Lie Group Of Transformations Research Articles

Similarity solutions using Lie group theoretic method for cylindrical shock wave in self-gravitating perfect gas with axial magnetic field: isothermal flow

Propagation of cylindrical shock wave in a self-gravitating perfect gas under the influence of axial magnetic field using Lie group of transformation method is investigated. The flow is considered to be isothermal. Density and magnetic field are assumed to be varying in the undisturbed medium. Two different cases of solutions are brought out by the arbitrary constants appearing in the expressions of infinitesimals of local Lie group of transformations. One is with a power law shock path and the other one is with an exponential law shock path. Numerical solutions are obtained for both the cases of power law and exponential law shock paths. The effects of variation in Alfven-Mach number, gravitational parameter and ambient density variation index for power law shock path and effects of variation in Alfven-Mach number, gravitational parameter and ambient magnetic field variation index on the flow variables in the case of exponential law shock path are studied. Also the effects of increase in value of gravitational parameter and in the strength of ambient magnetic field on the shock strength are investigated. The increase in value of Alfven-Mach number leads to the increase in the density ratio which infers to the decrease in shock strength.

Simulations of surface corrugations of graphene sheets through the generalized graphene thermophoretic motion equation

In the current decade, nanomaterials have attracted great attention due to the wide range of applications in various disciplines and nanotechnology. Graphene is the best nanoscale material and one of the thinnest elastic films and has various applications. The thermophoretic motion system describes the diffusion of solitaries into substrate-supported graphene sheets. Lie group transformation of the motion equation is used to reduce the equation into solvable equation which is solved through the Bernoulli approximation method and some properties of the solutions are discussed.

Converging strong shock waves in magnetogasdynamics under isothermal condition

The similarity solutions to the problem of cylindrically symmetric strong shock waves in an ideal gas with a constant azimuthal magnetic field are presented. The flow behind the shock wave is assumed to spatially isothermal rather than adiabatic. We use the method of Lie group invariance to determine the possible class of self-similar solutions. Infinitesimal generators of Lie group transformations are determined by using the invariance surface conditions to the system and on the basis of arbitrary constants occurring in the expressions for the generators, four different possible cases of the solutions are reckoned and we observed that only two out of all possibilities hold self-similar solutions, one of which follows the power law and another follows the exponential law. To obtain the similarity exponents numerical calculations have been performed and comparison is made with the existing results in the literature. The flow patterns behind the shock are analyzed graphically.

The Combination of Conservation Laws and Auxiliary Equation Method

In this paper, appropriate Lie group transformations are considered as an alternative to a travelling wave transformation, in the auxiliary equation method to solve the nonlinear partial differential equations. In addition, Bernoulli differential equation has been considered as an auxiliary equation and using the approach presented.

Dual solutions in biomagnetic fluid flow and heat transfer over a nonlinear stretching/shrinking sheet: Application of lie group transformation method.

Of concern in the paper is a theoretical investigation of boundary layer flow of a biomagnetic fluid and heat transfer on a stretching/shrinking sheet in the presence of a magnetic dipole. The problem has been treated mathematically by using Lie group transformation. The governing nonlinear partial differential equations are thereby reduced to a system of coupled nonlinear ordinary differential equations subject to associated boundary conditions. The resulting equations subject to boundary conditions are solved numerically by using bvp4c function available in MATLAB software. The plots for variations of velocity, temperature, skin friction and heat transfer rate have been drawn and adequate discussion has been made. The study reveals that the problem considered admits of dual solutions in particular ranges of values of the suction parameter and nonlinear stretching/shrinking parameter. A stability analysis has also been carried out and presented in the paper. This enables one to determine which solution is stable that can be realized physically, and which is not. The results of the present study have been compared with those reported by previous investigators to ascertain the validity/reliability of the computational results.

Complete Geometric Analysis Using the Study SE(3) Parameters for a Novel, Minimally Invasive Robot Used in Liver Cancer Treatment

The paper presents a complete geometric analysis of a novel parallel medical robotic system designed for minimally invasive treatment of hepatic tumors using brachytherapy, ablation or targeted chemotherapy. An algebraic method based on the study parameters of the special Euclidean transformation Lie group SE(3) was used to determine the mechanism kinematics singularities and workspace. Moreover, two particular medical tool manipulations for the minimally invasive medical procedures are defined in terms of the Study parameters. The first manipulation of the medical tool refers to the linear insertion (of e.g., needles) and the second one is the remote center of motion manipulation of specific medical instruments (e.g., ultrasound probes). The constraint equations of the robotic system are derived and then, the operational workspace is illustrated for the novel parallel robotic system. Lastly, a numerical simulation is presented showing the behavior of the robotic system manipulating the ultrasound probe constrained by the remote center of motion. The geometric analysis of the operational workspace and the numerical simulation show promising results that validate the novel robotic system (safe-wise) for the medical procedure.

Passive control nanoparticles in double‐diffusive free convection nanofluid flow over an inclined permeable plate

AbstractThe present study has been conducted to acquire the solutions for the flow problem of an incompressible nanofluid past a permeable inclined plate implanted in a porous medium. In this study, double‐diffusivity, Brownian motion, and thermophoresis as well as passive control nanoparticles have been studied. We employ Lie group transformation on the ruling equations to extract nonlinear ordinary differential equations and solve them numerically using the fourth‐order Runge‐Kutta method and shooting approach. The supremacy of affined parameters on temperature and velocity distributions has been exposed by means of tables and graphs. This investigation suggests that both fluid velocity and nanoparticle concentration are enhanced by the modified Dufour parameter and the thermophoresis parameter. The assistance of the Lewis number intensifies the heat transport for suction.

Some symmetries, similarity solutions and various conservation laws of a type of dispersive water waves

We investigate the point symmetries, Lie–Bäcklund symmetries for a type of dispersive water waves. We obtain some Lie transformation groups, various group-invariant solutions, and some similarity solutions. Besides, we produce different formats of conservation laws of the dispersive water waves by using different schemes. Finally, we consider some special solutions of the stationary dispersive water-wave equations.

Magnetized and non-magnetized two layer liquids: A Lie symmetry analysis

It is well trusted that the physical believe subject to non-Newtonian fluid models in a magnetized frame always yield the complex mathematical equations and an exact solution in this direction seems impossible. Therefore, it remains challenging task for the researchers to report acceptable solutions. The current pagination contains a systemic remedy in this direction. To be more specific, the Carreau fluid flow towards flat surface is considered along with an externally applied magnetic field. The flow regime is manifested with pertinent physical believes namely, the heat source/sink and chemical reaction. The flow field is additionally supported with both the velocity and thermal slip effects. The Lie group of transformations is proposed for the present problem by means of group theoretic approach. The symmetry generator is constructed against developed Lie group of transformations. The characteristic equation subject to symmetry generator gives suitable set of variables. Such set of variables are used to step down the flow narrating differential equations in terms of an independent variables. Further, the reduced system is solved by numerical algorithm and the observations are offered by way of graphs. It is observed that the Carreau fluid velocity is suppressed for the case of magnetized flow field as compared to non-magnetized Carreau flow field. Further, the temperature of flow regime is higher in strength when the heat source is present. Finally, the results are validated by constructing comparison with an existing work.

Optical solitons and other solutions with anti-cubic nonlinearity by Lie symmetry analysis and additional integration architectures

In this work, symmetry reduction for nonlinear Schrödinger's equation with anti-cubic nonlinearity is determined by using the invariance of equations under Lie group of transformations. Using the similarity transformations the equation is reduced into system of ordinary differential equations. Corresponding to reduced ordinary differential equations, some exact solutions are obtained. These yields bright solitons, dark solitons, singular solitons and other explicit solutions. It is also pointed out that some of the results reported earlier are particular forms of current solutions.

Some new periodic solitary wave solutions of (3+1)-dimensional generalized shallow water wave equation by Lie symmetry approach

Many important physical situations such as fluid flows, marine environment, solid-state physics and plasma physics have been represented by shallow water wave equations. In this article, we construct new solitary wave solutions for the (3+1)-dimensional generalized shallow water wave (GSWW) equation by using optimal system of Lie symmetry vectors. The governing equation admits twelve Lie dimension space. A variety of analytic (closed-form) solutions such as new periodic solitary wave, cross-kink soliton and doubly periodic breather-type solutions have been obtained by using invariance of the concerned (3+1)-dimensional GSWW equation under one-parameter Lie group of transformations. Lie symmetry transformations have applied to generate the different forms of invariant solutions of the (3+1)-dimensional GSWW equation. For different Lie algebra, Lie symmetry method reduces (3+1)-dimensional GSWW equation into various ordinary differential equations (ODEs) while one of the Lie algebra, it is transformed into the well known (2+1)-dimensional BLMP equation. It is affirmed that the proposed techniques are convenient, genuine and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Under the suitable choices of arbitrary functions and parameters, 2D, 3D and contour graphics to the obtained results of GSWW equation are also analyzed.

Lie Group Analysis of Nanofluid Slip Flow with Stefan Blowing Effect via Modified Buongiorno’s Model: Entropy Generation Analysis

This article presents a detailed theoretical and computational analysis of alumina and titania-water nanofluid flow from a horizontal stretching sheet. At the boundary of the sheet (wall), velocity slip, thermal slip and Stefan blowing effects are considered. The Pak-Cho viscosity and thermal conductivity model is employed together with the non-homogeneous Buongiorno nanofluid model. The equations for mass, momentum, energy and nanoparticle species conservation are transformed via Lie-group transformations into a dimensionless system. The partial differential boundary value problem is therefore rendered into nonlinear ordinary differential form. With appropriate boundary conditions, the emerging normalized equations are solved with the semi-numerical homotopy analysis method (HAM). To consider entropy generation affects a second law thermodynamic analysis is also carried out. The impact of some physical parameters on the skin friction, Nusselt number, velocity, temperature and entropy generation number (EGM) are represented graphically. This analysis shows that diffusion parameter is a key factor to retards the friction and rate of heat transfer at the surface. Further, temperature of fluid decreases for the higher value of thermal slip parameter. In addition, EGM enhances with nanoparticles ambient concentration and Reynolds number. A numerical validation of HAM results is also included. The computations are relevant to thermodynamic optimization of nano-material processing operations.

Lie point symmetries of Stratonovich stochastic differential equations

This paper considers Lie point symmetries of stochastic differential equations (SDEs) in Stratonovich form. First, we derive the determining equations of the random symmetries, which correspond to Lie group transformations acting in the space of the independent variable (time), the dependent variables and the Brownian motion. Then, the determining equations of the random symmetries are used to obtain the determining equations of the deterministic symmetries, which correspond to Lie group transformations acting in the space of the independent variable and dependent variables. In the case of the deterministic symmetries the transformation of the Brownian motion is induced by the random time change. It is proved that the corresponding Itô and Stratonovich SDEs admit the same random symmetries and the same deterministic symmetries.

LIE GROUP ANALYSIS OF SORET AND DUFOUR EFFECTS ON RADIATIVE INCLINED MAGNETIC PRESSURE-DRIVEN FLOW PAST A DARCY-FORCHHEIMER MEDIUM

The study examines Soret and Dufour effects on steady convective heat and mass transfer of magnetohydrodynamic (MHD) pressure-driven flow in a Darcy-forchheimer porous medium with inclined uniform magnetic field and thermal radiation. The governing partial differential equations of the model are reduced to a system of coupled non-linear ordinary differential equations by applying a Lie group of transformations. The resulting coupled differential equations are solved using weighted residual method (WRM). The results obtained are presented graphically to illustrate the influence of various fluid parameters on the dimensionless velocity, pressure drop, temperature and concentration. Finally, the effects of Skin friction, Nusselt and Sherwood numbers results are presented and discussed accordingly

Unified model for the output accuracy of open-chain manipulators that considers joint clearance and structural parameters

A unified model is proposed for the output accuracy of open-chain manipulators in consideration of joint clearance and structural parameters. First, the operator of the finite-displacement screw matrix and the combination operation are presented. Second, the joint clearance and structural parameters are described and analyzed with screw theory. A virtual screw is established for the joint clearance and structural parameter errors. Third, a unified model is built through the adjoint transformation of Lie groups in consideration of the two effectors of the virtual screw. The error pose is decomposed into orientation and position errors, which are obtained through the virtual screw. Finally, an open-chain manipulator with six degrees of freedom is analyzed based on the proposed model. The position and orientation errors are obtained with the trajectory that provided an intuitive geometric insight into the accuracy and exact maximal position and orientation errors.

Lie symmetry analysis, complex and singular solutions of ( $$2+1$$ 2 + 1 )-dimensional combined MCBS–nMCBS equation

In this work, we study Lie symmetry analysis for integrable equation of combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative order (MCBS–nMCBS) that is one of the applications of symmetries. Symmetry analysis and similarity reduction of the ( $$2+1$$ )-dimensional combined MCBS–nMCBS equation are investigated by using the invariance of the equations under the Lie group of transformations. We have found Lie point symmetries of this equation and then we reduced combined MCBS–nMCBS equation to ordinary differential equation through two successive reductions. The same integrable equation was constructed and solved by Wazwaz (Nonlinear Dyn 91:877–883, 2018) using the tanh/coth method. However, we find out the complex and singular solutions using Lie group of transformation method. Some of the derived solutions are analyzed graphically. Eventually, we also obtained conservation laws of the equation with corresponding Lie symmetry.

Random Lie symmetries of Itô stochastic differential equations

Lie point symmetries of stochastic differential equations (SDEs) which include transformations depending on the Brownian motion are considered. The corresponding Lie group transformations, acting in the space of the independent variable (time), the dependent variables and the Brownian motion, preserve the differential form of SDEs and the properties of the Brownian motion. Such symmetries are called random symmetries.The focus of the paper is a definition of the random symmetries and their properties. It is shown that the random symmetries form a Lie algebra. We also investigate relations between the random symmetries and the conserved quantities of SDEs. For several stochastic differential equations we find their random symmetries and compare them with the deterministic symmetries, which correspond to Lie group transformations in the space of the independent and dependent variables.

Mei symmetry and new conserved quantities for non-material volumes

This paper is focused on the Mei symmetry of non-material volumes. The Mei symmetrical definition and criterion are given. The criterion equations of the system are presented by introducing the invariance of dynamical functions of the system under infinitesimal transformations of Lie groups. The structure equations and the form of new conserved quantities are proposed. An example is given to illustrate the application of the method; a new conserved quantity is obtained under the Mei symmetrical transformations.

Lie Group Transformation for Double-Diffusive Free Convection Nanofluid Flow Over an Inclined Plane

This paper concerns with a steady two-dimensional double-diffusive free convection flow of an incompressible nanofluid past an inclined plate embedded in a porous medium. The model incorporates the effects of cross diffusion and regular diffusion as well as Brownian motion and thermophoresis. A special form of Lie-group transformations, identified as scaling group of transformation is applied to the governing equations of the boundary value problem. The reduced non-linear ordinary differential equations subject to the boundary conditions are tackled numerically using Runge–Kutta Gill and Shooting methods. The impacts of various related parameters on dimensionless temperature, solutal concentration and nanoparticle volume fraction are represented graphically as well as Nusselt number and Sherwood number are elucidated through a table. It is found that, by increasing Brownian motion parameter the thermal boundary layer thickness enhances. It is also noticed that the reduced Sherwood number increases with the increase of Dufour solutal parameter and nanofluid Lewis number.

Symmetry reduction and numerical solution of a nonlinear boundary value problem in fluid mechanics

PurposeThe purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics.Design/methodology/approachThe authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods.FindingsFirst, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions.Originality/valueThe application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.