Quasi 3D Finite Element Algorithm for Rotating Mixing Flows

The present research article presents numerical simulations of rotating of Newtonian fluid mixing flows in a cylindrical container through single rotating stirrer with agitator, where stirrer is located on the lid of container in concentric position. For this purpose a Quasi 3D (Three-Dimensional) FEA (Finite Element Algorithm) has been developed. The numerical algorithm is based on fractional stages semiimplicit Taylor-Galerkin/Pressure-Correction scheme. The simulation has been carried out to analyze the effects of agitator on mixing behavior. The numerical results show that Quasi 3D FEA is an accurate mathematical tool and able to achieve good results for flow structure in laminar regime.

The Pseudo 3D Finite Element approach was used to investigate unsteady free surface water profile in channel [9]. The numerical technique was based on fractional stages predictor-corrector Taylor-Galerkin/ Pressure-Correction scheme in Cartesian coordinate system.
In this study, the mixing behavior of Newtonian fluid is investigated in a cylindrical container through single rotating stirrer with agitator, where stirrer is located on the lid of container and positioned concentrically. In CFD (Computational Fluid Dynamics), the flow of Newtonian fluid in cylindrical shaped container is mathematically modelled through non-linear system of partial differential equations, such as conservation of mass and conservation of momentum transport equations in cylindrical polar coordinate system to determine the flow structure, velocity gradient and pressure differential. FEM (Finite Element Method) has been used as a powerful mathematical tool by many researchers to reduce the complexities in mixing processes, like accurate model of domain boundaries [10], spatial discretization of domain, time discretization of transient flows, use of suitable material properties and boundary conditions [11].
The objective of this study is to develop a Quasi 3D FEA based on fractional stages semi-implicit Taylor-Galerkin/ Pressure-Correction scheme [10] for the rotating mixing flows in cylindrical container through single agitated rotating stirrer and to investigate the effects of agitator on flow structure and pressure gradient. The problem specification is described in section 2, the details of basic governing equations and numerical scheme used is presented in section 3 and section 4 respectively. The numerical predictions are demonstrated in section 5 and conclusion is described in section 6.

PROBLEM SPECIFICATION/DEFINITION
The problem in the present study is to investigate mixing flow of Newtonian fluid, relevant to the pharmaceutical and cosmetic industrial applications. The concentric configuration of a single rotating stirrer with agitator has been adopted, where the stirrer will be located on the lid of the cylindrical container [4]. The mixing is performed between rotation of stirrer with agitator and stationary cylindrical container for numerical simulations. A cylindrical polar coordinate system has been used due to the nature of problem.
In order to present the well posed problem specification for the mixing flows in cylindrical container, it is essential to use the suitable initial and boundary conditions. At initial time, the motionless values have been taken for components of velocity such as:

Quasi 3D Finite Element Algorithm for Rotating Mixing Flows
No slip boundary conditions have been imposed at the round disc, solid wall and base of cylindrical container

GOVERNING EQUATIONS
The and the momentum transport equation as: .
is the velocity vector field of fluid,  is the fluid density, t represents time,  is the Cauchy stress tensor which can be described as and  is the spatial differential operator.
Where p is isotropic fluid pressure, Kronecker delta  is the unit tensor and T is the total stress tensor. For Newtonian fluids, according to Newton's viscous law, the total stress tensor T is proportional to rate-ofdeformation tensor and can be expressed as:

FIG. 1. COMPUTATIONAL DOMAIN OF INTEREST AND FINITE ELEMENT MESH
For an incompressible flow without body force, the Quasi 3D component wise system of governing equations without asterisk notation can be expressed non- Where, the non-dimensional Reynolds number is defined as:

4.
NUMERICAL SCHEME Step-1(a): Compute the velocity component at half time stepn + ½ from initial velocity field at time step t n .
Step-1(b): Calculate an intermediate non-divergence-free velocity field u * at full time step, using solutions at levels n and n + ½.
Step-2: Calculate pressure difference using u * from step-1(b) via Poisson equation using Choleski method at full time step interval [n, n+1].
Step-3: Compute a divergence-free velocity field u n+1 using Jacobi iterative method from pressure difference field of step-2. Near cylindrical vessel wall fluid is almost stagnant, whereas, below agitator a small recirculation starts to mix the material. It is observed that the Re=25 is found to be a critical value where the fluid starts to be slightly pushed below agitator due to small recirculation. Physically, it

Effects of Inertia on Flow Structure
shows that the movement of material is less near wall of vessel. As inertia increases, mixing process becomes better due to which the material is pushed towards the vessel wall, stirrer and agitator.
As the inertial level approaches the critical value, i.e.
Re=76 and onwards, the recirculation increases below the agitator, this flow phenomenon is displayed in Fig. 3.
Vortex centre appears in front of the lower corner of agitator almost at 45 0 and becomes stronger as the inertial level reaches at Re=500. This phenomena continues when the inertial level turns out to be equal from O(2) to O(3) (i.e. from Re=750 to Re=3000) and the vortex eye is moved to the side wall of container as displayed in Fig. 4 which shows that material is shift towards wall and creates space at centre. The increase in inertia again pushes material in vacuum. The material collides with agitator and this process is continued till homogenised mixture is obtained.

Effects of Inertia on Pressure
The fluid flow becomes asymmetric and instability starts after Re=3000, also vortex centre changes its location.  Where P max and P min are used for maximum and minimum values of pressure respectively. The graph of relative pressure drop against increasing Re is shown in Fig. 8. It is observed that the magnitude of pressure drop decreases rapidly as value of Reincreases. Therefore, the curve is horizontal asymptotic in nature.