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

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 three-dimensional 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 non-dimensional 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.