a single critical transonic speed, and for all supersonic speeds. Continuity of contact zone traction is lost for the latter two cases. Calculations for five representative materials indicate that contact zone width achieves minimum values at high, but not critical, subsonic sliding speeds. A key feature of the analysis is the factorization that gives, despite anisotropy, solution expressions that are rather simple in form. In particular, a compact function of the Rayleigh-type emerges that leads to a simple exact formula for the Rayleigh speed itself.

Analytical Modeling of Mechanical Properties and Behavior of Nanotube-Based Nanocomposites

2013 ◽  
Author(s):  
Davood Askari ◽  
Mehrdad N. Ghasemi-Nejhad ◽  
Alexander L. Kalamkarov
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Axial Strain ◽  
Stress Distributions ◽  
Closed Form Solutions ◽  
Loading Case ◽  
Analytical Formulae ◽  
Plain Strain ◽  
And Behavior ◽  
Radial Loading

The objective of this paper is to introduce analytical closed form solutions for the prediction of effective axial and transverse Young’s modulus and Poisson ratios of a matrix-filled nanotube (i.e., a representative element of nanotube reinforced nanocomposites) as well as its mechanical behavior (i.e., displacements, strains and stress distributions) when it is subjected to externally applied uniform axial and radial loads. In this work, both the nanotube and its filler material are considered to be generally cylindrical orthotopic. For the derivation of exact solutions for radial loading case, no plain strain condition is assumed and effects of axial strain is taken into consideration to obtain a more precise set of solutions. Analytical formulae are developed based on the principles of linear elasticity and continuum mechanics and then exact solutions are obtained for displacements, strains and stress distributions within the domain of each individual constituent. To validate and verify the accuracy of the closed form solutions obtained from the analytical approach, a 3-D model of a matrix-filled nanotube is generated and solved for displacements, strains and stresses, numerically, using a finite element method. Excellent agreements were achieved between the results obtained from the analytical and numerical methods.

Generalized Couette Flow of a Third-Grade Fluid with Slip: The Exact Solutions

2010 ◽  
Vol 65 (12) ◽  
pp. 1071-1076 ◽  
Author(s):  
Rahmat Ellahi ◽  
Tasawar Hayat ◽  
Fazal Mahmood Mahomed
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Closed Form ◽  
Analytical Solutions ◽  
Present Note ◽  
Nonlinear Problems ◽  
Third Grade ◽  
Third Grade Fluid ◽  
Flow Problems ◽  
Grade Fluid

The present note investigates the influence of slip on the generalized Couette flows of a third-grade fluid. Two flow problems are considered. The resulting equations and the boundary conditions are nonlinear. Analytical solutions of the governing nonlinear problems are found in closed form.

Exact Solutions to the Thermal Entry Problem for Laminar Flow Through Annular Ducts

2020 ◽  
Vol 142 (5) ◽  
Author(s):  
T. D. Bennett
Keyword(s):  
Separation Of Variables ◽  
Outer Wall ◽  
Series Solutions ◽  
Eigenvalues And Eigenfunctions ◽  
Nusselt Numbers ◽  
Wide Range ◽  
Flow Through ◽  
Laminar Forced Convection ◽  
Thermal Entrance ◽  
Annular Tube

Abstract The thermal entrance region for laminar-forced convection of a Newtonian fluid in an annular tube is solved by separation of variables using as many eigenvalues and eigenfunctions as needed to report exact results for a specified range of Graetz numbers. Results for the local and average Nusselt numbers are calculated for a wide range of inner to outer wall radius ratios and for convection to either the inner or outer wall, when the opposing wall is adiabatic. The present benchmark results are utilized to critically examine the accuracy of previous extended Lévêque series solutions that are convergent for short axial distances, and Graetz series solutions that are convergent for long axial distances, and to examine the performance of a new correlation for convection in annular tubes.

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.

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