Three-Dimensional Axisymmetric Solidification of a Viscous Incompressible Flow in the Stagnation Point Region

Abstract

The history of the study of fluid solidification in stagnation flow is very limited. Among these studies, only one two-dimensional Cartesian coordinate case has considered fluid viscosity and pressure variation along the boundary layer. In the present paper, the solidification process of an incompressible viscous fluid in a threedimensional axisymmetric coordinate system is considered. The solidification is modeled by solving the momentum equations governing a problem in which a plate is moving toward an impinging fluid with a variable velocity and acceleration. The unsteady momentum equations are transformed to ordinary differential equations by using properly introduced similarity variable. Furthermore, pressure variations along the boundary layer thickness are taken into account. The energy equation is solved by numerical method as well as similarity solution. Interestingly, similarity solution of the energy equation is used for validation of the numerical solution. In this research, distributions of the fluid temperature, transient distributions of the velocity components and, most importantly, the solidification rate are presented for different values of nondimensional governing parameters including Prandtl number and Stefan number. A comparison is made between the solidification processes of axisymmetric three-dimensional and two-dimensional cases to justify the achieved results in a better way. The obtained results reveal that there is a difference between the final solid thickness, when the process has reached to its steady condition, of three-dimensional axisymmetric and two-dimensional cases. Also the results show that increase the Prandtl number up to 10 times or increase the heat diffusivity ratio up to 2 times lead to decrease the ultimate frozen thickness almost by half. While, the Stefan number has no effect on the value of thickness and its effect is captured only on the freezing time. Prediction the ultimate thickness of solid before obtaining solution and introducing a new method for validation of numerical results are achievements in this research.

An Experimental Study of Water Freezing in Cylindrical Stagnation Flow

“…Shokrgozar and Rahimi [22] presented the solidification of incompressible fluid in twodimensional stagnation flow. Shokrgozar [23] also showed increasing the Prandtl number up to 10 times or increasing the heat diffusivity ratio up to 2 times leads to a reduction of the ultimate frozen thickness almost by half, While the Stefan number does not affect on the thickness and its effect is captured only on the freezing time in axisymmetric stagnation flow. Alizadeh et al [24] investigated the steady-state viscous flow and heat transfer in the vicinity of an unaxisymmetric stagnation-point of an infinite stationary cylinder.…”

Vapor Solidification of Saturated Air in Two-Dimensional Stagnation Flow

“…Solidi cation of incompressible uid in the two-dimensional stagnation ow was considered by Shokrgozar and Rahimi [31]. The freezing of incompressible uid in unsteady axisymmetric stagnation ow was also investigated by Shokrgozar [32]; however, the solidi cation of vapor from saturated gas has not been investigated so far. In this study, saturated air vapor freezing in stagnation ow is closely analyzed.…”