A Novel Version of HPM Coupled with the PSEM Method for Solving the Blasius Problem

This work studies the nonlinear differential equation that models the Blasius problem (BP) which is of great importance in fluid dynamics. The aim is to obtain an approximate analytical expression that adequately describes the phenomenon considered. To find such approximation, we propose a new method denominated powered homotopy perturbation (PHPM). Unlike HPM algorithm, the successive integration process generated by PHPM will consider zero the constants of integration in each approximation, except the last one. In the same way, PHPM will propose an adequate initial trial function provided of some unknown parameters in such a way that it will not evaluate the initial conditions in the iterations of the process; therefore, this set of parameters will be employed with the purpose of adjusting in the best accurate way the proposed approximation at the final part of the process. As a matter of fact, we will note from this analysis that the proposed solution is compact and easy to evaluate and involves a sum of five exponential functions plus a linear part of two terms, which is ideal for practical applications. With the purpose to get a better approximation, we find useful to combine PHPM with the power series extender method (PSEM) which implies to add to the PHPM solution one rational function with parameters to adjust. From this proposal, we find an approximate solution competitive with others from the literature.

Numerical Approximate Solution to Flow over a Flat Plate ( the Balusis Problem) By Perturbation Iteration- Algorithm

This paper applied perturbation iteration- algorithm (PI-A) to solve the problem of the incompressible two-dimensional laminar boundary layer flow over a flat plat as wall as called the Blasius problem (BP). BP is governed by Navier- Stokes equation (NSE) and continuity equation which were transformed into ordinary differential equation using similarity transforms. The results presented are tabulated for similarity stream function and can be seen highly of accuracy through comparable with that obtained by Ganji et al.[3] who studied BP using Homotopy perturbation technique (HPT) , Research results for the same problem using the variationally This paper applied perturbation iteration- algorithm (PI-A) to solve the problem of the incompressible two-dimensional laminar boundary layer flow over a flat plat as wall as called the Blasius problem (BP). BP is governed by Navier- Stokes equation (NSE) and continuity equation which were transformed into ordinary differential equation using similarity transforms. The results presented are tabulated for similarity stream function and can be seen highly of accuracy through comparable with that obtained by Ganja et al.[3] who studied BP using Homotopy perturbation technique (HPT) , Research results for the same problem using the variational iteration technique (VIT) before Aiyesimi and Niyi[5] and results numerical by Blasius[1]. Finally, The method that is efficient and widely applicable for solving ODE.

Solution for Celebrated Blasius Problem Using Homotopy Perturbation Method

In this manuscript, we have analyzed Celebrated Blasius boundary problem with moving wall or high speed 2D laminar viscous flow over gasifying flat plate. To find the way out of this nonlinear differential equation, a version of semi-analytical homotopy perturbation method has been applied. It has been observed that the precision of the solution would be achieved with increasing approximations. On comparison with literature, our solution has been proven highly accurate and valid with faster rate of convergence. It has been revealed that the second order approximate solution of Blasius equation in terms of initial slope is obtained as 0.33315 reducing the error by 0.32%.

The granular Blasius problem

We consider the steady flow of a granular current over a uniformly sloped surface that is smooth upstream (allowing slip for $x<0$) but rough downstream (imposing a no-slip condition on $x>0$), with a sharp transition at $x=0$. This problem is similar to the classical Blasius problem, which considers the growth of a boundary layer over a flat plate in a Newtonian fluid that is subject to a similar step change in boundary conditions. Our discrete particle model simulations show that a comparable boundary-layer phenomenon occurs for the granular problem: the effects of basal roughness are initially localised at the base but gradually spread throughout the depth of the current. A rheological model can be used to investigate the changing internal velocity profile. The boundary layer is a region of high shear rate and therefore high inertial number $I$; its dynamics is governed by the asymptotic behaviour of the granular rheology for high values of the inertial number. The $\unicode[STIX]{x1D707}(I)$ rheology (Jop et al., Nature, vol. 441 (7094), 2006, pp. 727–730) asserts that $\text{d}\unicode[STIX]{x1D707}/\text{d}I=O(1/I^{2})$ as $I\rightarrow \infty$, but current experimental evidence is insufficient to confirm this. We show that this rheology does not admit a self-similar boundary layer, but that there exist generalisations of the $\unicode[STIX]{x1D707}(I)$ rheology, with different dependencies of $\unicode[STIX]{x1D707}(I)$ on $I$, for which such self-similar solutions do exist. These solutions show good quantitative agreement with the results of our discrete particle model simulations.

Numerical Solution of Falkner-Skan Equation by Iterative Transformation Method

In this paper, we study the nonlinear boundary-layer equation of Falkner- Skan defned on a semi-infnite domain. An iterative finite difference (IFD) scheme is proposed to numerically solve such nonlinear ordinary differential equation. A computational iterative scheme is developed based on Newton-Kantorovich quasilinearization. At every iteration, the obtained linearized differential equation is numerically solved using the standard finite difference method. Numerical experiments show the accuracy and efficiency of the method compared to existing solvers. The computation is performed for different parameter values, including the special case of Blasius problem.

Modeling the motion of a spherical drop into a laminarboundary layer using Ansys Fluent

The article provides a formulation to the problem of spherical drop motion into laminar boundary layer over a flat semi-infinite plate (Blasius problem). The paper presents the solution in the package Ansys Fluent without taking into account deformation and rotation of drops. The article describes building of geometric area, construction of the grid models, boundary and initial conditions statement, methods of solution, course of computation. The article describes the examples of problem solving for various initial conditions of a drop, compares the results obtained using different methods. There is good agreement between the results. A combination of different methods for solving similar problems is the guarantee of a successful solution.