Chapters authored
Nonlinear Oxygen Transport with Poiseuille Hemodynamic Flow in a Micro-Channel
In a recent paper by the authors, a well-known governing nonlinear PDE used to model oxygen transport was formulated in a generalized coordinate system where the Laplacian was expressed in metric tensor form. A reduction of the PDE to a simpler problem, subject to specific integrability conditions, was shown, and in the present work, a novel approximate analytical solution is obtained in terms of the degenerate Weierstrass P function using a compatibility relation through the factorization of the reduced almost linear ode and subject to similar boundary conditions for a microfluidic channel used in recent work by the authors. A specific form of the initial equation which was reduced has been used by Nair and coworkers describing the intraluminal problem of oxygen transport in large capillaries or arterioles and more recent work by the corresponding author describing the release of adenosine triphosphate (ATP) in micro-channels. In the present problem, a channel with a central core, rich in red blood cells, and with a thin plasma region near the boundary wall, free of RBCs is considered.
Part of the book: Nonlinear Systems
Periodic Navier Stokes Equations for a 3D Incompressible Fluid with Liutex Vortex Identification Method
The Incompressible Navier-Stokes Equations (NSEs) are on the list of Millennium Problems, to prove their existence and uniqueness of solutions. The NSEs can be formulated in a periodic 3D domain, where they are termed the Periodic Navier Stokes (PNS) Equations, and can be treated on a subspace spanning a 3-dimensional torus, or T3. Treating the PNS Equations in T3-space, this article demonstrates that a decaying of turbulence occurs in the 3D case for the z component of velocity when non-smooth initial conditions are considered for x, y components of velocity and that ‘vorticity’ sheets in the small scales of 3D turbulence dominate the flow to the extent that non-smooth temporal solutions exist for the z velocity for smooth initial data for the x, y components of velocity. Unlike the Navier-Stokes equations, which have no anti-symmetric vorticity tensor, there are new governing equations which have vorticity tensor and can be decomposed into a rotational part(Liutex), antisymmetric shear and compression and stretching. It is shown that under these recent findings, that there is a strong correlation between vorticity and vorticies for (PNS).
Part of the book: Vortex Simulation and Identification