Interaction of a singular surface with a strong shock in the interstellar gas clouds

We investigate the interaction between a singular surface and a strong shock in the self-gravitating interstellar gas clouds with the assumption of spherical symmetry. Using the method of the Lie group of transformations, a particular solution of the flow variables and the cooling–heating function for an infinitely strong shock is obtained. This paper explores an application of the singular surface theory in the evolution of an acceleration wave front propagating through an unperturbed medium. We discuss the formation of an acceleration, considering the cases of compression and expansion waves. The influence of the cooling–heating function on a shock formation is explained. The results of a collision between a strong shock and an acceleration wave are discussed using the Lax evolutionary conditions. doi:10.1017/S1446181121000328

Elements of Classical Field Theory

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

Impact of solitons on the progression of initial lesion in aortic dissection

Aortic dissection (AD) is the most common catastrophic disease reported at cardiovascular emergency in hospitals. Herein, a tear in the tunica intima results into separation of layers of aortic wall leading to rupture and torrential bleed. Hypoxia and oxidative stress are associated with AD. The release of hypoxia inducible factor (HIF)-1[Formula: see text] from the initial flap lesion in the tunica intima is the basis for aneurysmal prone factors. We framed a boundary value problem (BVP) to evaluate homeostatic saturation for oxygen dynamics using steady-state analysis. We prove uniqueness and existence of the solution of the BVP for gas exchange at capillary–tissue interface as a normal physiological function. Failure of homeostatic mechanism establishes hypoxia, a new quasi-steady-state in AD. We model permeation of two-layer fluid comprised of blood and HIF-1[Formula: see text] through tunica media as a generalized [Formula: see text]-dimensional nonlinear evolution equation and solve it using Lie group of transformations method. We note that the two-layer fluid permeates the tunica media as solitary wave including solitons such as bright soliton, dark soliton, peregrine soliton, topological soliton, kink soliton, breather soliton and multi-soliton complex. Also, we introduce the main result and discuss the implications of soliton solution, using graphic interpretation, to describe the early stage of progression of AD.

INTERACTION OF A SINGULAR SURFACE WITH A STRONG SHOCK IN THE INTERSTELLAR GAS CLOUDS

We investigate the interaction between a singular surface and a strong shock in the self-gravitating interstellar gas clouds with the assumption of spherical symmetry. Using the method of the Lie group of transformations, a particular solution of the flow variables and the cooling–heating function for an infinitely strong shock is obtained. This paper explores an application of the singular surface theory in the evolution of an acceleration wave front propagating through an unperturbed medium. We discuss the formation of an acceleration, considering the cases of compression and expansion waves. The influence of the cooling–heating function on a shock formation is explained. The results of a collision between a strong shock and an acceleration wave are discussed using the Lax evolutionary conditions.

SIMILARITY SOLUTIONS FOR CYLINDRICAL SHOCK WAVE IN SELF-GRAVITATING NON-IDEAL GAS WITH AXIAL MAGNETIC FIELD: ISOTHERMAL FLOW

The purpose of this study is to obtain the solution using the Lie group of symmetry method for the problem of propagating magnetogasdynamic strong cylindrical shock wave in a self-gravitating non-ideal gas with the magnetic field which is taken to be axial. Here, isothermal flow is considered. In the undisturbed medium, varying magnetic field and density are taken. Out of four different cases, only three cases yield the similarity solutions. Numerical computations have been performed for the cases of power-law and exponential law shock paths, to find out the behavior of flow variables in the flow-field immediately behind the shock. Similarity solutions are carried out by taking arbitrary constants in the expressions of infinitesimals of the Lie group of transformations. Also, the study of the present work provides a clear picture of whether and how the variations in the non-ideal parameter of the gas, Alfven-Mach number, adiabatic exponent, ambient magnetic field variation index and gravitational parameter affect the propagation of shock and the flow behind it. Software package “MATLAB” is used for all the computations.

Solitary wave solutions of mKdV–Calogero–Bogoyavlenskii–Schiff equation by using Lie symmetry analysis

In this paper, we introduced and established some group invariant results of [Formula: see text]-dimensional mKdV–Calogero–Bogoyavlenskii–Schiff equation. Using the one-parameter Lie-group of transformations, we explored various closed-form solutions. The procedure minimizes the number of independent variables by one in every proceeding stage leading to form a system of the ordinary differential equations. The nature of solutions is investigated both analytically and physically through their evolutionary profiles by considering adequate choices of arbitrary functions and constants. The obtained results have been plotted with the aid of numerical simulation to obtain a significant appearance of the traced results. Simulation is carried out by taking an adequate option of arbitrary constants and functions, applying MATLAB code through progressing profiles. Wave solutions derived here are positons, multiple solitons, negaton and kink types which are shown through graph analysis.

A comparison of the modern Lie scaling method to classical scaling techniques

In the past 2 decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics, including but not limited to overland flow, groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non-dimensionalization, and the Vaschy–Buckingham-Π theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is made with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables, which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

Similarity Solutions of Cylindrical Shock Waves in Magnetogasdynamics With Thermal Radiation

Using Lie group of transformations, here we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of an ideal gas in the presence of radiative cooling and idealized azimuthal magnetic field. The similarity solutions are investigated behind a cylindrical shock wave which is produced as a result of a sudden explosion or driven out by an expanding piston. The shock is assumed to be strong and propagates into a medium which is at rest, with nonuniform density. The total energy of the wave is assumed to be time dependent obeying a power law. Indeed, with the use of the similarity solution, the problem is transformed into a system of ordinary differential equations (ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.

A comparison of the modern Lie scaling method to classical scaling techniques

In the past two decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics including but not limited to groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non dimensionalization, and the Buckingham Π theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is done with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.