A Steady Flow of The Viscous Compressible Liquid in Vertical Pipe

In the study of the flow of gas-liquid mixture over circular vertical pipe rising from the deep zone to ground level, it is observed that, the velocity on surface of tube is much more than in the center of flow. Such a picture is seen also in the water filtration process in sands ordering from high permeability zone to lower. Same phenomena occur in transportation of the water carbon nana-tubes. In order to predict behavior of those processes in this paper, we have studied the compressible liquid flow over the circular vertical pipe ordered from the deep zone to the ground surface for some concrete inlet and outlet regimes of the pipe. The Navier-Stokes equations system as a model of study. For certain inlet and outlet regimes of the flow splitting the equation into the cross section and axes variables, the pressure and velocity distribution are found. The Lane-Emden equation arises for determining the pipe cross section velocity distribution, which is also justified by our calculations on the used model.

Stellar Structure in a Newtonian Theory with Variable G

A Newtonian-like theory inspired by the Brans–Dicke gravitational Lagrangian has been recently proposed by us. For static configurations, the gravitational coupling acquires an intrinsic spatial dependence within the matter distribution. Therefore, the interior of astrophysical configurations may provide a testable environment for this approach as long as no screening mechanism is evoked. In this work, we focus on the stellar hydrostatic equilibrium structure in such a varying Newtonian gravitational coupling G scenario. A modified Lane–Emden equation is presented and its solutions for various values of the polytropic index are discussed. The role played by the theory parameter ω, the analogue of the Brans–Dicke parameter, in the physical properties of stars is discussed.

Non-Radial Pulsations of Distorted Stellar Structures: Effect of Mass Variation

Eigen-frequencies (EF) of non–radial modes (NRM) of pulsations of differentially rotating (D R) and tidally distorted (T D) stellar models by considering the effect of mass variation (MV) on its equi-potentials surfaces inside a star. The method utilizes an averaging proposal of Kippenhahn and Thomas (K and T) with conjunction of the concept of Roche-equipotential. The study accolades and corrects earlier studies of non-radial (NR) pulsations of DR and TD stellar structures of different natures such as radial and non-radial oscillations, X-ray, gamma ray and other electromagnetic disturbances. The reflection of the work comes from the requirements of the inclusion of non-uniform densities that yield Lane-Emden equation to have reliable results up to second order disturbances.

Approximate solution of Lane-Emden problem via modified Hermite operation matrix method

Lane-Emden equations are singular initial value problems and they are important in mathematical physics and astrophysics. The aim of this present paper is presenting a new numerical method for finding approximate solution to Lane-Emden type equations arising in astrophysics based on modified Hermite operational matrix of integration. The proposed technique is based on taking the truncated modified Hermite series of the solution function in the Lane-Emden equation and then transferred into a matrix equation together with the given conditions. The obtained result is system of linear algebraic equation using collection points. The suggested algorithm is applied on some relevant physical problems as Lane-Emden type equations.

Lane-Emden equations perturbed by nonhomogeneous potential in the super critical case

Our purpose of this paper is to study positive solutions of Lane-Emden equation − Δ u = V u p i n R N ∖ { 0 } $$\begin{array}{} -{\it\Delta} u = V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\} \end{array}$$ (0.1) perturbed by a non-homogeneous potential V when p ∈ [ p c , N + 2 N − 2 ) , $\begin{array}{} p\in [p_c, \frac{N+2}{N-2}), \end{array}$ where pc is the Joseph-Ludgren exponent. When p ∈ ( N N − 2 , p c ) , $\begin{array}{} p\in (\frac{N}{N-2}, p_c), \end{array}$ the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution wk of −Δ u = up in ℝ N ∖ {0} by authors in [9]. While the fast decaying solution wk is unstable for p ∈ ( p c , N + 2 N − 2 ) , $\begin{array}{} p\in (p_c, \frac{N+2}{N-2}), \end{array}$ so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V. A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of − Δ u = u N + 2 N − 2 $\begin{array}{} -{\it\Delta} u = u^{\frac{N+2}{N-2}} \end{array}$ in ℝ N and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V.

Quadratic First Integrals of Time-dependent Dynamical Systems of the Form q¨a=-Γbcaq˙bq˙c-ω(t)Qa(q)

We consider the time-dependent dynamical system q¨a=−Γbcaq˙bq˙c−ω(t)Qa(q) where ω(t) is a non-zero arbitrary function and the connection coefficients Γbca are computed from the kinetic metric (kinetic energy) of the system. In order to determine the quadratic first integrals (QFIs) I we assume that I=Kabq˙aq˙b+Kaq˙a+K where the unknown coefficients Kab,Ka,K are tensors depending on t,qa and impose the condition dIdt=0. This condition leads to a system of partial differential equations (PDEs) involving the quantities Kab,Ka,K,ω(t) and Qa(q). From these PDEs, it follows that Kab is a Killing tensor (KT) of the kinetic metric. We use the KT Kab in two ways: a. We assume a general polynomial form in t both for Kab and Ka; b. We express Kab in a basis of the KTs of order 2 of the kinetic metric assuming the coefficients to be functions of t. In both cases, this leads to a new system of PDEs whose solution requires that we specify either ω(t) or Qa(q). We consider first that ω(t) is a general polynomial in t and find that in this case the dynamical system admits two independent QFIs which we collect in a Theorem. Next, we specify the quantities Qa(q) to be the generalized time-dependent Kepler potential V=−ω(t)rν and determine the functions ω(t) for which QFIs are admitted. We extend the discussion to the non-linear differential equation x¨=−ω(t)xμ+ϕ(t)x˙(μ≠−1) and compute the relation between the coefficients ω(t),ϕ(t) so that QFIs are admitted. We apply the results to determine the QFIs of the generalized Lane–Emden equation.

Improved Numerical Computation to Solve Lane-Emden type Equations

Lane-Emden equation is also of fundamental importance in mathematical physics, celestial mechanics,and computer science. It can be used to describe stellar structures, equilibrium density distribution in polytrophicisothermal gas, thermal behavior in mutual attraction of its molecules. An improved numerical method is developed for solving Lane-Emden type differential equations. The method is based on power series solutions of differential equations and Maclaurin series expansion. A python program is written to carry out numerical calculations. Five examples are solved and shown in this paper, the solutions obtained by the program are compared with the exact solutions of differential equation, an excellent agreement is found between them. The present method improves runtime.