Mathematical Modeling of Oxygen Diffusion from Capillary to Tissues during Hypoxia through Multiple Points Using Fractional Balance Equations with Memory

The diffusion of oxygen through capillary to surrounding tissues through multiple points along the length has been addressed in many clinical studies, largely motivated by disorders including hypoxia. However relatively few analytical or numerical studies have been communicated. In this paper, as a compliment to physiological investigations, a novel mathematical model is developed which incorporates the multiple point diffusion of oxygen from different locations in the capillary to tissues, in the form of a fractional dynamical system of equations using the concept of system of balance equations with memory. Stability analysis of the model has been conducted using the well known Routh-Hurwitz stability criterion. Comprehensive analytical solutions for the differntial equation problem in the new proposed model are obtained using Henkel transformations. Both spatial and temporal variation of concentration of oxygen is visualized graphically for different control parameters. Close correlation with simpler models is achieved. Diffusion is shown to arise from different points of the capillary in decreasing order along the length of the capillary i.e. for the different values of z. The concentration magnitudes at low capillary length far exceed those further along the capillary. Furthermore with progrssive distance along the capillary, the radial distance of diffusion decreases, such that oxygen diffuses only effectively in very close proximity to tissues. The simulations provide a useful benchmark for more generalized mass diffusion computations with commercial finite element and finite volume software including ANSYS FLUENT.

Computation of Melting Dissipative Magnetohydrodynamic Nanofluid Bioconvection with Second-order Slip and Variable Thermophysical Properties

This paper studies the combined effects of viscous dissipation, first and second-order slip and variable transport properties on phase-change hydromagnetic bio-nanofluid convection flow from a stretching sheet. Nanoscale materials possess a much larger surface to volume ratio than bulk materials, significantly modifying their thermodynamic and thermal properties and substantially lowering the melting point. Gyrotactic non-magnetic micro-organisms are present in the nanofluid. The transport properties are assumed to be dependent on concentration and temperature. Via appropriate similarity variables, the governing equation with boundary conditions are converted to nonlinear ordinary differential equations and are solved using the BVP4C subroutine in the symbolic software MATLAB. The non-dimensional boundary value features a melting (phase change) parameter, temperature-dependent thermal conductive parameter, first as well as second-order slip parameters, mass diffusivity parameter, Schmidt number, microorganism diffusivity parameter, bioconvection Schmidt number, magnetic body force parameter, Brownian motion and thermophoresis parameters. Extensive computations are visualized for the influence of these parameters. The present simulation is of relevance in the fabrication of bio-nanomaterials for bio-inspired fuel cells.

BIOPHYSICAL HEAT TRANSFER STUDY FOR THE CONTRACTION AND EXPANSION OF MUSCLES WITH MULTI-WALL CARBON NANOTUBES

The present investigation is associated with the contemporary study of viscous flow in a vertical tube with temperature dependent viscosity. The main flow problem has been modeled using cylindrical coordinates and flow equations are simplified to ordinary differential equations using longwave length and low Reynold's number approximation and exact solutions have been obtained for velocity, pressure gradient and temperature. Results acquired are discussed graphically for better understanding. It is seen that with an increment in the Grashof number, the velocity of the governing fluids start to decrease significantly and pressure gradient is higher for pure water as compared to multi-walled carbon nanotubes (MWCNTs) due to low density. As the specific heat is very high for pure water as compared to the multi-wall CNTs, it rise temperature of the muscles for the case of pure water as compared to the MWCNTs. Furthermore, it is noted that the trapped bolus starts to decrease in size as the buoyancy forces are dominant as compared to viscous force.

MODELING NANOPARTICLE GEOMETRY EFFECTS ON PERISTALTIC PUMPING OF MEDICAL MAGNETOHYDRODYNAMIC NANOFLUIDS WITH HEAT TRANSFER

Magnetic nanofluid technologies are emerging in numerous areas including medicine, lubrication (smart tribology), pharmacology, etc. In this paper, we examine heat diffusion in hydromagnetic nanofluids in a peristaltic system, motivated by applications in medical drug delivery systems and gastric magnetographic monitoring. The mathematical formulation encompasses momentum and heat conservation equations with appropriate boundary conditions for compliant walls. Sophisticated correlations are employed for thermal conductivity of the nanoparticles. The nonlinear boundary value problem is normalized with appropriate variables and closed-form solutions are derived for stream function, pressure gradient and temperature profile. Analytical solutions are evaluated numerically with MATHEMATICA symbolic software. Validation of computations is performed for the nonlinear moving boundary value problem via a Chebyschev spectral collocation method (CSM). A detailed study is performed for the influence of various nanoparticle geometries (bricks, cylinders and platelets). With greater magnetic field, flow velocity is enhanced for platelet nanoparticles whereas it is depressed for brick particles. Temperature is dramatically modified with nanoparticle geometry and greater thermal conductivity is achieved with brick-shaped nanoparticles in the fluid, with implications for optimized medical device systems.