A simple numerical methodology for BFD problems using stream function vorticity formulation

Abstract

SUMMARYThe fundamental problem of biomagnetic fluid dynamics (BFD) in a 2D rectangular channel is numerically studied. The physical problem is described by a coupled, non-linear system of partial differential equations, with appropriate boundary conditions. For the mathematical formulation, the stream function-vorticity formulation is used and the numerical solution is obtained by developing a pseudotransient numerical technique. A boundary condition for the vorticity is also constructed and grid stretching is used. The efficiency of the method is verified by comparison with other results documented in the literature. New results are also obtained for high values of the magnetic number which is the dominant factor for the determination of the flow field in biomagnetic fluid flow problems.

Biomagnetic fluid flow in a channel with stenosis

“…This increment of the magnetization can be achieved either by adding artificially created nanoparticles (in which case the biofluid behaves like a ferrofluid), which is a common technique in experimental applications like targeted drug delivery, or by decreasing the Reynolds number. Clearly, the magnetic number expresses somehow the ratio of the magnetization forces to the viscous forces; when the Reynolds number is decreased, for a specific fluid and magnetic field strength, the magnetic number is increased [11,23,28].…”

“…The development of a more stable and simpler in the application (than that in [23]) numerical methodology for BFD problems was presented in [28]. The primary elements of the method in [28] are the following: The numerical formulation is made using the stream function-vorticity formulation and the numerical solution is obtained by developing a pseudotransient numerical methodology, where the time t plays the role of an iteration parameter.…”

“…Moreover, the grid is stretched in such a way that it is configured to be dense in the area where the major disturbances of the flow field are expected. Finally, a new methodology for the construction of the boundary condition for the vorticity equation on the solid walls is also given [28].…”

“…As far as the mathematical model is concerned, the mathematical model of [11] is taken into account. Consequently, the biofluid is blood and it is considered as a homogeneous, Newtonian and electrically conducting fluid sustained equilibrium magnetization [11,23,28]. The physical problem is described by a coupled, non-linear system of PDEs resulting from the mathematical model presented in [11].…”

“…The primary elements of the method in [28] are the following: The numerical formulation is made using the stream function-vorticity formulation and the numerical solution is obtained by developing a pseudotransient numerical methodology, where the time t plays the role of an iteration parameter. This methodology is based on the development of a semi-implicit numerical technique for the estimation of the solution at the next time step (iteration).…”