Existence of Solutions of a Two-Point Boundary Value Problem

Existence of Solutions of a Two-Point Boundary Value Problem A two-point boundary value problem with a non-negative parameter Q arising in the study of surface tension induced flow of a liquid metal or semiconductor is studied. We prove that the problem has at least one solution for Q ≥ 0. This improves a recent result that the problem has at least one solution for 0 ≤ Q ≤ 13.21. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Existence of Solutions of a Two-Point Boundary Value Problem

Existence of Solutions of a Two-Point Boundary Value Problem

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 77–80 Existence of Solutions of a Two-Point Boundary Value Problem Shu-hong Wu Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China (E-mail: hustwsh@21cn.com) Abstract A two-point boundary value problem with a non-negative parameter Q arising in the study of surface tension induced flow of a liquid metal or semiconductor is studied. We prove that the problem has at least one solution for Q ≥ 0.This improves a recent result that the problem has at least one solution for 0 ≤ Q ≤ 13.21. Keywords two-point boundary value problem; Brouwer degree; existence of solution 2000 MR Subject Classification 34B07; 34B08 1 Introduction We consider solutions of the following two-point boundary value problem on [0,1]: (a)[η(f /η) ] + Q[f (f /η) − η(f /η) ]= βη ( = d/dη), (1) (b) f (0) = f (1) = 0 and (f /η) | =(f /η) | − 1= 0. η=0 η=1 This problem arises in the study of surface-tension induced flows of a liquid metal or semicon- ductor in a cylindrical floating zone of length 2L and radius R. In dimensionless coordinates (x, r), points of the cylinder are given by −1 ≤ x = X/L ≤ 1, 0 ≤ η = r/R ≤ 1, with free surface η =1. The (x, r)-components of dimensionless velocity (u.v) are respectively, given 3 3  3 by u =2A (Re)f/η and v = −2A (Re)f /η, where Re is the Renolds number (Q =2A Re), and A = L/R is the aspect ratio. Assuming that the dimensionless pressure p is a quadratic function of x, we find that the r-component of the acceleration equation in the Navier-Stokes energy system describing the flow of a fluid and its temperature in the cylinder becomes (1)(a). The physical boundary conditions reduce to the conditions (1)(b) if we make the assumption that the free boundary is time-independent but not “flat”. Numerical solutions of...
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Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0217-z
Publisher site
See Article on Publisher Site

Abstract

A two-point boundary value problem with a non-negative parameter Q arising in the study of surface tension induced flow of a liquid metal or semiconductor is studied. We prove that the problem has at least one solution for Q ≥ 0. This improves a recent result that the problem has at least one solution for 0 ≤ Q ≤ 13.21.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

References

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