In this article, we review previous studies of modeling problems for blood flow with or without transport of a solute in a section of arterial blood flow and in the presence of atherosclerosis. Moreover, we review problems of bio-fluid dynamics within the field of biophysics. In most modeling cases, the presence of red blood cells in the plasma is taken into account either by using a two-phase flow approach, where blood plasma is considered as one phase and red blood cells are counted as another phase, or by using a variable viscosity formula that accounts for the amount of hematocrit within the blood. Both analytical and computational methods were implemented to solve the governing equations for blood flow in the presence of solute transport, which, depending on the type of the investigated problem, could contain momentum, mass conservation, and solute concentration, which were mostly subjected to reasonable approximations. The form of atherosclerosis implemented in the modeling system either was either based on the experimental data for an actual human or was due to a reasonable mathematical modeling for both steady and unsteady atherosclerosis cases. For the wall of the artery itself, which is elastic in nature, modeling equations for the displacement of the artery wall has previously been used, even though their effects on the blood flow inside the artery were shown to be rather small. In some cases, thermal effects were also taken into account by including a temperature equation in the investigation. In the case of the presence of solutes in the blood, various blood flow parameters such as blood pressure force, blood speed, and solute transport were mostly determined or approximated for different values of the parameters that could represent hematocrit, solute diffusion, atherosclerosis height, solute reaction, and pulse frequency. In some studies, available experimental results and data were used in the modeling system that resulted in a more realistic outcome for the blood flow parameters, such as blood pressure force and blood flow resistance. Results have been found for variations of blood flow parameters and solute transport when compared to different values of the parameters. Effects that can increase or decrease the blood flow parameters and solute transport in the artery have mostly been determined, with particular applications for further understanding efforts to improve the patients' health care.

We consider a nonlinear three-dimensional viscoelastic fiber jet that is generated during a forcespinning process. We provide a particular case for such a rotating jet at a high rotation rate. We use a viscoelastic constitutive model for the jet equations and then applying a new slender body approach, we continue with proper scaling and perturbation technique to develop a new model for such a jet system. We find that the profiles for jet quantities versus arc length are notably different from all those in related studies reported before for either high or low rotation rates. In particular, jet radius first rapidly decreases as the arc length decreases and then reaches its macro-or nano-scale size not far away from its exit section. The present model can predict a nano-fiber jet that is entirely based on proper scaling, perturbation technique and full fluid mechanics laws and equations.

We investigated a physical system for unsteady blood flow and solute transport in a section of a constricted porous artery. The aim of this study was to determine effects of hematocrit, stenosis, pulse oscillation, diffusion, convection and chemical reaction on the solute transport. The significance of this study was uncovering combined roles played by stenosis height, hematocrit, pulse oscillation period, reactive rate, blood speed, blood pressure force and radial and axial extent of the porous artery on the solute transported by the blood flow in the described porous artery. We used both analytical and computational methods to determine blood flow quantities and solute transport for different parametric values of the described physical system. We found that solute transport increases with increasing stenosis height, blood pulsation period, convection and blood pressure force. However, transportation of solute reduces with increasing hematocrit, chemical reactive rate and radial or axial distance.

We investigate three-dimensional nonlinear rotating viscoelastic curved jets in the presence of gravity force. Applying the Giesekus model for the viscoelastic stress parts of the jet flow system and using perturbation methods with a consistent scaling, a relatively simple system of equations with realistic three-dimensional centerlines is developed. We determine numerically the relevant solution quantities of the model in terms of the radius, speed, tensile force, stretching rate, strain rate and the jet centerline versus arc length and for different parameter values associated with gravity, viscosity, rotation, surface tension and viscoelasticity. Considering the jet flow system in full 3-dimensions and in the presence of gravity can be significant, impacting the jet speed, strain rate, tensile force, stretching rate and the centerline curvature are notably increased in magnitude and the jet radius size is reduced and this becomes more dominant with larger values of the arc length, gravity, rotation and viscoelasticity. In particular, for a typical value of the gravity and for an order one value of the arc length, we found that gravity makes jet speed higher by at least a factor of 2 and makes jet radius lower by a factor of 0.6 or smaller as we compare to the corresponding values when gravity is not considered.

We consider rotationally driven nonlinear polymeric fiber jet, whose centerline is in an inclined plane, in the presence of gravity force. An empirical viscosity model is used for the polymeric fluid flow to investigate properties of the rotating inclined polymeric fiber jet. The aim of the study is to understand properties of such inclined fiber jet shape, which can be due to orientation change of an orifice in a rotating spinneret that generates such jet. Perturbation, asymptotic, scaling and numerical techniques are used to determine the nonlinear steady solutions for the jet quantities for different values of the parameters due to gravity, rotation, viscosity, surface tension and relaxation time. In contrast to the horizontal jet case, presence of gravity and jet inclination increase values of the jet speed, strain rate, stretching rate and the centerline curvature and decrease the value of the jet radius and more so with increasing the arc length, gravity, rotation rate and the relaxation time. However, a non-inclined jet with no imposed restriction on its shape and in presence of gravity leads to smaller fiber radius and larger speed as compared to the ones for inclined jet case.

In this work we study the effect of hydraulic resistivity variation on a time-dependent hydrothermal convective flow in the presence of pollution concentration. Here we treat the aquifer layer as a porous medium where Darcy's law holds, subject to the condition that the porous layer's hydraulic resistivity varies in the vertical direction. Using the weakly nonlinear approach, we obtain the conditions at which oscillatory mode of flow is preferred over the steady case. The oscillatory solutions for convective flow quantities such as vertical velocity, temperature, and pollution concentration that arise as the Rayleigh number exceeds its critical value are computed and presented for different values of the parameters.

We study the problem of drug delivery in a catheterized artery in the presence of atherosclerosis. The problem is modeled in the context of a two-phase flow system which consists of red blood cells and blood plasma. The coupled differential equations for fluid (plasma) and particles (red cells) are solved for the relevant quantities in the reasonable limits. The drug delivery problem is modeled with a partial differential equation that is developed in terms of the drug concentration, blood plasma velocity, hematocrit value and the diffusion coefficient of the drug/fluid. A conservative-implicit finite difference scheme is develop in order to numerically solve the drug concentration model with an atherosclerosis region. We find that the evolution of the drug concentration varies in magnitude depending on the roles played by the convection and diffusion effects. For the cases where the diffusion coefficient is not too small, then convection effect is not strong enough and drug was delivered mostly in the central part of the blood flow region and could not reach effectively the atherosclerosis zone. However, for sufficiently small values of the diffusion coefficient, the convective effect dominates over the diffusion effect and the drug was delivered effectively over the blood flow region and on the atherosclerosis zone.

We consider the problem of nonlinear rotating non-Newtonian jets in the presence of ambient flows. Using the original governing system of equations for such jet flows, we use scaling and perturbation techniques to reduce such system to a simplified one where the stress tensor is governed by Giesekus constitutive equations. We develop a method to take into account the effect of the ambient flow, which usually exists due to the externally imposed rotational constraint, on the formation of the non-Newtonian jet. We compute numerically the nonlinear steady solutions for the jet and determine expressions for the jet quantities like speed, stretching rate, radius, strain rate, and tensile force. These quantities are calculated for different values of the parameters such as those due to the ambient flow and viscoelasticity. We find that the ambient flow is stabilizing so that the strain rate, stretching rate, speed, and tensile force decrease with increase in the ambient flow effect, while the jet radius increases with such effect. The viscoelasticity of the fiber jet reduces notably the stabilizing effect of the ambient flow on the jet quantities, but the stabilizing effect of the ambient flow increases with the arc length of the jet centerline.

We consider rotationally driven three-dimensional nonlinear polymeric fiber jets with the effect of gravity force. We implement an empirical viscosity model for the polymeric fluid of such flows to investigate the properties of the three-dimensional polymeric fiber jets generated by the imposed rotational forces. We apply theoretical and numerical techniques to determine the expressions for the leading order nonlinear solutions for the jet quantities such as radius, speed, stretching rate, strain rate and jet centerline versus arc length, and we calculate these quantities for different values of the parameters that represent the effects due to gravity, rotation, viscosity, surface tension and relaxation time. We find, in particular, that three-dimensionality of the jet system and the presence of the gravity force on the curved polymeric fiber jet can be significant in the sense that the values of the jet speed, strain rate, stretching rate and the centerline curvature are notably raised up and the jet radius size is notably dropped down and more so with increasing the arc length and the parameters due to gravity, rotation and the relaxation time of the polymeric jet.

We consider a hydrothermal convective flow in a porous medium to investigate the effect of the vertical rate of change in thermal diffusivity. Using a weakly nonlinear approach, we derive the linear and first-order systems assuming a no-flow basic state system. The solutions for the linear and first-order systems are computed numerically using both the fourth-order Runge-Kutta and shooting methods. Numerical results obtained in this study show a stabilizing effect on the dependent variables for the case of a positive vertical rate of change in diffusivity, whereas a destabilizing effect is noticed for the case of a negative vertical rate of change in diffusivity. The present results indicate that convective flow driven by the buoyancy force is more effective if thermal diffusivity is weaker, while the opposite result holds for a stronger diffusivity effect. In particular, both velocity and convective temperature decrease with increasing diffusivity, while they increase with decreasing diffusivity. At the middle of the layer (z = 0) for x = 0, the contribution of the linear and first-order solutions to the velocity component are 0.3345, 0.3031, and 0.3679 for the respective values 0.0, 0.6, and -0.4 of the diffusivity parameter. For temperature, these contributions are 0.0167, 0.0116, and 0.0229, respectively. Some other quantitative results are provided in tabular form.

We consider mathematical modeling of blood glucose-insulin regulatory system with the additional effect of the secreted insulin by the pancreatic beta cells and in the presence of an external energy input to such system. Such modeling system is investigated to determine the time-dependent nonlinear dynamics that take place by the quantities, which represent the glucose and insulin concentrations in the blood, insulin action as well as in the absence or presence of secreted insulin due to the pancreatic beta cells. Using both analytical and numerical procedures, we determine such quantities versus time for both diabetes patients and normal human and for different values of the parameters. We find that the nonlinear effect of the dynamics of the investigated regulatory system increases the values of the insulin action and the glucose and insulin concentrations. In the absence of the beta cells effects, which can correspond to the case of severe type 1 diabetes, the plasma glucose is higher and the insulin action and the insulin concentration are less active than the corresponding ones for the case in the presence of beta cells, which is relevant for type 2 diabetes or moderate type 1 diabetes patients. For the present system, smaller values of the parameters of the model, which represent kinetics of the glucose and insulin action, insulin sensitivity, insulin secretion enhancement and the plasma insulin decay rate, can lead to notably lower values of the glucose concentration. In the presence of the secreted insulin by the pancreatic beta cells the insulin action and the insulin concentration are more effective to reduce the blood glucose, which can help to improve the diabetes patient's health.

Recently Riahi (2018) studied steady nonlinear rotating viscoelastic jet, which was subjected to the Giesekus constitutive equations for the stress tensor. He applied scaling, perturbation and numerical techniques for the viscoelastic jet with sufficiently small aspect ratio and determined the nonlinear steady solutions for the jet. He then calculated and described the results for the jet quantities such as radius, speed, stretching rate, strain rate and tensile force. In this paper we investigate stability of the nonlinear steady solutions of the time dependent form of such jet system, by superimposing small amplitude disturbances in the form of travelling waves that can grow in time, in space or simultaneously in time and in space. We find, in particular, a main condition for the existence of instability is that the growth rates of the disturbances should depend on the arc length of the jet. Under such condition, the only possible instability of the steady solution is temporal in nature and is due only to disturbances that simultaneously grow in time but decay in space. The growth rates of these disturbances increase with increasing the rotation rate, viscoelasticity and the arc length of the jet. However, the magnitude of such growth rate decreases with increasing the fluid viscosity and surface tension.

We consider a convective flow in a horizontal porous medium to investigate the marginal stability due to linear variation in resistivity in the vertical direction. The flow in the porous medium can be described by a system of partial differential equations consisting of the continuity equation for conservation of mass, the heat equation for conservation of energy, and the momentum-Darcy equation for conservation of momentum. The resistivity of the medium is defined as the ratio of dynamic viscosity to permeability. Assuming a vertically varying basic state, we apply a linear stability approach to compute the critical Rayleigh and wave numbers from the linear system. Marginal stability curves and linear solutions are obtained numerically using fourth-order Runge-Kutta and shooting methods for different values of the resistivity parameter. This work is important for applications as in the case of a porous layer of underground water whose permeability varies vertically.

We consider forcespinning (FS) of nonlinear three-dimensional rotating viscoelastic jets of Boger fluids and in the presence of gravity effect. In FS process, a fluid jet is forced through an orifice of a rotating spinneret leading to the formation of a curved jet. Applying the upper-convected Maxwell constitutive model for the stress tensor part of the governing non-Newtonian jet system and using scaling and perturbation methods, we develop a model for the jet system. We determine the nonlinear solutions for the jet quantities such as the radius, speed, stretching force, compressive force, tensile force and the jet centerline, and we calculate these quantities for different values of the jet arc length and the parameters representing rotation, viscoelasticity, gravity, surface tension and polymer viscosity. We find that the effects of rotation, relaxation time, gravity and three-dimensionality of the modeling system on the jet of Boger fluid can be significant. Beyond a very short distance from the jet exit, the jet speed, tensile force, stretching force and the jet curvature are notably increased and the jet radius is much reduced with increasing the viscoelasticity, rotation rate and the jet arc length. Strong tensile force or stretching force corresponds to weak compressive force and very thin jet of Boger fluid.

Heat and mass transfer through porous media has been a topic of research interest because of its importance in various applications. The flow system in porous media is modelled by a set of partial differential equations. The momentum equation which is derived from Darcy’s law contains a resistivity parameter. We investigate the effect of hydraulic resistivity on a weakly nonlinear thermal flow in a horizontal porous layer. The present study is a realistic study of nonlinear convection flow with variable resistivity whose rate of variation is arbitrary in general. This is a first step for considering more general problems in applications that involve variable resistivity that may include both variations in permeability and viscosity of the porous layer. Such problems are important for understanding properties of underground flow, migration of moisture in fibrous insulations, underground disposal of nuclear waste, welding process, petrochemical generation, drug delivery in vascular tumor, etc. Using weakly non-linear procedure, the linear and first-order systems are derived. The critical Rayleigh number and the critical wave number are obtained from the linear system using the normal mode approach for the two-dimensional case. The linear and first-order systems are solved numerically using the fourth-order Runge-Kutta and shooting methods. Numerical results for the temperature are presented in tabular and graphical forms for different resistivities. Through this study, it is observed that a stabilizing effect on the dependent variables occurs in the case of a positive vertical rate of change in resistivity, whereas a destabilizing effect is noticed in the case of a negative vertical rate of change in resistivity. The results obtained indicate that the convective flow due to the buoyancy force is more effective for weaker resistivity.

In the usual forcespinning (FS) process, a fluid jet is forced through an orifice of a rotating spinneret leading to the formation of a jet with curved centreline. In this paper, we investigate the properties of nonlinear viscoelastic jets during the FS process. We apply scaling and perturbation techniques to determine the modelling system for the viscoelastic jets, subjected to the Giesekus constitutive equations for the stress tensor. We calculate numerically the expressions for the nonlinear steady solutions for the jet quantities such as radius, speed, stretching rate, strain rate, tensile force and trajectory. We determine these quantities for different values of the parameters such as those representing the effects due to rotation, surface tension, viscosity, jet drag and viscoelasticity. We find, in particular, that the fibre jet radius decreases and the tensile force increases with the jet arc length as well as with increasing the effects of rotation and viscoelasticity. Both viscosity and surface tension indicate stabilizing effect on the viscoelastic jet. In addition, strain rate, stretching rate and the jet speed increase with the arc length, viscoelasticity and rotation rate.

In this paper we apply an empirically generated relation to investigate strain hardening of nonlinear polymeric jets, such as those due to polymeric melts like Boger fluids, which can be generated by the forcespinning process (FS). We apply scaling and perturbation analyses to determine a simplified modeling FS system for the nonlinear fiber jets and calculate numerically the expressions for the nonlinear steady solutions of jet quantities such as radius, speed, stretching rate and strain rate. We find, in particular, that although such jet quantities are not affected upstream by the strain hardening, in the downstream away from the jet exit, the strain rate, stretching rate and the jet speed are reduced and the fiber jets are thickened by the strain hardening effect.

We consider a nonlinear non-Newtonian rotating jet flow whose centerline is curved and is elongated by an electric force due to an imposed electric field. The jet is driven both by the imposed electric and rotational forces. We introduce a non-Newtonian viscosity model, which, in particular, takes into account both extension thinning and thickening of the jet. From the governing equations and the boundary conditions of such jet flow, we construct a modeling system based on the slender-body theory for such nonlinear jet and calculate numerically the expressions for the nonlinear steady solutions for the jet quantities such as radius, speed, stretching rate, induced electric field and surface charge versus arc length. We determine these quantities for different values of the parameters that represent effects due to rotation, electric field, electric conductivity, viscosity and relaxation time. We find, in particular, that the jet speeds up, stretches up, and its diameter reduces significantly with increasing the imposed electric force, rotational forces and the jet relaxation time. The notable jet radius reduction that is due to strong electric and rotational effects is found to be for jet dominated by negative surface charge whose magnitude enhances with the rotation rate and the electrical conductivity.

In the usual forcespinning (FS) process, a meso-scale fluid jet is forced through an orifice of a rotating spinneret, where the ambient fluid is air. This leads to the formation of a jet with a curved centerline. In this study we make use of a phenomenological viscosity model for polymeric fluid to investigate the properties of nonlinear polymeric fiber jets during FS process. We apply multi-scale and perturbation techniques to determine the governing modeling systems for such nonlinear rotating jets and their stabilities. First, we calculate numerically the expressions for the leading order nonlinear steady solutions for the jet quantities such as radius, speed, stretching rate, strain rate and trajectory versus arc length, and we determine, in particular, these quantities for different values of the parameters that represent effects due to rotation, viscosity and relaxation time. Next, we calculate the stability of the nonlinear jet versus different types of perturbations. We find that the nonlinear fiber jet flow can be stable in most cases and uncover conditions for which fiber radius reduces and the jet speed or stretching rate increases.

Nanofibers are known to have incredible properties and applications in diverse field such as defense, energy, aerospace, filtration, and biotechnology, to name a few. Forcespinning (FS) is a new experimental process that can produce nanoscale fibers under the action of rotating forces. In the usual FS process, a mesoscale fluid jet is forced through an orifice of a rotating spinneret, where the ambient fluid is air. This leads to the formation of a jet with a curved centerline. In this study we apply multiscale and perturbation techniques to investigate rotating fiber jets and their stabilities in the presence of aerodynamic drag force due to the ambient air that is known to exist experimentally in the FS process. First, we calculate numerically the expressions for the leading-order steady solutions for jet quantities such as radius, speed, and trajectory versus arc length, and we determine, in particular, the results for such quantities and their variations in the presence of aerodynamic effect. Next, we calculate the stability of the fiber jet versus temporally growing, spatially growing, or spatiotemporally growing perturbations and determine the results for the growth rates of the effective perturbations versus jet flow parameters.