Closed-form exact solutions of MHD viscous flow over a shrinking sheet

2009 ◽  
Vol 14 (7) ◽  
pp. 2853-2857 ◽  
Author(s):  
Tiegang Fang ◽  
Ji Zhang
Keyword(s):  
Exact Solutions ◽  
Viscous Flow ◽  
Closed Form ◽  
Shrinking Sheet

On the Approximate Solution of Viscous-Flow Problems

1963 ◽  
Vol 30 (2) ◽  
pp. 263-268 ◽  
Author(s):  
J. A. Schetz
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Viscous Flow ◽  
Closed Form ◽  
New Method ◽  
Two Dimensional ◽  
General Technique ◽  
Closed Form Solutions ◽  
Flow Problems

The need for a general technique for the approximate solution of viscous-flow problems is discussed. Existing methods are considered and a new method is presented which results in simple closed-form solutions. The accuracy of the method is demonstrated by comparisons with the results of known exact solutions, and finally the general technique is employed to determine a new solution for the fully expanded two-dimensional laminar nozzle problem.

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.

scholarly journals Closed-Form Non-Stationary Solutionsfor Thermo and Chemovibrational Viscous Flows

Fluids ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 175 ◽  
Author(s):  
Dmitry Bratsun ◽  
Vladimir Vyatkin
Keyword(s):  
Viscous Flow ◽  
Closed Form ◽  
General Procedure ◽  
Fluid Velocity ◽  
Fluid Motion ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Chemical Transformations ◽  
Navier Stokes Equation ◽  
Harmonic Vibrations

A class of closed-form exact solutions for the Navier–Stokes equation written in the Boussinesq approximation is discussed. Solutions describe the motion of a non-homogeneous reacting fluid subjected to harmonic vibrations of low or finite frequency. Inhomogeneity of the medium arises due to the transversal density gradient which appears as a result of the exothermicity and chemical transformations due to a reaction. Ultimately, the physical mechanism of fluid motion is the unequal effect of a variable inertial field on laminar sublayers of different densities. We derive the solutions for several problems for thermo- and chemovibrational convections including the viscous flow of heat-generating fluid either in a plain layer or in a closed pipe and the viscous flow of fluid reacting according to a first-order chemical scheme under harmonic vibrations. Closed-form analytical expressions for fluid velocity, pressure, temperature, and reagent concentration are derived for each case. A general procedure to derive the exact solution is discussed.

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.

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