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A Treatise on Hydrodynamics, Dissertation – Hypothesis Example

Pages: 12

Words: 3190

Dissertation - Hypothesis

The study intends to draw analytical solutions from the fourth order boundary-value problem for ψ(r, θ). This is a critical value in relation to how spheres and cylinders impact fluid sows, specifically in relation to when a cylinder or a sphere is placed in a uniform and shear flow occurs from basic fields. The study attempts to assess the impact of the permeability k and the slip coefficient ξ on the velocity and pressure fields. The anticipated results are that the approach taken to the boundary condition will reveal new distinct results for the velocity and pressure fields. It is expected that these results will provide for analysis of permeability k and slip coefficient ξ and their impact on the flow field in the presence of boundaries. Boundaries tested by the equations in this study put in the place essential factors related to permeability, which functions as a too for operations in the science and engineering industry. The study believes that these mathematical results are a critical tool in optimizing, as well as understanding, the modelling of porous mediums, and that taking this approach will further intelligence in design.

The study expects that approaches to the boundary-value problems will ultimately produce new exact results for the velocity and pressure fields in respect to fluid flow transport in porous mediums. It is believed the findings of this research will supplement advancements in engineering sciences such the exploration of geothermal power, areas as oil and gas recovery. Various modes of assessment that give expression to fluid flow and transport are important to these areas of science and research. The study hypothesizes that advancing knowledge in permeability techniques in fluid flows through the use of acute mathematical models can enhance the efficiency of engine dynamics and performance, as well as how oil is harvested for supply and demand. These exact results will better enhance science related to permeability for engineers in their daily application, but also in how they can be applied in academic areas to further knowledge on the topic. The hypothesis takes the position that through distinct calculations and the use of proven mathematical models measuring porous mediums, this in turn can be used to analyze the effect of permeability k and slip coefficient ξ on the flow fields in the presence of boundaries. Of course, such mathematical results are essential for a better understanding of modeling in porous media with inclusions especially, for design purposes. The hypothesis of this study extends beyond the belief of increased performance, control, and manipulation of permeable mediums to reach desired goals, but there is also a logical progression that will evolve out of this finding that ultimately will contribute to the field of science and engineering ways that have rarely been seen. For example, improved understanding in how permeability works can enhance functionality and performance of valued methods of fluid extractions in important industries. The optimum governing equation, measuring aspects of energy, continuity, and momentum can be used to calculate the necessary pressure fields for ideal or premium velocity. This requires an acute understanding of boundary conditions which impact the velocity statements.  The Brinkman and Darcy model both provide insight into the method needed to accomplish this goal, especially as it relates to the application of gas and oil recovery. Some distinct features can be identified within the function of slow viscous flow, specifically when applying Stoke’s paradox, the idea that no slow fluid flow can exist around a disk. The resistance experienced represents the key boundaries which the research in this study intends to overcome.

The theoretical framework for the current study, based on the epistemologies acquired within previous research on fluid flow and solid inclusions embedded in porous mediums. The study seeks to form analytical solutions for the fourth order boundary-value problem for ψ(r, θ) when a cylinder or a sphere is placed in a uniform and shear flow basic fields. Through the analysis of the permeability k and the slip coefficient ξ on the velocity, flow past solid inclusions embedded in a porous medium is evaluated based on the mathematical solutions through which their functionality can be understood in the form of vector partial differential equations. In this thesis, we propose to 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. The vector partial differential equations are transformed in terms of a scalar function ψ(r, θ), known as the Stokes stream function. In addition to the impenetrability condition at the boundary surface, we use a ‘velocity slip’ at the surface – a physical condition that has not been exploited in the porous media context.

The study intends to drive analytical solutions for the fourth order boundary-value problem for ψ(r, θ). The problem arises when a sphere or cylinder that is embedded in a uniform and shear flow basic field. The impact of the permeability k and the slip coefficient ξ on the velocity and pressure fields are addressed in this study through the following problem design:

The set of boundary-value problems (7), (9) and (14), (15) impose the following challenges:

  • the presence of viscous term ∇2u in (1) leads to fourth order PDE for the stream function ψ as compared to Darcy flow where ∇2u is absent;
  • the consideration of velocity slip introduces mixed boundary conditions at the surface of the cylinder or the sphere;
  • in the two-dimensional case, the limit of large permeability coefficient k produces no solution (Stokes paradox).

The partial differentia equations entail the mathematical solutions relative to vector partial differential equations. To solve for the above problems, the optimal models are incorporated to achieve desired results.

The considerations of this study consider steady viscous flow of fluids that are traditionally incompressible in regards to flowing through porous mediums. The research assumes that slow motion creeping due to external forces, driven by the governing equation of motion, based on Brinkman’s model can be utilized to identify analytical solutions. As previously mentioned, the authors believe these results can be acquired using the Brinkman flow model which will better allow the study to determine flow past spherical or cylindrical inclusions in porous media, including slip boundary conditions at the surface of solid inclusions. Choosing suitable methods of solutions to overcome difficulties imposed by the set of boundary-value, is the main obstacle the hypothesis of this study seeks to overcomes and for this it needs accurate calculations to incorporate as models are tested and compared against one another. A key limitation that could potentially impact this study can be the chosen equation model used. For instance, momentum inserted into a porous medium flow can be measured by defining factors that influence how fluids will function. There can be a decline or regression that occurs based on design and this can reduce or improve performance depending on the intended result. The way permeability functions is largely reliant on natural laws related to linear equation factors.

Linear interpolations are part of the evaluation process used in this study to assess values for the solution of boundaries. As Abramowitz and Stegun, (1964) note, “with linear interpolation there is no difference in principle between direct and inverse interpolation. In cases where the linear formula provides an insufficiently accurate answer, two methods are available. We may interpolate directly, for example, by Lagrange’s formula to prepare a new table at a fine interval in the neighborhood of the approximate value”, the authors further notes that the inverses of this material is then backed and then apply accurate inverse linear interpolation to the sub-tabulated values (Abramowitz and Stegun, 1964). The authors’ findings demonstrate the critical importance of establishing sound mathematical models to find effective solutions. Due to the fact that solid inclusions are genuinely bigger, with circular, or spherical configurations, which are embedded in a porous incompressible fluids which are composed of smaller fine particle-like matrix). The motion of the fluid is assumed to be steady and flows around the inclusion. Two types of theoretical models are commonly use to describe flow problems in porous mediums. These are the Darcy model • the Brinkman model The Darcy model is employed for small permeability situations [7], while the model proposed by Brinkman [3], [4] is used for moderate to high porosity systems. Mathematically, the difference is that the Brinkman model uses higher order partial derivatives of the fluid velocity, whereas the Darcy model does not. It has been shown in [7], that the Brinkman equations are more appropriate and realistic for describing the flow past solids in a porous medium. Therefore, the study uses the Brinkman model which is also referred to as the Darcy-Brinkman model.

Permeability levels are measured based on high levels and low levels which is primarily influenced by geological factors, which define whether a medium is geologically porous. Hewitt (2014) notes that “Geological porous formations are commonly interspersed with thin low permeability layers… The flow is found to depend only on the ratio of the height and relative permeability of the interior layer, given by the impedance Ω. As Ω is increased, the dominant horizontal length scale of the flow increases, and, surprisingly, Nu can increase before decreasing markedly for larger values of Ω”(Hewitt, 2014). The author finds that permeability, like convection, works within a closed domain and it’s pushed forward by dense buoyancy associated with the upper boundary or layer of the medium.  There are key ways to acquire permeability that can be reached through the use of the Brinkman and Darcy model. The geological permeability can be measured through these models using a range of differential linear equations which keep the performance of fluid in these environments in a realistic light.

Brinkman’s studies are closely analyzed and incorporated within this study for their intrinsic value. It is the belief of this author that Brinkman can provide something within the field of study that best supports the initial theory on porous mediums. For example, his (1947) work on “a Calculation of The Viscous Force Exerted by a Flowing Fluid Dense Swarm of Particles,” contains research applicable to the acquisition of an effective mathematical model. Likewise, his work on medium permeability as it relates to studies of packed porous particles has relevant applications within this study as well. Through the initial works of Brinkman, in collaboration with his models for the theoretical framework of this study it has become a clear cut case to identify the research needed in the field but also the possibility for guaranteed results. As previously noted, in the literature review, Brinkman’s findings have been invaluable in the forming of the current hypothesis as they enable researchers to have finite data related to both calculations on permeability as well as experimental and theoretical epistemology related to dense swarms of particles as they behave within flowing fluids exerting viscous forces. The most valuable aspect of Brinkman’s work can be seen in the calculations which his research provides the field of study. The steps through which the hypothesis of this study were acquired start with the Brinkman model but modern application of more contemporary research has positioned this work with a conscious understanding of its place within its field. Follow up works to the Brinkman model which take into account the influence of boundaries to porous mediums have brought this research closer to more effectively finding solutions that can supplement a wide range of technical functions. Specifically, the works of Lauga, Brenner, and Stone (2007) in their study of microfluidics, has been applied incorporated here in the Brinkman model to evaluate no-slip boundary condition. Furthermore, the works of Leont’ev and his works on fluid dynamics reveal the state of fluids as they flow past a cylinder and a sphere in a porous medium. The authors acknowledge how this functions within the Brinkman model, through applying the Brinkman equation to their research. Their data and the framework are also applied within the hypothesis of this research as Leont’ev applies a Navier boundary condition. This is done to make the study more efficient in using data that has already been established.

Key aspects of the hypothesis are based on other contributing factors as well, such as analytical theories behind bio-heat transport Fan and Wang (2011). The way heat functions within an experiment where permeability is measured, can be attributed to how fluids function with natural law. Another study on which the hypothesis is based is the work of Lowe, Koseff, and Monismith, (2005) on how fluids with submerged canopies have oscillatory flow. The authors breakdown velocity structure of this relationship noting that “any benthic organisms form very rough surfaces on the seafloor that can be described as submerged canopies. Recent evidence has shown that, compared with a unidirectional current, an oscillatory flow driven by surface waves can significantly enhance biological processes such as nutrient uptake” (Lowe, Koseff, and Monismith, 2005). The research on which the hypothesis in this study is grounded can be attributed to the works of Lowe et al. (2005), in that it reveals gaps in knowledge that have yet to be established to date regarding fluid oscillation and velocity measures.  Like in our study, where we are aware there is a clear indication of potential permeability enhancements through the use of optimum mathematical models, but can’t clearly identify what those models are without quantitative analysis. The authors note that, “to date, the physical mechanisms responsible for this enhancement have not been established. This paper presents a theoretical model to estimate flow inside a submerged canopy driven by oscillatory flow. To reduce the complexity of natural canopies, an idealized canopy consisting of an array of vertical cylinders is used” (Lowe, Koseff, and Monismith, 2005). The exact measurements of the canopies, without clearly defining it, represent optimum measurements based on mathematical solutions aware of the distinct boundaries necessary for oscillatory flow. Lowe, Koseff, and Monismith, (2005) reveal something very telling on which this hypothesis in fluid permeability research and velocity can be applied. The authors note that the attenuation of the in-canopy oscillatory flow is shown to be governed by three dimensionless parameters defined on the basis of canopy geometry and flow parameters. The model predicts that an “oscillatory flow will always generate a higher in-canopy flow when compared to a unidirectional current of the same magnitude, and specifically that the attenuation will monotonically increase as the wave orbital excursion length is increased” (Lowe, Koseff, and Monismith, 2005). They acquired this knowledge through a section of laboratory experiments which they conducted for a variety of different oscillatory and unidirectional flow conditions, in which it was found that that oscillatory flow does in fact increases water motion inside a canopy. Based on the hypothesized findings of their research, we have devised our own hypothesis regarding this study in permeability models and their effective application for fluid controls and manipulation.

The focus on calculations, permeability with respect to spheres and cylinders and the mathematical cause of this flow all play a pivotal role  in the establishing how the hypothesis is formulated. The works of Pop and Cheng (1992), provides the foundation of data on which the calculations measuring Brinkman model flows ins this study are used, as they measure past studies in circular cylinders embedded in porous mediums and the way the fluid functions based on the Brinkman model. Value boundary problem measurements are what create the utilization for this study and they represent the fundamental factors of mathematical analysis essential for finding study parameters. AS the authors note, “historically, partial differential equations originated from the study of surfaces in geometry and a wide variety of problems in mechanics. During, the second half of the nineteenth century, a large number of famous mathematicians became actively involved in the investigation of numerous problems presented by partial differential equations” (Pop and Cheng, 1992). The analysis of partial differential equations enables researchers to better understand mathematical findings. These partial fundamental equations allow for the solutions necessary to assess boundaries as they represent fundamental laws of nature in collaboration with the mathematical equations used in engineering and science. Their use this study, specifically in respect to how they are applied to the hypothesis makes research efficient in calculating the Brinkman model in this study. In addition to these studies in mathematical. In addition to the use of fundamental differential equations the research of Myint-U and Debnath (2007) as they apply linear partial differential equations for their used by engineers and scientists and engineers is also taken thoroughly into account to assess whether or not new achievements can be reached. 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. Through the application of these mathematical models, it is believed that this research can produce the solutions for the fourth order boundary-value problem for ψ(r, θ). Furthermore, through the analysis of the permeability k and the slip coefficient ξ on the velocity, flow past solid inclusions. These cylinders are embedded in a porous medium and all factors are mathematically based on linear equations.

Hypothesis Conclusion

In sum, the goal of this research is to reveal new distinct results prevalent within the measure of pressure and velocity fields, and to do so with the use of boundary-value problems that develop challenges in respect to viscous flow, mixed boundary conditions, flow seepage problems, velocity slip, PDEs, cylinder or sphere surfaces, and the Stokes paradox. The study expects that approaches to the boundary-value problems will be uncovered and eventually produce new results for improved mathematical models that can function in the management of fluids within permeable mediums. This can be applied to studies related to increased velocity and pressure fields. Fluid flow transport in porous mediums is believed to play a critical role in a wide range of industrial operations. The findings of this research will supplement advancements in engineering sciences such the exploration of geothermal power, areas as oil and gas recovery. The hypothesis of this study reveals that there are distinct models that can enhance modes of assessment that give expression to areas of science and research. The study hypothesizes that advancing knowledge in permeability techniques in fluid flows through the use of acute mathematical models can enhance the efficiency of engine dynamics and performance, as well as how oil is harvested for supply and demand. It is further formulated, that these exact findings when applied models are infused can reveal inefficiencies in current practices as they relate to permeability for engineers in their daily application, but also in how they can be applied in academic areas to further knowledge on the topic. The hypothesis takes the position that through distinct calculations and the use of proven mathematical models measuring porous mediums, this in turn can be used to analyze the effect of permeability k and slip coefficient ξ on the flow fields in the presence of boundaries. This can better help design purposes within industries that distribute fluids mathematically for an increased understanding of modeling in porous media with alternate chemical inclusions. The hypothesis of this study furthers the belief of increased performance, control, and manipulation of permeable mediums to reach desired goals, while additionally progressing knowledge science and engineering fields.

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