Adomian Decomposition of the Flowfield in a Simulated Rocket Motor

The work presents an analytic, approximate solution to an internal flowfield for a solid rocket motor. The flowfield is modeled as a wall-normal injection or suction in a symmetric porous channel with laterally expanding or contracting walls. From the effective speeds that gases are ejected into the combustion chamber of typical rocket motors, the flowfield is modeled to be incompressible. Since the flame zone occurs in a very thin space above the propellant grain surface, it will be disregarded. Assuming linearly varying axial velocity and uniform expansion (or contraction), the Navier-Stokes equations will be reduced into a single nonlinear equation that can be solved asymptotically. The Adomian Decomposition Method is used to solve this problem. Its systematic approach to solving differential equations makes it ideally suited for the present application. With this approximate method one can recover an exact solution for problems that allow an analytic treatment, it can also be used to arrive at approximate solutions for problems that cannot be solved exactly. The governing equation that describes the bulk fluid motion within the rocket chamber and the solution provided here will take into account the viscosity, wall regression, and wall permeability

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed