A group invariant solution for a pre-existing fluid-driven fracture in permeable rock

2011 ◽  
Vol 12 (1) ◽  
pp. 767-779 ◽  
Author(s):  
A.G. Fareo ◽  
D.P. Mason
Keyword(s):  
Invariant Solution ◽  
Group Invariant ◽  
Permeable Rock

Group invariant solution for a pre-existing fracture driven by a power-law fluid in permeable rock

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640010 ◽  
Author(s):  
A. G. Fareo ◽  
D. P. Mason
Keyword(s):  
Newtonian Fluid ◽  
Power Law ◽  
Numerical Solutions ◽  
Invariant Solution ◽  
Two Dimensional ◽  
Initial Length ◽  
Power Law Fluid ◽  
Group Invariant ◽  
Permeable Rock ◽  
Fluid Leak

Group invariant analytical and numerical solutions for the evolution of a two-dimensional fracture with nonzero initial length in permeable rock and driven by an incompressible non-Newtonian fluid of power-law rheology are obtained. The effect of fluid leak-off on the evolution of the power-law fluid fracture is investigated.

scholarly journals Symmetry Reduction, Exact Solutions, and Conservation Laws of the ZK-BBM Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Wenbin Zhang ◽  
Jiangbo Zhou ◽  
Sunil Kumar
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Similarity Reduction ◽  
Group Invariant ◽  
New Exact Solutions ◽  
Reduction Group ◽  
The Given ◽  
Bbm Equation

Employing the classical Lie method, we obtain the symmetries of the ZK-BBM equation. Applying the given Lie symmetry, we obtain the similarity reduction, group invariant solution, and new exact solutions. We also obtain the conservation laws of ZK-BBM equation of the corresponding Lie symmetry.

scholarly journals Group-Invariant Solutions for Two-Dimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
R. Naz
Keyword(s):  
Fluid Velocity ◽  
Invariant Solution ◽  
Jet Flows ◽  
Order Partial Differential Equation ◽  
Free Wall ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Third Order ◽  
Group Invariant ◽  
Group Invariant Solutions

The group-invariant solutions for nonlinear third-order partial differential equation (PDE) governing flow in two-dimensional jets (free, wall, and liquid) having finite fluid velocity at orifice are constructed. The symmetry associated with the conserved vector that was used to derive the conserved quantity for the jets (free, wall, and liquid) generated the group invariant solution for the nonlinear third-order PDE for the stream function. The comparison between results for two-dimensional jet flows having finite and infinite fluid velocity at orifice is presented. The general form of the group invariant solution for two-dimensional jets is given explicitly.

Nonclassical Symmetry Analysis of Heated Two-Dimensional Flow Problems

2015 ◽  
Vol 70 (12) ◽  
pp. 1031-1037
Author(s):  
Imran Naeem ◽  
Rehana Naz ◽  
Muhammad Danish Khan
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Invariant Solution ◽  
Heat Transfer Analysis ◽  
Two Dimensional ◽  
Group Invariant ◽  
Boundary Layer Equations ◽  
Flow Problems ◽  
Nonclassical Symmetry ◽  
Nonclassical Symmetries

AbstractThis article analyses the nonclassical symmetries and group invariant solution of boundary layer equations for two-dimensional heated flows. First, we derive the nonclassical symmetry determining equations with the aid of the computer package SADE. We solve these equations directly to obtain nonclassical symmetries. We follow standard procedure of computing nonclassical symmetries and consider two different scenarios, ξ1≠0 and ξ1=0, ξ2≠0. Several nonclassical symmetries are reported for both scenarios. Furthermore, numerous group invariant solutions for nonclassical symmetries are derived. The similarity variables associated with each nonclassical symmetry are computed. The similarity variables reduce the system of partial differential equations (PDEs) to a system of ordinary differential equations (ODEs) in terms of similarity variables. The reduced system of ODEs are solved to obtain group invariant solution for governing boundary layer equations for two-dimensional heated flow problems. We successfully formulate a physical problem of heat transfer analysis for fluid flow over a linearly stretching porous plat and, with suitable boundary conditions, we solve this problem.

Group Invariant Solutions of the Full Plastic Torsion of Rod with Arbitrary Shaped Cross Sections

2012 ◽  
Vol 4 (03) ◽  
pp. 382-388 ◽  
Author(s):  
Kefu Huang ◽  
Houguo Li
Keyword(s):  
Symmetry Group ◽  
Cross Sections ◽  
Lie Group ◽  
Equilibrium Equation ◽  
Yield Criterion ◽  
Invariant Solution ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Full Symmetry

AbstractBased on the theory of Lie group analysis, the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied. Full symmetry group admitted by the equilibrium equation and the yield criterion is a finitely generated Lie group with ten parameters. Several subgroups of the full symmetry group are used to generate invariants and group invariant solutions. Moreover, physical explanations of each group invariant solution are discussed by all appropriate transformations. The methodology and solution techniques used belong to the analytical realm.

Application of Lie Group Analysis to the Plastic Torsion of Rod with the Saint Venant–Mises Yield Criterion

2012 ◽  
Vol 461 ◽  
pp. 265-271
Author(s):  
Hou Guo Li
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Yield Criterion ◽  
Invariant Solution ◽  
Group Analysis ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Lie Group Analysis ◽  
Group Invariant ◽  
Group Invariant Solutions

Based on Lie group and Lie algebra theory, the basic principles of Lie group analysis of differential equations in mechanics are analyzed, and its validity in theory of plasticity is explained by example. For the plastic torsion of rod with variable cross section that consists in non-linear Saint Venant-Mises yield criterion, the 10-dimensional Lie algebra admitted by the equilibrium equation and yield criterion is completely solved, and invariants and group invariant solutions relative to different sub-algebras are given. At last, physical explanations of each group invariant solution are discussed by some types of transformations.

Similarity solutions of the two-dimensional unsteady boundary-layer equations

1990 ◽  
Vol 216 ◽  
pp. 537-559 ◽  
Author(s):  
Philip K. H. Ma ◽  
W. H. Hui
Keyword(s):  
Boundary Layer ◽  
Stokes Equations ◽  
Invariant Solution ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Nonlinear Superposition ◽  
Group Invariant ◽  
Boundary Layer Equations ◽  
Special Cases

The method of Lie group transformations is used to derive all group-invariant similarity solutions of the unsteady two-dimensional laminar boundary-layer equations. A new method of nonlinear superposition is then used to generate further similarity solutions from a group-invariant solution. Our results are shown to include all the existing solutions as special cases. A detailed analysis is given to several classes of solutions which are also solutions to the full Navier–Stokes equations and which exhibit flow separation.

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