In continuum mechanics, a Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly proportional to the local strain rateâthe rate of change of its deformation over time. Since most of the differences among the different categories of non-Newtonian fluids are related to their viscosity, which is a dominant physical property within the boundary layer region, a thorough understanding of the flow in the boundary layer is of considerable importance in a range of chemical and processing applications. For n = 1, the consistency factor reduces to the Newtonian viscosity μ; in general, the units of K depend on the value of n. (Both n and K can be determined from viscometer measurements using standard laboratory techniques.). The flow of Newtonian fluids is studied in hydrodynamics and aerodynamics. ; The liquids have the ability to vary depending on the tension; Their viscosity value is not defined or constant. Rheology is the study of such flows. After the fluid starts to flow there is a linear relationship between shear stress and shear rate. In the general case of a three-dimensional flow, for a Newtonian fluid a linear relation holds between the stress tensor and the tensor of the rates of strain. An exact annular flow solution, however, is available for nonrotating drillpipes. Surface viscometer values for fluid parameters having questionable scientific merit often find routine field usage. Eq. It is defined as the sum of Potential energy head, Pressure energy head and Kinetic velocity energy head is constant when the liquid is flowing from one end to another end in a tube or pipe. Leye M. Amoo, R. Layi Fagbenle, in Applications of Heat, Mass and Fluid Boundary Layers, 2020. We will suppose that the x, y, and z components of V are, respectively, u, v, and w. The unit vectors in the x, y, and z directions will be written x, y, and z. and t and l subscripts indicate turbulent and laminar flow conditions respectively. Characteristics of non-Newtonian fluid. Fluids are divided into several categories according to their rheological behaviors as observed in shear stress-shear rate plots. 1. The problem of concentric, nonrotating, annular flow was solved using numerical methods in Fredrickson and Bird (1958). For Newtonian fluids the ratio of the shear stress to the shear rate is constant. The fluid which follows the Newtonian equation is called the Newtonian fluid and which does not follow is called a non-Newtonian fluid. Therefore a constant coefficient of viscosity cannot be defined. Y and λ in Equations 17-59 and17-60, known in chemical engineering as the Fredrickson-Bird Y and λ functions, respectively, depend on n and Ri/Ro only. 1) A Newtonian fluid's viscosity remains constant, no matter the amount of shear applied for a constant temperature. For example, the axial velocity vz(r) in our cylindrical radial flow satisfies, which, despite its simple appearance, is difficult to solve because it is nonlinear. The nature of boundary layer flow influences not only the drag at a surface or on an immersed object, but also the rates of heat and mass transfer when temperature or concentration gradients exist. This model is a two parameter model that includes yield stress and plastic viscosity of the fluid. The static pressure P is the actual pressure of the fluid. Fig. Its viscosity is proportional to the ratio of drag force to velocity. Oobleck isnât the only shear-thickening non-Newtonian fluid. Dynamic viscosity of a fluid is defined as the shear stress applied divided by the velocity gradient achieved when a shear force is applied to a fluid. Viscosity varies greatly among fluids. In other words, the apparent viscosity of a power law flow varies from problem to problem, whereas n and K do not. (17.59), (17.60), known in chemical engineering as the Fredrickson-Bird Y and λ functions, respectively, depend on n and Ri/Ro only. Fluids that exhibit gelling property are called thixotropic. 17.13. s). The hydrostatic pressure Ïgz is not pressure in a real sense since its value depends on the reference level selected, and it accounts for the effects of fluid weight on pressure. If this alignment develops more or less instantaneously for a given shear rate and depends significantly on shear rate, we will have a ‘shear-thinning’ material for which the apparent viscosity decreases with shear rate (Fig. The substance that has a tendency to flow is called as fluid. This fact is not appreciated in drilling engineering. For a discussion on three-dimensional effects and a rigorous analysis of the stress tensor, the reader should refer to Computational Rheology. Newtonian fluids are described by Navier–Poisson constitutive equations: where σ is Cauchy stress tensor, D = (L + LT)/2 is the strain rate tensor, and p(J, T) is the hydrostatic pressure, related to the density ρ and temperature T through the equation of state (EOS). In the above equations, if Fann 35 dial readings are multiplied by constant 1.0678, the unit of shear stress is lbf/100 ft2. Non-Newtonian fluids are fluids for which the relations indicated above are not linear, for example, for the rectilinear flow. As a consequence we can distinguish two types of effects on the mechanical behaviour. The application of the power law and the Herschel-Bulkley models are described in an example at the end of this section. the apparent viscosity for a given shear rate varies in time: From this example we see that shear-thinning and thixotropy can be confused because they may find their origin in the same physical effect. 21. Most drilling fluids do not behave like Newtonian fluids, and the study of rheology focuses on the stress behavior of different fluids acting at different shear rates. Another type of non-Newtonian fluids is shear-thickening fluid which the viscosity of the fluid increases as the shear rate increases. (17.62) can be evaluated using n, K, Rc, and the prescribed annular volume flow rate Q. Figure 1: Fly Ash Shear Rate vs Shear Stress â Power Law Fluid. The apparent viscosity of the flow, however, will vary throughout the cross-section of the flow geometry and additionally varies with the pressure gradient, or equivalently, the total flow rate. The general form of power law model as given in Eq. Non-Newtonian in nature, its constitutive equation is a generalised form of the Newtonian fluid. The behavior of a Herschel-Bulkley fluid is described as. Such fluids are characterized by the following rheological law: uy()n K y âââ Ï= ââ ââ â (1) where n is the flow behaviour index and K is the consistency of the fluid. In 2006 API recommended using the Herschel-Bulkley to predict the fluid behavior and pressure drop calculations more accurately for deep and complex wells. Thus, in principle, a formula analogous to Equation 17-51, which relates mudcake edge shear stress, total volume flow rate, pipe radius, and fluid properties, is available. A shear thinning fluid is easier to pump at high shear rates. If n is equal to 1, then the Herschel-Bulkley reduces to the Bingham plastic model. In addition, shear-thinning effects may occur in moderate or concentrated suspensions as a result of variations in colloidal interactions with shear rate. Liquid 3. Matter around us exists in three phases (excluding plasma) 1. After the value of n is determined, K is calculated as. For now, we shall continue our discussion of mudcake shear stress, but turn our attention to power law fluids. 14.4. High gel strength may cause excessive pressure surge when the circulation starts and fractures the formation. See Fluid flow, Fluids, Viscosity. The non-Newtonian fluid used in this study is the power-law model (Ostwald-de Waele fluid). 14.3, followed by a brief overview of future research prospects in this area in Sect. However, the power law model for the low shear rate section still passes through the origin and does not explain the thixotropic behavior of the drilling fluid. The Bingham plastic model became widely used because it is simple and estimates pressure loss in a turbulent condition with accuracy close to the other models. (17.59), (17.60), we obtain the required result, which relates mudcake edge shear stress, volume flow rate, pipe radius, and fluid properties. In shear experiements, all such fluids under constant pressure and temperature conditions show a constant resistance to flow, i.e., there is a linear relationship between the viscous stress and the strain rate. Where stress is proportional to rate of strain, its higher powers and derivatives (basically everything other than Newtonian fluid). Drilling fluids initially resists flowing as shown in Figure 2-15. Fredrickson-Bird λ Function (condensed). The equilibrium mudcake thickness is defined by the condition τ(Rc) = τyield as before, and the procedure for the critical invasion rate discussed earlier carries through unchanged. Newtonian fluids exhibit constant viscosity at different shear rates and constant temperature. The main difference between fluids and solid lies in their ability to resist shear stresses. There are other classes of fluids, such as Herschel-Bulkley fluids and Bingham plastics, that follow different stress-strain relationships, which are sometimes useful in different drilling and cementing applications. The concept of the τ0 and τy are very different. The result can be interpreted either as the motion of a test particle immersed in the fluid or as the motion of the fluid itself. Solid 2. Thus, it is not surprising that, at least in cuttings transport analyses, they cannot be correlated with measurable events such as hole cleaning efficiency. A simple example, often used for measuring fluid deformation properties, is the steady one-dimensional flow u(y) between a fixed and a moving wall (see illustration). That is equivalent to saying those forces are proportional to the rates of change of the fluid's velocity vector as one moves away from the point in question in various directions. The literature shows that there is a significant amount of research with the goal of understanding non-Newtonian flows through pipes and channels due to its relevance to the applications mentioned previously [2,3]. Fredrickson-Bird X Function (condensed). This behavior enables drilling fluid to suspend the drilling cuttings and solids within the drilling fluid when the circulation stops. Newtonian fluids also have predictable viscosity changes in response to temperature and pressure changes. For more information, readers are referred to API RP 13D released in 2003. https://encyclopedia2.thefreedictionary.com/Newtonian+Fluid. The flow of a dusty and electrically conducting fluid through a circular pipe in the presence of a transverse magnetic field has important applications such as MHD generators, pumps, accelerators, and flowmeters. If Ri and Ro are inner and outer radii, where ΔP is a pressure drop, L is a characteristic length, and Q is the annular volume flow rate, these authors show that, while the shear stress at the outer wall r = Ro is given by. Finally the relative importance of Brownian motion and hydrodynamic dissipations may be appreciated from the Peclet number (Pe): where b is the particle size, kB the Boltzmann constant and T the temperature. The synovial fluid that coats the knee and elbow joints is a shear-thickening non-Newtonian fluid. The rheological behavior of Newtonian fluids can be written as, Figure 2-15. Thixotropy is dealt with in more detail in Section 1.6. The Bingham plastic model is the most common rheological model used in the drilling industry. Introduction A non-Newtonian fluid is a fluid whose flow properties differ in many ways from those of Newtonian fluids.