Purely accelerating and decelerating flows within two flat disks
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 \(\lambda( r)\) and \(F( r)\) to be constant. They call \(\lambda(r)\) modified or local Reynolds number. Under such an assumption, the equation can be surely solved for \(\Phi( z)\) numerically or analytically, e.g., as the authors did by using the obvious boundary conditions and power series expansion of the inverse function \(z=z(\Phi)\), after a first integration with respect to \(z\) and whatever the value of \(r\). Furthermore, it is not shown in the paper, but most probably in the same manner as above, the authors assume \(z= \text{const}\) and obtain an expression for the dimensionless pressure difference \(\Delta p=p( r)-1\). The results so obtained for the velocity and the pressure drop are analyzed as functions of \(\lambda( r)\) and presented in graphical or tabular form versus experimental results up to \(\lambda( r)\approx 1\). It is not surprising that for \(\lambda( r)<1\) this comparison shows good agreement with experimental data, because in that case \(\lambda^2\ll 1\) (i.e., \(\text{Re}\ll 1\), \(\text{Eu}>1\)), and the flow can be seen to be governed by the solvable Stokes model. Comment: This approach is a violation of the assumed kinematik of the flow (radial streamlines, \(v_z=0\)), which is not shown in the paper. For the solution obtained in the paper \(\Phi( r,z)=\partial\Psi( r,z)/\partial z\), so that \(v_z=r^{-1}\partial\Psi( r,z)/\partial r\neq 0\), which is in contradiction with the starting assumption. It could only make a sense when \(\Delta r=r_o-r_i\approx 0\), \(h\approx 0\) are infinitesimally small, which is not the case for a device of the assumed real dimensions. This is how this reviewer understands and interprets the paper. However, he may possibly be wrong, because the journal where the paper has been published is known to him as a very serious one.
scientific article; zbMATH DE number 1143216