The swimming of a flagellar micro-organism by the propagation of helical waves along its flagellum is analysed by a boundary-element method. The method is not restricted to any particular geometry of the organism nor does it assume a specific wave motion for the flagellum. However, only results for an organism with a spherical or ellipsoidal cell body and a helically beating flagellum are presented here.With regard to the flagellum, it is concluded that the optimum helical wave (amplitude α and wavenumber k) has αk ≈ 1 (pitch angle of 45°) and that for the optimum flagellar length L/A = 10 (L being the flagellar length, A being the radius of the assumed spherical cell body) the optimum number of wavelengths Nλ is about 1.5. Furthermore there appears to be no optimal value for the flagellar radius a, with the thinner flagella being favoured. These conclusions show excellent quantitative agreement with those of slender-body theory.For the case of an ellipsoidal cell body, the optimum aspect ratios B/A and C/A of the ellipsoid are about 0.7 and 0.3 respectively; A, B and C are the principal radii of the ellipsoid. These and all of the above conclusions show good qualitative agreement with experimental observations of efficiently swimming micro-organisms.
Abstract. This paper presents mesh-free procedures for solving linear di erential equations (ODEs and elliptic PDEs) based on Multiquadric (MQ) Radial Basis Function Networks (RBFNs). Based on our study of approximation of function and its derivatives using RBFNs that was reported in an earlier paper (Mai-Duy and Tran-Cong, 1999), new RBFN approximation procedures are developed in this paper for solving DEs, which can also be classi ed into two t ypes: a direct (DRBFN) and an indirect (IRBFN) RBFN procedure. In the present procedures the width of the RBFs is the only adjustable parameter according to a (i) = d (i) , where d (i) is the distance from the ith centre to the nearest centre. The IRBFN method is more accuarte than the DRBFN one and experience so far shows that can be chosen in the range 7 10 for the former. Di erent c o m binations of RBF centres and collocation points (uniformly and randomly distributed) are tested on both regularly and irregularly shaped domains. The results for a 1D Poisson's equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1:0e ; 4) and O(1:0e ; 8), respectively, with a centre density of 50. Similarly, the results for a 2D Poisson's equation show t h a t t h e DRBFN and the IRBFN procedures achieve a norm of error of at least O(1:0e;3) and O(1:0e ; 6), respectively, with a centre density o f 1 2 12.
This paper presents a numerical approach, based on Radial Basis Function Networks (RBFNs), for the approximation of a function and its derivatives (scattered data interpolation). The approach proposed here is called the indirect radial basis function network (IRBFN) approximation which is compared with the usual direct approach. In the direct method (DRBFN) the closed form RBFN approximating function is rst obtained from a set of training points and the derivative functions are then calculated directly by di erentiating such closed form RBFN. In the indirect method (IRBFN) the formulation of the problem starts with the decomposition of the derivative of the function into RBFs.
SUMMARYA numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a 'universal approximator' based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of 'finite element-type' discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re= 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of 11×11, Re=400 for the driven cavity flow with a density of 33×33 and Rayleigh number Ra= 1 000 000 for the natural convection flow with a density of 27 ×27.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.