Article pdf available in journal of nonnewtonian fluid mechanics 951. Let us consider a fluid medium and use cartesian tensors to derive the. Another type of non newtonian fluid is viscoplastic or yield stress fluid. Non newtonian fluids are the opposite of newtonian fluids. The cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. Suppose, further, that the sides of the cube are aligned parallel to the principal axes of the local stress tensor. In the general case of a threedimensional flow, for a newtonian fluid a linear relation holds between the stress tensor and the tensor of the rates of strain. The normal stress for onedimensional loading is given by. Stabilisation of computations for incompressible and compressible liquid flows. Non newtonian fluid mechanics is a field of study which is growing in prominence and importance as the years progress, not least because many of the fluids one encounters in everyday life are non newtonian in their behavior. Heat transfer to newtonian and nonnewtonian fluids in. For the vast majority of practical problems, where the fluid can be assumed to be newtonian, the components of the viscous stress tensor are defined by the relations 3, 4. Encyclopedia of life support systems eolss all stress applied to a material produces a deformation.
Newtonian fluids, defined as fluids for which the shear stress is linearly proportional to the shear strain rate. Derivation of the stress tensor relation for a newtonian fluid 3. Ii newtonian and nonnewtonian flow ibarz, albert, castellperez, elena, barbosacanovas, gustavo v. However, some experimental results show that in nature there exist stronger dissipative mechanisms not captured by the classical stokes law. Chhabra abstract the objective of this chapter is to introduce and to illustrate the frequent and wide occurrence of nonnewtonian. Franco and pedro partal encyclopedia of life support systems eolss given that the force is exerted to the area perpendicular to the y axis in the positive direction of the x axis and, consequently, the only component of the stress tensor is. Chapter 3 the stress tensor for a fluid and the navier stokes. Temperature dependence of viscosity of nonnewtonian. A traceless stress tensor formulation for viscoelastic fluid flow article pdf available in journal of nonnewtonian fluid mechanics 951. The study of the deformation of flowing fluids is called rheology. The response characteristics of the fluid are embodied in the constitutive relation for the stress. In a non newtonian fluid the relation between the shear stress and the shear rate is different and can even be timedependent time dependent viscosity a constant coefficient of viscosity cannot be defined for non newtonian fluids as its possible for newtonian fluids. We shall see that the stress on any surface anywhere in the fluid can in general be specified in terms of six scalar functions of x, y, z, and t. The deformation can be angular or longitudinal, depending on the direction of the applied stress.
Lectures of transport phenomena course at olin college the newtonian stress tensor and connection to fluid kinematics. There are some fluids that when a given shear stress, the produced shear rate is always the same regardless how was this fluid previously handled. The fluids which do not follow such a linear relationship are called non newtonian fluid. Newtonian fluids in industries, processing molten plastics, polymers, pulps, slurries, emulsion, etc. Consider the surface forces exerted on some infinitesimal cubic volume element of a static fluid. Spagnolie abstract in this chapter we introducethe fundamentalconceptsin newtonian and complex. This defines a newtonian fluid and both air and water. Rate of deformation is proportional to the applied stress a linear viscous fluid w.
The motion of a fluid under incompressible and isothermal conditions is governed by the balance of linear momentum 2. But, for the vast majority of practical problems, where the fluid can be assumed to be newtonian, the components of the viscous stress tensor are defined by the relations 3, 4. Hence, viscoplastic fluids behave like solids when the applied shear stress is less. Eduard feireisl, in handbook of mathematical fluid dynamics, 2002. Considerable efforts have been conducted to understand the behavior of non newtonian fluids. I have big problems in understanding the stress tensor for newtonian fluids. Chapter 3 the stress tensor for a fluid and the navier. Newtonian and non newtonian fluids linkedin slideshare. For newtonian fluids see text for derivation, it turns out that now we plug this expression for the stress tensor ij into cauchys equation. The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point the viscous stress tensor is formally similar to the elastic stress tensor cauchy tensor that describes internal forces in an elastic material due to its deformation. A fluid where there is no linear relationship between stress and strainrate is called a nonnewtonian fluid. These fluids do not obey the newtonian postulate that the stress tensor is directly proportional to the deformation tensor. The complete stress tensor has been measured using a computer simulation of an assemblage of rough, inelastic spheres in an imposed simple shear flow.
Pdf a traceless stress tensor formulation for viscoelastic fluid flow. These functions involve two material parameters taken as constants throughout the fluid, although depending on ambient temperature and pressure. Newtonian fluid article about newtonian fluid by the free. But the viscosity of some fluids is affected by factors other than temperature. Interpret the physical meaning of different terms in the deformation tensor, including dilation, shear strain, and rotation topicsoutline. Derivation of the viscous flow equations to obtain the equation for viscousdominated inertialfree flow, we need to start with the local force balance in the fluid, which is the same expression we used previously in a solid, ij 0 1. A newtonian fluid is one where there is a linear relationship between stress and strainrate. For a newtonian fluid the stress strictly, the deviatoric stress.
Another type of nonnewtonian fluid is viscoplastic or yield stress fluid. In the absence of shear stresses, therefore, the stress. Chapter 5 stress in fluids cauchys stress principle and the conservation of momentum the stress tensor the symmetry of the stress tensor hydrostatic pressure principal axes of stress and the notion of isotropy the stokesian fluid constitutive equations of the stokesian fluid the newtonian fluid interpretation of the constants. The fluid which does not follow the newtons law of viscosity and relationship is depend upon all the three quantity i. When shear is applied to non newtonian fluids, the viscosity of the fluid changes. The flow of newtonian fluids is studied in hydrodynamics and aerodynamics. The equations of mass and momentum conservation are derived, and the cauchy stress tensor makes its. On the other hand, fluids like, for example, melted plastic or blood behave in a different mannerthey are non newtonian fluids. A shear stress will not affect the viscosity of a newtonian fluid. More generally, in a nonisotropic newtonian fluid, the coefficient that relates internal friction stresses to the spatial derivatives of the velocity field is replaced by a nineelement viscous stress tensor. Closed captioning is not yet available for this video. These six are the independent components of a quantity called the stress tensor. The result is the famous navierstokes equation, shown here for incompressible flow.
The fluid is said to be newtonian if these matrices are related by the equation. Conservation equations and stress tensor example 1. Suppose that the components of the stress tensor are approximately constant across the element. As the stability of numerical schemes depends strongly. Only five components of the stress tensor were found to be significantly different from zero. Up to now, we have considered only newtonian fluids where the viscous stress tensor. This assumption is almost always true expect for materials. Adapting the fluid classification suggested by chabbra and richardson 2008, the non newtonian fluid can be categorized into three groups.
A simple fluid is immediately deformed by a shear force, whereas a simple solid may. Given the cauchy stress tensor, the following may be defined. A fluid in which the components of the stress tensor are linear functions of the first spatial derivatives of the velocity components. The stress tensor for a fluid and the navier stokes equations 3. Recall from mechanics of materials, hookes law for elastic bodies states that stresses are directly proportional to strains. The viscosity of a non newtonian fluid will change due to agitation or pressuretechnically known as shear stress. The question is quite fundamental and more on a beginners level not sure if good in this highlevelforum, but i try. The tensor consists of nine components that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. This isotropy, combined with the assumption that the fluid does not carry surface couples or internal torques, will kill off any remaining extra terms and immediately generates the newtonian fluid model. More precisely, a fluid is newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. Chapter 3 nonnewtonian fluid almustansiriya university. A traceless stress tensor formulation for viscoelastic fluid flow. Institute of non newtonian fluid mechanics swansea, u.
Expressing flow and deformation in terms of strain and strain rates. The property of the fluid that causes this stress is its viscosity. Luis lehner, huan yang and aaron zimmerman also began to theori ze that space time. To discuss the various aspects of research in the non newtonian fluid mechanics field. The viscosity of a nonnewtonian fluid will change due to agitation or pressuretechnically known as shear stress. Jan 29, 2014 expressing flow and deformation in terms of strain and strain rates. List and apply the basic assumptions used in classical fluid dynamics for ocean engineering 2. On the other hand, fluids like for example melted plastic or blood behave in a different manner they are nonnewtonian fluids. These fluids have a linear relationship between viscosity and shear stress.
The shear stress must exceed a critical value known as the yield stress for the fluid to start flowing. Non newtonian fluids are generally divided in to two categories like viscoinelastic fluids and viscoelastic fluids. Velocity profiles, viscosity data, and laminar flow friction factor equations for flow in a circular duct conference paper pdf available january 2008 with. By the combination of newtonian and simple viscoelastic maxwell. A polymer is composed of long chains of repeated units known as monomers that are strung. A fluid where there is no linear relationship between stress and strainrate is called a non newtonian fluid for a newtonian fluid the stress strictly, the deviatoric stress. Boundarylayer flows of nonnewtonianpower law fluids.
Review of nonnewtonian mathematical models for rheological. This is a fluid that will not flow when only a small shear stress is applied. San andreas fault palmdale california state university. You can probably guess that non newtonian fluids are the opposite of newtonian fluids. Leaving aside pathological non newtonian fluids the resisting stress is generally proportional to the deformation rate. Under such circumstances, special treatment is necessary, not only for effective discretisation of these equations. In continuum mechanics, the cauchy stress tensor, true stress tensor, or simply called the stress tensor is a second order tensor named after augustinlouis cauchy. Oct 23, 2014 lectures of transport phenomena course at olin college the newtonian stress tensor and connection to fluid kinematics. The newtonian fluid encyclopedia of life support systems. This data item, the first of a group on the flow properties of non newtonian fluids, provides an introductory guide to the classification and characteristics of homogeneous fluids, polymeric.
The stress tensor was total stress tensor, invented to make the calculation of fluid stress easier. Thus the strain and velocity gradient tensor have the following form. A newtonian fluid is one in which the components of the stress tensor are related linearly to the components of the strain rate tensor. A graphical representation of this transformation law is the mohrs circle for stress. For our newtonian fluid zero yield stress with a 10cp viscosity, the unsteady formulation in figure 9.
Newtonian and nonnewtonian flow encyclopedia of life. Here we assume that we have a two dimensional problem with only a shear force and strain. Chapter 1 introduction to complex fluids alexander morozovand saverio e. Abstract in this chapter we introducethe fundamentalconceptsin newtonian and complex. Any fluid in which the deviatoric stress tensor takes the previous form is termed a newtonian fluid, because newton was the first to postulate a linear relationship between shear stresses and velocity gradients. Where stress is proportional to rate of strain, its higher powers and derivatives basically everything other than newtonian fluid. The ratio of stress to strainrate is the viscosity of the fluid. Most liquids, including water and lubricating oil, and all gases have the properties of a newtonian fluid. Constitutive equations in fluids fluid models newtonian fluids constitutive equations of newtonian fluids relationship between thermodynamic and mean pressures components of the constitutive equation stress, dissipative and recoverable power dissipative and recoverable powers thermodynamic considerations. How can i derive the stress tensor for a newtonian fluid in. A fluid where there is no linear relationship between stress and strainrate is called a non newtonian fluid. Since the relationship between the shear rate and shear stress is nonlinear in non newtonian fluids, the turbulent flow behavior is generally more. Non newtonian fluids while rather extensive studies have been conducted for non newtonian fluids in turbulent circular pipe flow, less attention has been paid to these fluids in noncircular flow.
Analogous to the newtons formula for fluid friction, stokes formulated the symmetric stress tensor, i. The stress tensor for simple shear flows of a granular. Newtonian fluid article about newtonian fluid by the. It is customary to split the total stress into an isotropic part pressure and a deviatoric part as. Doing the math, this kills off a great deal of elements in the above tensor. How can i derive the stress tensor for a newtonian fluid. When the deformation gradient is given as a second order tensor. An internet book on fluid dynamics the stress tensor the general state of stress in any homogeneous continuum, whether. Here, is a fourthorder tensor this follows from the quotient rule because and are both proper secondorder tensors. Pdf numerical study of effect of stress tensor for viscous and. In non newtonian fluids, viscosity can change when under force to either more liquid or more solid. For a newtonian material, the viscosity is defined as follows.
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