Conservation laws and conserved quantities for the two-dimensional laminar wake

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640014 ◽  
Author(s):  
L. N. Kokela ◽  
D. P. Mason ◽  
A. J. Hutchinson
Keyword(s):  
Conservation Laws ◽  
Physical Significance ◽  
Order System ◽  
Conserved Quantities ◽  
Order Partial Differential Equation ◽  
Two Dimensional ◽  
Third Order ◽  
Partial Differential ◽  
Laminar Wake ◽  
Unified Method

A systematic and unified method is presented for the derivation of the conserved quantities for the laminar classical wake and the wake of a self-propelled body. The multiplier method for the derivation of conservation laws is applied to the far downstream wake equations expressed in terms of the velocity components which gives rise to a second-order system of two partial differential equations, and in terms of the stream function which results in one third-order partial differential equation. Once the corresponding conservation laws are obtained, they are integrated across the wake and upon imposing the boundary conditions and the condition that the drag on a self-propelled body is zero, the conserved quantities for the classical wake and the wake of a self-propelled body are derived. In addition, this method results in the discovery of a new laminar wake which may have physical significance.

scholarly journals Conservation Laws for a Variable Coefficient Variant Boussinesq System

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Variable Coefficient ◽  
Boussinesq System ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
Partial Differential

We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.

scholarly journals Conservation Laws for a Generalized Coupled Korteweg-de Vries System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Daniel Mpho Nkwanazana ◽  
Ben Muatjetjeja ◽  
Chaudry Masood Khalique
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Fourth Order ◽  
Nonlinear Partial Differential Equations ◽  
Order System ◽  
Conserved Quantities ◽  
Third Order ◽  
The Third ◽  
New System

We construct conservation laws for a generalized coupled KdV system, which is a third-order system of nonlinear partial differential equations. We employ Noether's approach to derive the conservation laws. Since the system does not have a Lagrangian, we make use of the transformationu=Ux,v=Vxand convert the system to a fourth-order system inU,V. This new system has a Lagrangian, and so the Noether approach can now be used to obtain conservation laws. Finally, the conservation laws are expressed in theu,vvariables, and they constitute the conservation laws for the third-order generalized coupled KdV system. Some local and infinitely many nonlocal conserved quantities are found.

scholarly journals A note on the nonlocal boundary value problem for a third order partial differential equation

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 801-808 ◽  
Author(s):  
Kh. Belakroum ◽  
A. Ashyralyev ◽  
A. Guezane-Lakoud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Order Partial Differential Equation ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Partial Differential

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space with a self-adjoint positive definite operator is considered. Applying operator approach, the theorem on stability for solution of this nonlocal boundary value problem is established. In applications, the stability estimates for the solution of three nonlocal boundary value problems for third order partial differential equations are obtained.

One‐pass 3-D seismic extrapolation with the 45° wave equation

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1817-1824 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Wave Equation ◽  
Seismic Data ◽  
Correction Term ◽  
Synthetic Data ◽  
Second Order ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
Partial Differential ◽  
Numerical Tests ◽  
And Migration

One‐pass 3-D modeling and migration for poststack seismic data may be implemented by replacing the traditional 45° one‐way wave equation (a third‐order partial differential equation) with a pair of second‐ and first‐order partial differential equations. Except for an extra correction term, the resulting second‐order equation has a form similar to the Claerbout 15° one‐way wave equation, which is known to have a nearly circular impulse response. In this approach, there is no need to compensate for splitting errors. Numerical tests on synthetic data show that this algorithm has the desirable attributes of being second order in accuracy and economical to solve. A modification of the Crank‐Nicholson implementation maintains stability.

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.

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