Group classification for path equation describing minimum drag work and symmetry reductions

The path equation describing the minimum drag work first proposed by Pakdemirli is reconsidered （Pakdemirli, M. The drag work minimization path for a fly- ing object with altitude-dependent drag parameters. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 223（5）, 1113- 1116 （2009））. The Lie group theory is applied to the general equation. The group classi- fication with respect to an altitude-dependent arbitrary function is presented. Using the symmetries, the group-invariant solutions are determined, and the reduction of order is performed by the canonical coordinates.

Symmetries of boundary layer equations of power-law fluids of second grade

A modified power-law fluid of second grade is considered. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The equations of motion are derived for two dimensional incompressible flows, and from which the boundary layer equations are derived. Symmetries of the boundary layer equations are found by using Lie group theory, and then group classification with respect to power-law index is performed. By using one of the symmetries, namely the scaling symmetry, the partial differential system is transformed into an ordinary differential system, which is numerically integrated under the classical boundary layer conditions. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.

Structural Synthesis of Parallel Mechanisms with High Rotational Capability

Most parallel mechanisms(PMs) encountered today have a common disadvantage, i.e., their low rotational capability.In order to develop PMs with high rotational capability, a family of novel manipulators with one or two dimensional rotations is proposed. The planar one-rotational one-translational(1 R1 T) and one-rotational two-translational(1 R2 T)PMs evolved from the crank-and-rocker mechanism(CRM) are presented by means of Lie group theory. A spatial 2 R1 T PM and a 2 R parallel moving platform with bifurcated large-angle rotations are proposed by orthogonal combination of the RRRR limbs. According to the product principle of the displacement group theory, a hybrid 2 R3 T mechanism in possession of bifurcated motion is obtained by connecting the 2 R parallel moving platform with a parallel part, which is constructed by four 3 T1 R kinematic chains. The presented manipulators possess high rotational capability. The proposed research enriches the family of spatial mechanisms and the construction method provides an instruction to design more complex mechanisms.

Invariance analysis for determining the closed-form solutions, optimal system, and various wave profiles for a (2+1)-dimensional weakly coupled B-Type Kadomtsev-Petviashvili equations

In the case of negligible viscosity and surface tension,the B-KP equation shows the evolution of quasione-dimensional shallow-water waves,and it is growingly used in ocean physics,marine engineering,plasma physics,optical fibers,surface and internal oceanic waves,Bose-Einstein condensation,ferromagnetics,and string theory.Due to their importance and applications,many features and characteristics have been investigated.In this work,we attempt to perform Lie symmetry reductions and closed-form solutions to the weakly coupled B-Type Kadomtsev-Petviashvili equation using the Lie classical method.First,an optimal system based on one-dimensional subalgebras is constructed,and then all possible geometric vector yields are achieved.We can reduce system order by employing the one-dimensional optimal system.Furthermore,similarity reductions and exact solutions of the reduced equations,which include arbitrary independent functional parameters,have been derived.These newly established solutions can enhance our understanding of different nonlinear wave phenomena and dynamics.Several threedimensional and two-dimensional graphical representations are used to determine the visual impact of the produced solutions with determined parameters to demonstrate their dynamical wave profiles for various examples of Lie symmetries.Various new solitary waves,kink waves,multiple solitons,stripe soliton,and singular waveforms,as well as their propagation,have been demonstrated for the weakly coupled B-Type Kadomtsev-Petviashvili equation.Lie classical method is thus a powerful,robust,and fundamental scientific tool for dealing with NPDEs.Computational simulations are also used to prove the effectiveness of the proposed approach.

Soliton solutions,travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics

Many physical systems can be successfully modelled using equations that admit the soliton solutions.In addition,equations with soliton solutions have a significant mathematical structure.In this paper,we study and analyze a three-dimensional soliton equation,which has applications in plasma physics and other nonlinear sciences such as fluid mechanics,atomic physics,biophysics,nonlinear optics,classical and quantum fields theories.Indeed,solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour.We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time.Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function,elliptic functions,elementary trigonometric and hyperbolic functions solutions of the equation.Besides,various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique.These solutions comprise dark soliton,doubly-periodic soliton,trigonometric soliton,explosive/blowup and singular solitons.We further exhibit the dynamics of the solutions with pictorial representations and discuss them.In conclusion,we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula.We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.

Numerical solution of thermo-solutal mixed convective slip flow from a radiative plate with convective boundary condition

A mathematical model for mixed convective slip flow with heat and mass transfer in the presence of thermal radiation is presented. A convective boundary condition is included and slip is simulated via the hydrodynamic slip parameter. Heat generation and absorption effects are also incorporated. The Rosseland diffusion flux model is employed. The governing partial differential conservation equations are reduced to a system of coupled, ordinary differential equations via Lie group theory method. The resulting coupled equations are solved using shooting method. The influences of the emerging parameters on dimensionless velocity, tempera- ture and concentration distributions are investigated. Increasing radiative-conductive parameter accelerates the boundary layer flow and increases temperature whereas it depresses concentration. An elevation in convection-conduction parameter also accelerates the flow and temperatures whereas it reduces concentrations. Velocity near the wall is considerably boosted with increasing momentum slip parameter although both temperature and concentration boundary layer thicknesses are decreased. The presence of a heat source is found to increase momentum and thermal boundary layer thicknesses but reduces concentration boundary layer thickness. Excelle- nt correlation of the numerical solutions with previous non-slip studies is demonstrated. The current study has applications in bio- reactor diffusion flows and high-temperature chemical materials processing systems.

Type Synthesis and Dynamics Performance Evaluation of a Class of 5-DOF Redundantly Actuated Parallel Mechanisms

This paper presents a five degree of freedom(5-DOF)redundantly actuated parallel mechanism(PM)for the parallel machining head of a machine tool.A 5-DOF single kinematic chain is evolved into a secondary kinematic chain based on Lie group theory and a configuration evolution method.The evolutional chain and four 6-DOF kinematic chain SPS(S represents spherical joint and P represents prismatic joint)or UPS(U represents universal joint)can be combined into four classes of 5-DOF redundantly actuated parallel mechanisms.That SPS-(2UPR)R(R represents revolute joint)redundantly actuated parallel mechanism is selected and is applied to the parallel machining head of the machine tool.All formulas of the 4SPS-(2UPR)R mechanism are deduced.The dynamic model of the mechanism is shown to be correct by Matlab and automatic dynamic analysis of mechanical systems(ADAMS)under no-load conditions.The dynamic performance evaluation indexes including energy transmission efficiency and acceleration performance evaluation are analyzed.The results show that the 4SPS-(2UPR)R mechanism can be applied to a parallel machining head and have good dynamic performance.