Purely accelerating and decelerating flows within two flat disks
type: Item
instance of: scholarly article
title: Purely accelerating and decelerating flows within two flat disks
zbMATH Open document ID: 0919.76019
DOI: 10.1007/BF01178407
author: G. Zitouni
author: Georgios H. Vatistas
published in: Acta Mechanica
publication date: 1999-05-17T00:00:00Z
review text: The presentation is connected with the problem of steady, axisymmetric, fully developed, pressure-gradient driven, laminar flow of a Newtonian incompressible fluid between two coaxial circular disks of inner radius \(r_i\) and outer radius \(r_o\). In cylindrical \(r,\varphi,z\)-coordinate system the disks are parallel to the \(r,\varphi\)-plane and are placed at a distance \(\pm h\) from it. The authors assume existence of purely radial flow with the velocity components \(v_\varphi\equiv 0\), \(v_z\equiv 0\), \(v_r=v_r( r,z)\) in the entire domain between the disks. Introducing the stream function via \(v_r=r^{-1}\partial\Psi/\partial z\), \(v_z=-r^{-1}\partial\Psi/\partial r=0\), we necessarily have \(\Psi=\Psi( z)\) that satisfies the continuity equation identically. The vorticity transport equation yields \(\nu\Psi''''( z)+2r^{-2}\Psi'( z)\Psi''( z)=0\). There are no other differential equations that can be used in this problem. Equation (2) in the paper is only a dimensionless expression for the flow rate through the device considered, but not the continuity equation as stated in the paper. Obviously the equation is contradictory, and the problem as stated has no solution, except for the two unreal limiting cases \(\nu=0\) and \(\nu=\infty\). Introducing the notation \(\Psi'( z)=rv_r=\Phi( z)\), and once integrating the above equation with respect to \(z\), we have \(\nu\Phi''( z)+r^{-2}\Phi^2( z)=\rho^{-1}rp'( r)\). Making dimensionless with quantities at the outer radius \(r_o\) and once integrating the equation from 1 to \(r\), one obtains: \(\Phi''( z)+\lambda^2( r) F( r)=0\), where \(\lambda^2( r)= \text{Re}(r^2-1)/(2r^2\ln r)\), \(r=2\text{Eu }r^2/( r^2-1)\cdot( p( r)-1)\), \(\text{Re}=r_oV_o/\nu\), \(\text{Eu}=p_o/(\rho V_o^2)\). Of course, this equation is also contradicting and has no solution either. Nonetheless, the authors believe the equation to have a solution (s. page 157 and the conclusion in the paper) and seek for a solution by regarding \(\la…
Mathematics Subject Classification ID: 76D05
zbMATH DE Number: 1143216
zbMATH Keywords: stream function
zbMATH Keywords: vorticity transport equation
reviewed by: Tomislav Zlatanovski
MaRDI profile type: Publication
cites work: Q3242757
cites work: Laminar Radial Flow Between Parallel Plates
cites work: Radial inflow between two flat disks
cites work: Q5523565
full work available at URL: https://doi.org/10.1007/bf01178407
OpenAlex ID: W2037042376
description: scientific article; zbMATH DE number 1143216