Analytical Theory of Bioheat Transport, Essay Example

Literature Review

This literature review chapter provides the theoretical framework of the thesis and discussesrelevant literature on the subject of solid inclusions and fluid flows in a porous medium. This will be further elaborated upon later in the review.This research study proposes we study incompressible flows of a viscous fluid past a cylinder or a sphere suspended in a porous matrix of permeability k based on Brinkman equations. We existing research studies regarding solid inclusions embedded in porous media.The literature review is portioned into sections that contain the various literature works related to different areas of research related to the thesis topic.

The first section of this review describes how fluid flows through a porous medium. The section outlines this process and gives an example of the process and explains the purpose of analyzing fluid flows in porous media.The second section of the review highlights two theoretical model types used in analyzing fluid flow, which are: (1) the Darcy model and (2) the Brinkman model, and how these two models intersect to analyze flow over solid inclusions, for the purpose of examining transport modalities in porous media. The third section examines transient flows, which includesthe process of steady flows and unsteady flows through porous media, as well as issues that can arise with flows. The fourth section examines slip behavior, including no-slip boundary conditions to describe liquid-solid boundary motion. Finally, the fifth section examines the use of mathematical tools to analyze steady flow processes of viscous, incompressible fluids in a porous medium.

Fluid Flow in a Porous Medium

Fluid flow and fluid transport in porous media have numerous applications in various areas of science, such as energy science, engineering science, earth science, as well as many other scientific fields of research. This is particularly true for the fields that are concerned with such areas as oil and gas recovery and exploration of geothermal power, for example. Various modes of assessmentthat give expression to fluid flow and transport are important to these areas of science and research.Research on how fluid behaves when flowing through porous media has been ongoing, particularly during early oil production times when oil was harvested from far beneath the earth’s surface [11]. Darcy’s Law has been generally used for identifying how fluid flows through a solid body. Darcy’s Law is the principle that directs how fluid flows through a medium. Brinkman’s model is a variation of Darcy’s Law that accounts for the transition of flow between boundaries in a porous medium. Whereas typical modelling of solid inclusions in porous media includes identifying problems with both the Darcy model and the Brinkman model. Regarding this concept, a major application of modelling fluid flow in a porous medium is simulating the process of how water and oil flows in rock, for the purpose of optimizing the production of oil and gas. Simulating groundwater contamination is another application of modelling fluid flow in a porous medium.Groundwater sustainability is a significant factor for many who depend on it [11]. Therefore, it is important to mitigate environmental problems due to polluted groundwater, which can result from such issues as chemical leaks in tanks or pipelines.

In fluid (flow) dynamics, mathematical statements used are called governing equations, which consists of momentum equations, continuity equations, and energy equations of physics. Governing equations used to calculate the pressure fields and velocity of fluid flow are partial differential equations (PDEs), and these equations apply in a fluid at every point that is modelled as a continuum. When PDEs areintegrated, then constants or arbitrary functions appear inthe mathematical solutions. PDEs are evaluatedwith added velocity field statements and it is possible they are also evaluated with the addition of gradientsof the flow domains at natural boundaries. These velocity field statements are called boundary conditions (Nield & Bejan, 2006).

One way of modelling this type of scenario requires the use of two mediums such as clean groundwater and a toxic fluid, for the purpose of understanding how the toxic fluid spreads. This is a process that includes simulating the flow of fluids consideringthe effect of forces acting on it, such as viscous forces, gravity forces, and capillary forces.The specific mathematical model used consistsof partial differential equationsand specific boundary conditions, using therules of conservation of such factors as mass and momentum. In addition, Darcy’s law, which is used as a means of describing fluid flow through a porous medium,is employed for each phase to predict the progress and outcomes of flow scenarios [11].

When considering fluid flow, the term “porous medium” is relatively defined [11] as a solid medium (or matrix) that distributes miniature interconnected cavities that make up a quantifiable portion of its volume. In other words, porous media are characterized by a division of the total volume in a pore space (voids) and a solid matrix (frame), with one or more fluids (liquids or gases) filling the pore space. Yet, another way to describe a porous medium is it being a material made up of a solid matrix that has an interconnected void [11].

The larger cavities are responsible for the bulk of fluid flow through the medium and the smaller cavities do not affect bulk flow; however, the smaller cavities are instrumental in transport phenomena, such as diffusion. Pore structures in porous media are very complex; therefore, they are characterized by macroscopic physical properties, which are associatedwiththe underlying microscopic pore structures. Permeability and porosity are examples of these properties. Permeability relates to the medium’s quantity capacity for the purpose of transporting fluid. Porosity refers to the fractional volume of void space relative to total volume in the medium [12].

Slow Viscous Flow

According to Wang [19], many practical situations involve slow viscous flow in a porous medium. The examples given are oil and gas recovery, catalytic reactors, thermal insulation, and groundwater movement, which is often described by the Darcy equation, mainly due to low flow velocity. Scientific studies of slow fluid flow often relate to the Stokes’ paradox, which postulates that there can be no two-dimensional slow fluid flow around a disk. The paradox results from the non-existence of solutions to simplified equations for fluid velocity satisfying boundary conditions at infinity at the surface of a body. This is because a body in motion through a viscous fluid, at a constant speed, experiences at least some resistance.

The equation below (the Darcy-Brinkman equation) is well accepted for describing the flow in a porous medium [19]. It is a generalized version of the Darcy equation with the viscous drag terms retained, due to the Darcy equation not being able to “account for the tangential shear on the solid boundaries” [19] in a porous medium with solid inclusions.

p’ = _e²– / K(1)

Where p’ is the pressure, K is the permeability,  is the velocity vector, = _e is theeffective viscosity of the porous medium, and =  is the viscosity of the fluid.”It is important to note that if = _e is zero, then this isthe Darcy equation. However, if K, then it reduces to the Stokes’ equation (Wang, 2010, p. 261), resulting in the Stokes’ paradox and the predicted pressure drop caused by multiple inclusions. Consequently, the Darcy-Brinkman equation for flow over solid inclusions is essential in transportphenomena research involving mass transfer, heat, and momentum in porous media. However, the Stokes’ paradox is not seen in porous media.

According to Leont’ev [8], the Brinkman based model adequately approximates a description of slow flows of an incompressible fluid in a porous medium that is highly-penetrated when forces within a body are absent. This can produce analytical solutions to the problem of fluid flow past a sphere, as well as a cylinder, in a porous medium.[15] Pop and Ingham developed an exact solution for forced flow past an embedded sphere in a porous medium. They use just the Brinkman model and conclude that the flow configuration has no flow separation. In comparison, Pop and Cheng [14] analyze a circular cylinder in a porous medium with a steady incompressible flow past the cylinder, which also bases on the Brinkman model.

TransientFlows

Transient flow is a state of fluid flow pressure and velocity that changes over time [12].

Palaniappan [12] presents representations of general solutions with initial conditions given by transient flows as Stokes’ flows (Reynolds numbers) through porous media, by velocity and pressure fields, in addition to Brinkman model flows. This is significant to solving problems in Stokes flow theory. These problems include stream function techniques, analytical function methods, numerical computations, and differential representation techniques, which can be used to solve problems in fluid dynamics, elasticity, as well as problems with unsteady creeping flow and transient flow problems.The treatment of unsteady motions of particles in slow, creeping flows often includes the use of quasi-steady approximating. This process is based on the assumption that the motion of the particle is so slow that it is made up of a series of steady states, which renders the fluid in a steady Stokes’ flow.

According to Zhu, Waluga, Wohlmuth, and Manhart [21], unsteady flow can occur in porous media due to “unsteady boundary conditions or unsteady pressure gradients”. This unsteady flows can be found in the technical, environmental, or bio-mechanical fields, for instance. Examples of unsteady flows through porous media include a mass transfer between “the turbulent atmospheric boundary layer and a forest [6], a turbulent or wavy water stream and a plant canopy [9], or blood flow through organs [5] [21].

Regarding convection in porous media, researchers have taken particular interest in heat transfer convection in porous media. This is relative to applications in various disciplines, such as engineering, chemical catalytic reactors, groundwater hydrology, petroleum reservoirs, polluted ground water, and ?ltration processes [14] [15].

Additionally, Palaniappan proposes that there is a link between flow through a porous mediums and creeping flow. Regarding this, the Brinkman equations are efficient for modeling porous media and broadly apply to high porosity systems. In contrast, Darcy’s equations are used for modeling low porosity systems and are of lower merit than the Brinkman equations for this purpose, though some convection problems have been encountered due to some limitations of porous media equations [12].

Issues regarding flows relate to instabilities. Petrie and Denn [13] state that this is significant to examining instabilities that arise in shear and extensional flows in polymeric liquids processing, which usually occurs at very low Reynolds numbers. This is due to liquids with low molecular weight not showing unstable behavior, as there is much variation in the molecular structures of different polymers due to their different characteristics and qualitative similarities. According to Basset [2], research studies on viscous liquid instability concluded that motion was stable for small velocities but became unstable when velocity was gradually increased. It was noted that instability commenced at the point of increased velocity. This relates to the mechanics of fluid flows.

Nield and Bejan [11] examine the mechanics of how fluid flows through a porous medium, which they describe as material consisting of a solid matrix with an interconnected void, which allows the fluid to flow through the material. Nield and Bejan [11] examine the effects of forced convection, external natural convection, internal natural convection, mixed convection, double-diffusive convection, and convection with changes in phases on porous media. Brinkman [3] [4]argues that the equation for momentum in porous media flow must be reduced to that of the viscous flow limit, for fluid ?ow through highly permeable porous.

Brinkman illustrates the viscous force from flowing fluid on a swarm of dense particles embedded within a porous mass with the Stokes’ formula, which is

K = 6_o (2)

Where K equals the force from the fluid on the particle,  equals the fluid viscosity, R equals the spherical particle radius, and ?_o equals the flow velocity far from the particle. Brinkman further elaborates on the swarm of dense particles by examining their permeability when they are more closely packed and derived a formula for the permeability of the swarm of particles as a function of its porosity [4].

Slip Behavior

Engineers and scientists want to model fluid flow past a solid surface. This requires researchers to make assumptions about the nature of motion of the fluid flow (i.e., the boundary condition) at the edge of a solid interface. A boundary condition that is one of the simplest is called a no-slip condition. A no-slip conditionpostulates that a liquid element (that is in line witha surface)assumesthe surface’s velocity. The no-slip condition has been used successfully in modeling some types of flow characteristics. However, in some situations this leadsto unrealistic behavior. An example of this is a liquid spreading on a solid substrate.

Nield and Bejan [11]examined solid-liquid interfaces with no-slip boundary conditions.In that setting, the behavior of solid-liquid interfaces is complex due to the interplay of various physicochemical parameters. These parameters may include surface charge, surface roughness, impurities, dissolved gas, wetting, shear rate, and pressure [7]. Leont’ev [8] mentions that a no-slip condition between a porous medium and a solid, especially when using the Brinkman equation, must be replaced by a nonzero flow velocity condition at the boundary of the solid. Leont’ev [8] points out that both the Brinkman equation and the continuity equation approximate the description of slow flows of a Newtonian incompressible fluid in a highly-penetrated porous medium in the absence of body forces.

Research studies reveal that typical micron-size dimensions, or smaller, apparently demonstrate a violation of no-slip boundary conditions for Newtonian liquids flowing near a solid surface described in [7]. However, Leont’ev proposes that the use of non-approximated slip conditionsis inadequate. This is assumed because seepage velocity of flows in porous media is actually the fluid velocity averaged over a larger volume than with microstructure dimensions. Hence, an impermeable boundary condition must allow a nonzero flow velocity [8].Sbragaglia and Prosperetti [16] state that research in micro-fluidics originated from interest in slip boundary condition possibilities when a liquid and a solid wall contact each other.

According to Lauga, Brenner and Stone [7], a slip boundary condition can describe liquid-solid boundary motion, which includes three different slip types: (1) microscopic slip, which is seen at individual molecule scales; (2) actual continuum slip, which is seen at a liquid-solid boundary; and (3) apparent slip (effective slip), which occurs as a result of motion over complex, heterogeneous boundaries. It is significant to note that either real or apparent slips can have some reduced resistance to fluid motion.

The PDEs, or model equations, are supplemented by a set of boundary conditions at the surface of the inclusion. The no-slip boundary condition is usually applied at the surface of the solid. Pop and Cheng [14]derived analytical solutions, by using the non-slip conditions at the solid, for velocity and pressure fields for two-dimensional uniform flow around a circular cylindrical cross-section. Pop and Ingham [15] and Wang [19]also provided closed form results from their research to show slow flow past a sphere in a porous medium. Based on experiments [13], it has been found that a `velocity slip’ occurs at the solid surface during the solid-fluid interaction. One purpose of the current investigation is to draw attention to the fact that the use of no-slip condition is generally inadequate. Additionally it is significant to note that boundary-value problems with slip boundary conditions have been discussed in the content of viscous hydrodynamics[2] [7]. In this research and outlined in the thesis, the slip boundary conditions in the model for flow around solid inclusions in a porous medium is utilized as well as the effect of slip on velocity and pressure fields.

Mathematical Tools

When considering modeling the flow of fluids in a porous medium, basic continuum mechanics laws exist to describe fluid flows by utilizing governing equations. This is based on principles of conservation relating to momentum, energy, and mass. The equations show the relationship between changes in position and time with momentum, energy, and mass. To facilitate this, the equations must be accompanied by a appropriate rheologicalor constitutiveequationsthat define a specific fluid [11]. This specific fluid has an integral mathematical relationship relating to stress and rateofstrain tensors in the flow condition, resulting in the set of governing equations being closed and gives an indication of flow velocity and flow stress fields [21].

Constitutive equations should be used due to the complexity of various mathematical methods, which are usually used to satisfy the flow condition. This is due to the fact that it is not possible for only one type of constitutive equation to satisfy all purposes. When choosing an equation, various factors are to be considered. These include phenomena to be captured, flow type (e.g., transient or steady, extension or shear, etc.), computational resources availability, level of required accuracy, etc. Rheological equations, as mathematical tools, should usually be as simple as possible. These should involves only a few parameters and variables but should still be capable of predicting how complex fluids behave in complex flows [11].

Mathematical tools are required for generating mathematical results and solutions to PDEs with dependence on vector field boundary conditions. According to Abramowitz and Stegun [1], mathematical tools have proven valuable in various fields such as scientific, physical, and engineering fields. As it relates to PDEs, Myint-U and Debnath[10] point out that mathematical tools originated from the study of surfaces in geometry and a variety of problems in mechanics.

Mathematical tools are important to PDE research, as they are often a part of mathematical analysis in science and engineering problems, and they also express various fundamental laws of nature. With geometry, PDE research plays a pivotal role in the areas of analysis and physics as it pertains to modern mathematics. This is important because virtually “all physical phenomena obey mathematical laws that can be formulated by differential equations” [10].

Zhan and Xu [20] illustrate how mathematical equations can solve problems related to physical and physiological processes in solid tumors regarding fluid transport and drug transport. Solid tumors have varying sizes and microvessel branch patterns, depending on their size and type; therefore, research into the effectiveness of fluid and drug transport for the purpose of treating tumors utilizes mathematical modelling. Computational results from such studies can give insight into the most effective means of drug delivery to achieve the best outcomes for facilitating effective solid tumor cell death in the shortest period of time, which enhances therapeutic effects of the drugs.

According to Sciume et al. [17], time-dependent behavior of tumor masses can be analyzed using governing mathematical formulations, which uses mechanics of multiphase porous media to model solid tumor evolution. As it relates to fluid flow, solid tumors are often described as “homogeneous, viscous fluid and employ reaction-diffusion-advection equations for predicting the distribution and transport of nutrients and cells”. . . including “cell diffusion, convection and chemotactic motion” whereas momentum balance equations govern cell proliferation” [17].These types of mathematical tools are significant to such initiatives as cancer research.

Soltani and Chen [17] state that research on solid tumors considers how the size and shape of a tumor (solid mass) effects drug delivery to the tumor. This is based on applying governing mathematical equations for fluid flow in porous media, with appropriate boundary conditions. This relates to possible problems with seepage flow when considering drug delivery systems, as a main theme of the current thesis is the flow around solid inclusions that are embedded in porous media, which is important in the field of biological applications.

Conclusion

This review of the literature outlines a theoretical framework for the current study, based on fluid flow and solid inclusions embedded in porous media for the purpose of analyzing and discovering how new exact results for velocity and pressure fields may be yielded through the application of boundary-value problems that create challenges regarding such factors as viscous flow, seepage flow problems, PDEs, velocity slip, mixed boundary conditions at cylinder or sphere surfaces, and the Stokes paradox. It is important to note that the purpose of the thesis research includes overcoming the challenges by employing effective governing mathematical equations as tools and methods to derive analytical solutions to the challenges.

The two theoretical model types (the Darcy model and the Brinkman model) have also been highlighted in this review, as they are significant to analyzing flow over solid inclusions embedded in porous media, which is a main focus of the thesis. In addition, transient flows through porous media, slip conditions, and mathematical tools have been examined to analyze fluid interaction with solids in a porous medium. The purpose of which is to originate mathematical results for pressure fields and velocity of fluid flow past spherical or cylindrical inclusions in porous media, including slip boundary conditions at the surface of solid inclusions.

As mentioned in the thesis proposal, the objective is to overcome difficulties imposed by the set of boundary-value problems by choosing suitable methods of solutions. Since the governing equations are linear, we plan to use the method of separation of variables in two and three-dimensions (Myint-U & Debnath, 2007). The Bessel functions of integral and fractional orders, together with Chebyshev and Legendre polynomials (Abramowitz & Stegun, 1964) will be employed as mathematical tools in the derivation of analytical solutions. In addition, expected results include anticipating that the approach, used in this research, to the proposed boundary-value problems will yield new exact results for the velocity and pressure fields. This in turn can be used to analyze the effect of permeability and slip coefficient on the flow fields in the presence of boundaries. Such mathematical results are essential for a better understanding of modeling in porous media with inclusions especially, for design purposes.

Limitations and Further Research

It should be noted that the data and information presented in this literature review may have limitations due possible influence from bias or personal preferences of the researchers of the studies reviewed. With this in mind, future researchers should note the possibility of methodological flaws on the part of previous researchers so as not to replicate any flaws. In addition, the current research and thesis cannot possibly include all subjects and literature related to the topic.

In conclusion, this literature review and subsequent thesis research is fundamentally significant to various fields in science such as geology, engineering, chemistry, and biology to derive applications for current and future research. The importance of analytical solutions to problems and challenges relating to fluid flow past solid inclusions embedded in porous media are reviewed herein and should be considered for formulating effective solutions.

References

[1] Abramowitz, M., Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1964.

[2] Basset, A. B., A Treatise on Hydrodynamics, vol. 30, Ch. 21, 22, Cambridge, Deighton Bell, 1888.

[3] Brinkman, H. C., “A Calculation of The Viscous Force Exerted by a Flowing Fluid Dense Swarm of Particles,” Applied Science Research, vol. 1, pp. 27-34, 1947.

[4] Brinkman, H. C., “On The Permeability of Media Consisting of Closely Packed Porous Particles,” Applied Science Research, vol. 1, pp. 81-86, 1947.

[5] Fan, J., Wang, L., “Analytical Theory of Bioheat Transport,” J. Appl. Phys., vol. 109, no. 10, pp. 104702, 2011.

[6] Finnigan, J., “Turbulence in Plant Canopies,” Annu. Rev. Fluid Mech., vol. 32, no. 1, pp. 519-571, 2000.

[7] Lauga, E., Brenner, M. P., and Stone, H. A., Microfluidics:The No-Slip Boundary Condition, in: C.Tropea et al.(eds.), Handbook of Experimental Fluid Mechanics,Springer, Berlin, p.1219, 2007.

[8] Leont’ev, N. E., “Flow Past a Cylinder and a Sphere in a Porous Medium withinthe Framework of the Brinkman Equation withthe Navier Boundary Condition,” Fluid Dynamics, vol. 49, no. 2, pp. 232-237, 2014.

[9] Lowe, R. J., Koseff, J. R., Monismith, S. G., “Oscillatory Flow through Submerged Canopies: 1. Velocity Structure,” J. Geophys. Res., vol. 110, no. C10, 2005.

[10] Myint-U, T., and Debnath, L., Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition, Birkhauser, Boston, 2007.

[11] Nield, D. A., Bejan, A., Convection in Porous Media, Springer, New York, 2006.

[12] Palaniappan, D., “On Some General Solutions of Transient Stokes and Brinkman Equations,” Journal of Theoretical and Applied Mechanics, vol. 52, no. 2, pp. 405-415, 2014.

[13] Petrie, C. T., Denn, M. M., “Instabilities in Polymer Processing,” AIChEJ, vol. 22, pp. 209-236, 1976.

[14] Pop, I., Cheng, P., “Flow Past A Circular Cylinder Embedded In A Porous Medium Based On The Brinkman Model,” Int. J. Engng. Sci., vol. 30, no. 2, pp. 257-262, 1992.

[15] Pop, I., Ingham, D. B., “Flow Past A Sphere Embedded In A Porous Medium Based On The Brinkman Model,” In. Comm. Heat Mass Transfer, vol. 23, no. 6, pp. 865-874, 1996.

[16] Sbragaglia, M., Prosperetti, A., “Effective Velocity Boundary Condition at a Mixed Slip Surface,” J. Fluid Mech., vol. 578, pp. 435-451, 2007.

[17] Sciume, G., Shelton, S., Gray, W. G., Miller, C. T., Hussain, F., Ferrari, M. . . . Schrefler, B. A., “A Multiphase Model for Three-dimensional Tumor Growth,” New Journal of Physics, vol. 15, no. 1, pp. 015005, 2013.

[18] Soltani, M., Chen, P., “Effect of Tumor Shape and Size on Drug Delivery to Solid Tumors,” Journal of Biological Engineering, vol. 6, no. 4, pp. 1-15, 2012.

[19] Wang, C. Y., “Darcy-Brinkman Flow with Solid Inclusions,” Chem. Eng. Comm., vol. 197, pp. 261-274, 2010.

[20] Zhan, W., Xu, X. Y., “A Mathematical Model for Thermosensitive Liposomal Delivery of Doxorubicin to Solid Tumour,” Journal of Drug Delivery, vol. 2013, Article ID 172529, 2013.

[21] Zhu, T., Waluga, C., Wohlmuth, B., Manhart, M., “A Study of The Time Constant in Unsteady Porous Media Flow Using Direct Numerical Simulation,” Transp Porous Media, pp. 1-19, 2014.