Converging and diverging flow in narrow conical passages
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 equal to a constant like the authors did in the paper by putting \(r=\chi=const\). The solution of equation (B) with \(r=\chi=\text{const}\) has nothing to do with the solution of the problem under consideration because for every \(r\) different from \(\chi\) equation (A), and so the governing equations, would not be satisfied. Section 2.1 of the paper is superfluous because the solution of Stokes problem for a cone is not new and does not present any difficulty. The general solution of this problem, namely \[ \Psi\left(\zeta\right)=C_1+C_2\zeta+C_3\zeta^2+C_4\left[\zeta^2+{1\over 2}\zeta \left(1-\zeta^2\right)\ln{1+\zeta\over 1-\zeta}\right], \] is already given in the literature. So the solution to the conical gap Stokes problem cannot be regarded as a new one. Additionally, in section 2.1 and at several other places in the paper, the authors designate both the Legendre function of the first kind, \(P_n\left( x\right)\), and the Legendre function of the second kind, \(Q_n\left( x\right)\), as Legendre polynomials, though only \(P_n\left( x\right)\) are Legendre polynomials. The ensuing part of the paper is devoted to the solution of the ordinary third-order nonlinear differential equation that would be obtained when in equation (B) the independent variable \(r\) could be replaced by a constant \(r=\chi=\text{const}\). In section 2.2 the authors introduce a linearized solution for flows in cones. This linearization consists in the following: in equation (B) not only the independent variable \(r\) is set to a constant, but also the sought dependent variable \(f\left(\zeta\right)\) is declared to be constant, too. On the other hand, it is claimed that the derivatives of \(f\left(\zeta\right)\) should be different from zero, and so dependent on \(\zeta\). This does not make sense and is not (could not be) justified in the paper. In this way equation (B) is reduced to an ordinary third-order linear differential equation which can be once integrated to obtain \[ \left(1-\zeta^2\right)f''-2\zeta f'+\lambda f=K=\text{const}, \quad \lambda=\lambda\left(\nu,r,f\right)=\text{const.} \quad (C) \] The homogeneous part of equation (C) is the Legendre differential equation which has solutions (Legendre polynomials) that are regular on the axis only for the eigenvalues \(\lambda=n\left( n+1\right)\), \(n=1,2,\ldots\). On the other hand, \(\lambda\) should depend continuously on the kinematic viscosity \(\nu\). This contradiction is not seen and is not explained in the paper. Instead, the substitution \(z=\zeta^2\) is used which transforms the homogeneous part of equation (C) into Gauss's hypergeometric differential equation. However, due to the above-mentioned arguments, also the solution so obtained cannot be valid for arbitrary \(\lambda\left(\nu\right)\) in the case of the cone configuration. Section 3 of the paper is devoted to the numerical solution of the above nonlinear differential equation (equation (B) for \(r=\chi=const\)), and this is called ''numerical solution to the flow developed in conical passages'', which appears to this reviewer to be a strange formulation, since a solution of Navier-Stokes equations by using numerical methods means something different than that. Further, for the present time quite strange formulation is the following sentence at the end of section 2.2: ``Although the solution of the approximate system is intellectually stimulating, the evaluation of the associated infinite series requires the use of a digital computer that presents a higher degree of inconvenience than using the numerical technique which follows''. One should not be mislead by the seemingly acceptable appearance of the presentations of \(f\left(\varphi\right)\) in the paper. Such an appearance is only a consequence of the inherent nature of equation (B) for \(r=\chi=\text{const}\), and has little to do with the solution of the problem to the non-existing pure radial flow in conical passages, except for two limiting cases \(\nu=\infty\) and \(\nu=0\). There is a vast literature devoted to conical flow problems in general. For conical flows at high Reynolds numbers see, e.g., the recent article by \textit{R. Fernandez-Ferria} et al. [Q. J. Mech. Appl. Math. 52, No.~1, 1-53 (1999; Zbl 0948.76015)] and the references given there.
scientific article; zbMATH DE number 1356552