The Derivation of the Navier-Stokes Equation - Article Example

The Derivation of the Navier-stokes Equation The Navier-stokes Equations The Navier–Stokes equations have been formulated and rooted on the postulation that the liquid, at the measure of concern, is a continuum. That is, it is not comprised of distinct elements but rather an incessant matter. It is important to note that there are a series of Navier-stoke equations. One other fundamental assumption of these equations is that other areas that significant for instance, pressure and velocity can be differentiated. The principles that underlie these equations are those of momentum, mass, and energy conservation. The application of these principles requires that during testing and experimentation, a certain predetermined volume be assigned to the fluid. This volume is known as the control volume. The equation is applied on a number of fluids. These are the Compressible Newtonian fluids, incompressible Newtonian fluids and Non-Newtonian fluids (Gresho 414). These equations are the benchmark for viscous fluids and are derived by relating the Law of Motion by Newton to a fluid. It is important to note the aspects of the Compressible, Incompressible, and Non-Newtonian fluids. Compressible Newtonian Fluid The definition of compressibility is important in understanding what compressible fluids are. Compressibility refers to the decline in volume of the fluid because of outside forces exerted on it. It is imperative to note that, there are three basic assumptions that guide the application of these derivative functions to a number of fluids. The derivative function is shown below These suppositions are: The stress tensor should form a linear function of the stress rates The fluid in question should be isotropic For an immobile fluid, the Stress tensor T has to be zero (Morrison, 5). Application of the above assumptions will lead to a generic equation that has a number of elements. Important elements to note are two distinct proportionality constants that categorically denote that stress is determined linearly by stress rates. These constants are viscosity and the second coefficient of viscosity. The value of the second coefficient of viscosity generates a viscous stimulus that leads to volume change. However, the value is hard to ascertain in compressible fluids and is habitually negligible. It is stipulated that almost all fluids can be compressible to a certain extent. That is, variations in temperature and/or pressure will lead to variations in density. The influence of outside pressure will force a compressible fluid to diminish its volume. In this regard, the numerical extent of compressibility is denoted as the relative variation in volume of the fluid due to change in pressure. Gases are greatly compressible as opposed to fluids. There are two types of compressibility. Adiabatic compressibility refers to compressibility when the temperature of a system is persistent. On the other hand, isothermal compressibility occurs when compressibility is ascertained in situations when there is no transfer of energy. Incompressible-Newtonian Fluid It is important to note that these fluids are purely theoretical. These are referred to only for purposes of computations. An incompressible fluid is the type of fluid whose volume is not affected by variations in outside pressure (McDonough 10). Computations in fluid dynamics are founded on the assumption that the fluid is incompressible. In this instance, there are three fundamental assumptions. Viscosity is constant. The second derivative of viscosity is equal to zero The basic mass continuous equation is equal to zero Non-Newtonian Fluids A non-Newtonian fluid possesses flow characteristics that are fundamentally different from those attributed to Newtonian fluids. One outstanding aspect is the fact that their viscidness is not free from shear rate. Nevertheless, certain non-Newtonian fluids exhibiting shear autonomous viscidness display usual stress variances. Several saline solutions as well as melted polymers have been categorized as non- Newtonian fluids. The link between the shear stress and the shear rate is linear in Newtonian fluids (Malek and Rajagopal 16). This however, is not the same case with a non-Newtonian fluid. In this instance, the connection between the shear stress and the shear rate is dissimilar (Malek and Rajagopal 16). Conservation Laws It is important to look at the conservation laws since this will provide appropriate insights to this study. These are a number of ideologies which postulate that some physical attributes remain unchanged within a certain period in a secluded physical system. These conservation laws are particularly essential since they enable scrutiny and analysis of the macroscopic behavior of a system. The analysis does not have to focus on the infinitesimal details surrounding the process involved. Conservation of Energy The basic rule is that energy cannot be created or destroyed. It can however be transmitted in different forms. Therefore, in an isolated environment, every kind of energy remains constant. For example, an object thrown from above has the same energy. However, the kind of energy possessed fluctuates from potential to kinetic. Relativity theory states that mass and energy are equal. Therefore, the mass of an object at rest is a categorized as potential energy that can be converted into several energy forms. Conservation of Linear Momentum This law postulates that an object that is in motion maintains its entire momentum. This happens up to a point where there is interference from an outside force. In secluded systems, there are no outside interferences hence momentum is constantly maintained. This implies that the components of the object in every direction are maintained. Conservation of Mass This principle postulates that matter cannot be made or destroyed. That is, within a secluded system, certain processes that alter the chemical attributes of an object cannot alter its mass. For instance, when an element in liquid form is transformed into a gaseous state, the mass of the element remains unchanged through this transformation. Derivation of Continuity Equation A principle rule of Newtonian mechanics postulates that there is conservation of mass in a random object that has a control volume within a specified varying time. A material volume is composed of define quantities of fluid every time. A definite bounding surface (SM) encloses the fluid at a given time. This means that components contained in the fluid cannot get in or get out of the control volume. Movement on every location on the bounding surface is expressed by a certain local velocity U. The equation outlining this occurrence is shown below When the Reynolds transport hypothesis and divergence hypothesis are consequently applied and subsequently differentiated, a new equation is formed. The link deduced suggests that for the random volume (VM ),the integrand should be zero. The equation consequently derived is shown below This equation is true for every point in the fluid. In an incompressible fluid, the rate of variation of density is zero. The above equation can be further simplified into the equation below Stokes Hypothesis In retrospect, studies by Newton showed that a certain stress factor was linearly relative to the time frequency at which there was occurrence of the strain (Ladyzhenskaya 460). The factor causing proportionality was referred to as the coefficient of viscosity, µ. The velocity outline was found out to be linear, therefore leading to a general postulation shown in the equation below This is true for Newtonian fluids that exhibit a link between stress and rate of strain. Stokes hypothesis furthers Newtonian ideologies that postulated existence of only one component of velocity to a theory that postulates existence of multidimensional flows (Ladyzhenskaya 468). A series of relations formulated by Stoke including a requirement that three known strains add to zero are referred to as the Stoke hypothesis. This is the basic equation connoting stokes hypothesis This paper seeks to establish whether the assumptions made by Stoke will have an effect on the equations of incompressible fluids. As illustrated above, incompressible fluids have a number of properties that are consistent with the assumptions required for the application of the stoke hypothesis. The stoke hypothesis is a derivative of a series of Navier-stoke equations. These set of non-linear functions display the flow of fluid whose strain is linearly reliant on flow speed, incline, and pressure. Newtonian incompressible fluids exhibit the following equation where h is the difference in altitude (Xinwei and Hou 7). The application of the assumptions made by Stoke will therefore affect the equations of incompressible fluids due to the various components shared by both equations. For instance, the Stokes equations are applied to fluids that that exhibit a link between stress and rate of strain, an attribute possessed by Newtonian Incompressible fluids. However, the application of the Stokes hypothesis has stirred contentious debate over the years. It has been established that the real figures of the second coefficient of viscosity contrast often significantly and are not constant as suggested (Dulikravich 3). Moreover, findings showed that the quotient of the second and first coefficients of viscosity, λ /μ, pose a weighty influence on the magnitude of a compression waves and its thickness (Dulikravich 3). The study by Dulikravich displayed the internal configuration of regular stable shocks that rely on values of λ /μ which are different from those proposed by Stokes hypothesis (Dulikravich 3). Work cited Malek, Josef and K. Rajagopal. “Compressible Generalized Newtonian Fluids: Necas Center for Mathematical Modeling”. (2009). Print. Dulikravich, George. “Stokes’ Hypothesis and Entropy Variation within a Compression Shock: 23rd International Symposium on Shock Waves” .2001. Print Gresho, P. “Incompressible fluid dynamics: some fundamental formulation issues”, Annu. Rev. Fluid Mech. 23, (1991): 413–453 Print Ladyzhenskaya, O. “A dynamical system generated by the Navier–Stokes equations”. J. Soviet Math.3, (1973): 458–479 Print. McDonough J. “Computational Fluid Dynamics of Incompressible Flow: Mathematics, Algorithms and Implementations”, (2007). Print. Xinwei, Yu and Thomas, Hou. “Introduction to the Theory of Incompressible Inviscid Flows” (2010). Print Read More