Integral equation formulation and error estimates for radial flow between two flat disks

  • Authors:

  • J. I. Frankel;K. Taira

  • Affiliations:

  • Mechanical and Aerospace Engineering and Engineering Science Department, University of Tennessee, Knoxville, TN 37996-2210, USA;Engineering and Applied Science Division, California Institute of Technology, Pasadena, CA 91125, USA

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2005

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Abstract

A novel mathematical framework is developed for investigating steady, incompressible, laminar, radially accelerating or decelerating flows confined between two parallel flat disks. The mathematical description for this flow involves a nonlinear, second-order, boundary-value problem containing an unknown parameter. This equation is derivable from the cylindrical-polar coordinates form of the Navier-Stokes equations. In order to uniquely determine this parameter, the system is adjoined to an integral constraint. This paper presents a two-level resolution process involving the numerical solution of an equivalent nonlinear integral equation and a detailed error analysis based on an inverse method for developing error estimates for both the function and the unknown system parameter. First, the differential equation is converted into an equivalent Hammerstein-Fredholm equation that automatically incorporates the integral constraint and removes the explicit unknown parameter. This equation is accurately solved with the aid of the method of Kumar and Sloan using a Chebyshev basis. Second, an unusual nonlinear differential equation is constructed for the local error and resolved by an inverse method involving parameter estimation. The resulting nonlinear differential equation in the local error contains the second derivative of the error at an endpoint. Interrogation of the local error distribution overcomes the limitations associated with a posteriori error estimates based on classical functional analysis. Often the need rises for recognizing the local error character resulting from a numerical simulation. Numerical results indicate the merit of the approaches and re-iterate the need for error estimation in numerical studies.