This study investigates the effect of initial reactant concentration, residence time, and flow on the particle size and particle size distribution produced by single-phase reactive precipitation between BaCl2 and Na2SO4 in a helical coil reactor. Helical coils offer enhanced mixing due to the secondary flow generated by the imbalance of centrifugal forces. The particles produced in a helical coil reactor are found to have smaller sizes and narrower size distribution compared to a straight tube reactor under similar process conditions. It is shown that the average particle size in a helical coil reactor decreases with increasing Dean number, and particle size distribution becomes narrower at higher Dean number. For a reactant concentration of 0.5 M with the reactant concentration ratio of one, the average size changed from 580 to 263 nm when Dean number changed from 64 to 370. The initial reactant concentration is defined in terms of supersaturation ratio, and a decreasing trend in size is also observed with increasing supersaturation ratio. The particle size distribution curve also became narrower at higher supersaturation ratio. The residence time is observed to have no visible effects on the particle mean size and particle size distribution for the range of residence time, from 6 to 34 s, studied for the helical coil in this work.

A study of linear stability analysis of a surfactant-laden viscoelastic liquid flowing down a slippery inclined plane is carried out under the framework of Orr–Sommerfeld type eigenvalue problem. It is assumed that the viscoelastic liquid satisfies the rheological property of Walters' liquid B″. The Orr–Sommerfeld type eigenvalue problem is solved analytically and numerically based on the long-wave analysis and Chebyshev spectral collocation method, respectively. The long-wave analysis predicts the existence of two temporal modes, the so-called surface mode and surfactant mode, where the first order temporal growth rate for the surfactant mode is zero. However, the first order temporal growth rate for the surface mode is non-zero, which leads to the critical Reynolds number for the surface mode. Further, it is found that the critical Reynolds number for the surface mode reduces with the increasing value of viscoelastic coefficient and ensures the destabilizing effect of viscoelastic coefficient on the primary instability induced by the surface mode in the long-wave regime. However, the numerical result demonstrates that the viscoelastic coefficient has a non-trivial stabilizing effect on the surface mode when the Reynolds number is far away from the onset of instability. Further, if the Reynolds number is high and the inclination angle is sufficiently low, there exists another mode, namely the shear mode. The unstable region induced by the shear mode magnifies significantly even for the weak effect of viscoelastic coefficient and makes the transition faster from stable to unstable flow configuration for the viscoelastic liquid. Moreover, the slip length exhibits a dual role in the surface mode as reported for the Newtonian liquid. But it exhibits only a stabilizing effect on the shear mode. In addition, it is found that the Marangoni number also exhibits a dual nature on the primary instability induced by the surface mode in contrast to the result of the Newtonian liquid.

A linear stability analysis is carried out for a contaminated viscoelastic liquid flowing down an inclined plane in the presence of an imposed shear stress, where the elastic behavior of the liquid follows the upper-convected Maxwell model. The earlier work [Wei, “Stability of a viscoelastic falling film with surfactant subjected to an interfacial shear,” Phys. Rev. E 71, 066306 (2005)] conducted analytically in the long-wave regime is revisited again in exploring the results in the arbitrary wavenumber regime. An Orr–Sommerfeld-type eigenvalue problem is formed for the viscoelastic liquid and solved both analytically and numerically by using the long-wave expansion and Chebyshev spectral collocation technique, respectively. It is found that with increase in the value of the Weissenberg number, the critical Reynolds number for the interface mode reduces, but the stable region enhances in the finite wavenumber regime. Furthermore, the unstable domain induced by the interface mode reduces in the presence of insoluble surfactant but enhances in the presence of applied shear stress. If the Reynolds number is high, but the inclination angle is small, the shear mode arises in the numerical simulation, which becomes weaker in the presence of the Weissenberg number and Marangoni number but becomes stronger in the presence of applied shear stress. In a special case, it is demonstrated that the present study recovers the results of Walters's liquid B″ in the limit of low viscoelastic parameter.

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