Accurate benchmark results of Blasius boundary layer problem using a leaping Taylors series that converges for all real values

Anil Lal S; Martin Milin

Home/Archive/arXiv

arXiv · 2204.02215

Accurate benchmark results of Blasius boundary layer problem using a leaping Taylors series that converges for all real values

Anil Lal S·Martin Milin

View original source ↗Read source document ↗Export metadata ↓

Abstract

Blasius boundary layer solution is a Maclaurin series expansion of the function $f(\eta)$, which has convergence problems when evaluating for higher values of $\eta$ due to a singularity present at $\eta\approx-5.69$. In this paper we are introducing an accurate solution to $f(\eta)$ using Taylor's series expansions with progressively shifted centers of expansion(Leaping centers). Each series is solved as an IVP with the three initial values computed from solution of the previous series, so the gap between the centers of two consecutive expansions is selected from within the convergence disc of the first series. The last series is formed such that it is convergent for a reasonable high $\eta$ value needed for implementing the boundary condition at infinity. The present methodology uses Newton-Raphson method to compute the value of the unknown initial condition viz. $f''(0)$ in an iterative manner. Benchmark accurate results of different parameters of flat plate boundary layer with no slip and slip boundary conditions have been reported in this paper.

Explore related subjects

math.GM

Keep this discovery

BibTeX RIS

Anil Lal S, Martin Milin. 2022-03-22. Accurate benchmark results of Blasius boundary layer problem using a leaping Taylors series that converges for all real values. https://arxiv.org/abs/2204.02215

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗

arXiv2026

The decrease in sectional curvature of the Fisher--Rao manifold of beta distributions

In the paper, the author develops a general convexity lemma and applies it to demonstrate the monotonic decrease of a function involving polygamma functions, arising from the sectional curvature of the Fisher--Rao manifold of beta distributions.

math.GM↗

arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗