Converging and diverging flow in narrow conical passages
type: Item
instance of: scholarly article
title: Converging and diverging flow in narrow conical passages
zbMATH Open document ID: 0981.76018
DOI: 10.1007/BF01179258
published in: Acta Mechanica
publication date: 2002-03-11T00:00:00Z
review text: In this paper, in the same manner as in a previous paper [the authors, Comput. Fluids 26, No.~7, 683-695 (1997; Zbl 0919.76018)], the authors continue to present solutions to classical fluid dynamical problems for which it was long ago proven that they cannot have solutions. The authors consider an assumed and non-existent in reality two-dimensional viscous flow for which the streamline pattern and boundary conditions are given a priori, namely, a pure radial flow in a cone or in a conical gap. The authors claim to have obtained both analytical and numerical solutions to the Navier-Stokes equations that should govern such a kind of viscous flow. However, this viscous flow cannot be realized physically, and for it the Navier-Stokes equations reduce to conflicting and unsolvable differential relations. Using spherical coordinates (\(r,\varphi,\theta\)) and assuming \(v_{\varphi}=v_{\theta}=0\), \(\partial/\partial\theta=0\), \(v_r=f\left(\theta\right)/r^2\), the authors reduce the Navier-Stokes equations to the correct relation (9) in the paper. But, in the streamfunction formulation one can easily show that an assumed pure radial flow in a cone or conical gap is impossible. Indeed, for such assumed pure radial flow, with \(\zeta=\cos\varphi\), we necessarily have \(\Psi=\Psi\left(\zeta\right)\), \(v_r=\Psi'\left(\zeta\right)/r^2=f\left(\zeta\right)/r^2\). Introducing this into the vorticity transport equation in spherical coordinates, we obtain the relation \[ \nu r\left[\left(1-\zeta^2\right)\Psi''''-4\zeta\Psi'''+4\Psi''\right]+4\Psi '\Psi''=0 \quad (A) \] or \[ \left(1-\zeta^2\right)f'''-4\zeta f''+4\left(1+{1\over\nu r}f\right)f'=0 \qquad \quad (B) \] that corresponds to equation (13) in the paper. The only solution of this equation is \(f'\left(\zeta\right)=0\), i.e. a potential flow (\(\Psi=c_1\zeta+c_2\)) that cannot satisfy physical boundary conditions on the rigid wall(s). Note that in equation (A) \(r\) is an independent variable that cannot be simply set e…
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76D07
zbMATH DE Number: 1356552
zbMATH Keywords: Stokes flow in conical gap
zbMATH Keywords: Navier-Stokes equations
zbMATH Keywords: pure radial flow in cone
author: Georgios H. Vatistas
author: Wahid Ghaly
reviewed by: Tomislav Zlatanovski
MaRDI profile type: Publication
cites work: Purely accelerating and decelerating flows within two flat disks
cites work: On the pure polar flow within two concentric spheres
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description: scientific article; zbMATH DE number 1356552