Newtons Law of Viscosity

The shear strain rate (dy/d ) of a viscous material is a function of the applied shear stress t. The simplest dependence (for small stresses) is a linear function, i.e.. 

These relationships are known as Newton s Law of viscous flow a is termed the fluidity and -q the dynamical shear viscosity. Newton s Law is analogous to Hooke s Law, except shear strain has been replaced by shear strain rate and the shear modulus by shear viscosity. As shown later, this analogy is often very important in solving viscoelastic problems. In uniaxial tension, the viscous equivalent to Hooke s Law would be a=7] ds/dt), where q is the uniaxial viscosity. As v=0.5 for many fluids, this equation can be re-written as 7-=3Tj(de/dO using t7=t /[2(1+v)], the latter equation being the equivalent of the interrelationship between three engineering elastic constants, (fi=E/[2il + v)]). 

Newtonian viscosity leads to an interesting aspect for materials undergoing a drawing process. The normalized rate of change of the cross-sectional area for a linear viscous material is given by (dA/dt)IA = (deldt)=—a-/q or dAldt) = —Flq, where Fis the applied uniaxial force. This derivation shows that, for a constant F, dAldt) must be constant, i.e., the body can be reduced in cross-section at a constant rate. Thus, a section with an initially narrow section will not neck down faster than elsewhere. 

The viscosity of a glass depends on its particular composition. Indeed, the glass composition is chosen, in part, to ensure it has the appropriate viscosity for the various processes it must undergo during its production and use. For example, univalent and divalent ions are often used to decrease the viscosity, as these network modifiers tend to break up the silica network. The effect of various ions on the structural modification depends on the field strength and polarizabil- 

The coefficient of viscosity was introduced in Chapter 2, Section 2.3a, but this parameter is elusive enough to warrant further comment. In this section we examine the definition of the coefficient of viscosity—the viscosity, for short —of a fluid. This definition leads directly to a discussion of some experimental techniques for measuring viscosity these are discussed in the following sections. 

We can imagine within the fluid two layers separated by dy, over which distance the velocity changes by an amount dv. Therefore dv/dy defines a velocity gradient Newton s law of viscosity states that the shear stress, r = F/A, is proportional to dv/dy. The viscosity rj of the sandwiched fluid is the factor of proportionality  

1 The relationship between applied force per unit area and fluid velocity. 

2 Comparison of Newtonian liquids with several forms of non-Newtonian behavior. 

A second interpretation of 77 is as valid as Equation (1) and perhaps more illuminating. To arrive at this alternative, we multiply both sides of Equation (1) by dv/dy  

Newton s law of viscosity and the conservation of momentum are also related to Newton s second law of motion, which is commonly written Fx = max = d(mvx)/dt. For a steady-flow system, this is equivalent to 

It is important to distinguish between the momentum flux and the shear stress because of the difference in sign. Some references define viscosity (i.e., Newton s law of viscosity) by Eq. (1-8), whereas others use Eq. (1-9) (which we shall follow). It should be evident that these definitions are equvialent, 

It is also evident that this phenomenological approach to transport processes leads to the conclusion that fluids should behave in the fashion that we have called Newtonian, which does not account for the occurrence of non-Newtonian behavior, which is quite common. This is because the phenomenological laws inherently assume that the molecular transport coefficients depend only upon the thermodyamic state of the material (i.e., temperature, pressure, and density) but not upon its dynamic state, i.e., the state of stress or deformation. This assumption is not valid for fluids of complex structure, e.g., non-Newtonian fluids, as we shall illustrate in subsequent chapters. 

The upper plate is kept in motion in the x-direction at a velocity U by applying a force F to the plate in the x-direction. The lower plate is kept stationary. The force per unit area required to maintain the upper plate (of area A) in motion is given by 

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When this number is much less than unity, we have the usual viscous flow, where the Poiseuille flow is applicable. In this viscous flow, the Newton law of viscosity holds (that is the shear stress is proportional to the velocity gradient, and the proportionality constant is the viscosity), and the velocity at the wall is zero. 

Any fluid that does not obey the Newtonian relationship, i. e. the Newton law of viscosity, is termed as a non-Newtonian fluid. 

Equation (2.3) is called Newton s law of viscosity and those systems which obey it are called Newtonian. 

The elastic stress curve in figure perfectly follows elastic strain [2]. This constant is the elastic modulus of the material. In this idealized example, this would be equal to Young s modulus. Here at this point of maximum stretch, the viscous stress is not a maximum, it is zero. This state is called Newton s law of viscosity, which states that, viscous stress is proportional to strain rate. Rubber has some properties of a liquid. At the point when the elastic band is fully stretched and is about to return, its velocity or strain rate is zero, and therefore its viscous stress is also zero. 

Let us use a control volume approach for the fluid in the boundary layer, and recognize Newton s law of viscosity. Where gradients or derivative relationships might apply, only the dimensional form is employed to form a relationship. Moreover, the precise formulation of the control volume momentum equation is not sought, but only its approximate functional form. From Equation (3.34), we write (with the symbol implying a dimensional equality) for a unit depth in the z direction... 

This is a statement of Newton s law of viscosity and the constant of proportionality fi is known as the coefficient of dynamic viscosity or, simply, the viscosity, of the fluid. The rate of change of the shear strain is known as the rate of (shear) strain or the shear rate. The coefficient of viscosity is a function of temperature and pressure but is independent of the shear rate y. 

With the positive sign convention, Newton s law of viscosity is expressed as... 

The velocity profile must have a form like that shown in Figure 1.17. The velocity is zero at the pipe wall and increases to a maximum at the centre. From Example 1.8, it is known that the shear stress vanishes on the centre-line r = 0, so from Newton s law of viscosity (equation 1.45) the velocity gradient must be zero at the centre. 

A slightly different procedure is to substitute for rrx in equation 1.49 using Newton s law of viscosity. If this is done and the resulting equation integrated twice, equations 1.55 and 1.56 are obtained ... 

As before, in order to determine the velocity profile it is necessary to introduce Newton s law of viscosity but as the positive sign convention is now being used it is necessary to express Newton s law by equation 1.45a ... 

In general, with the different sign conventions, equations involving stress components have opposite signs in the two conventions. On substituting the appropriate form of Newton s law of viscosity, the sign difference cancels giving identical equations for the velocity profile. 

When a fluid flows past a solid surface, the velocity of the fluid in contact with the wall is zero, as must be the case if the fluid is to be treated as a continuum. If the velocity at the solid boundary were not zero, the velocity gradient there would be infinite and by Newton s law of viscosity, equation 1.44, the shear stress would have to be infinite. If a turbulent stream of fluid flows past an isolated surface, such as an aircraft wing in a large wind tunnel, the velocity of the fluid is zero at the surface but rises with increasing distance from the surface and eventually approaches the velocity of the bulk of the stream. It is found that almost all the change in velocity occurs in a very thin layer of fluid adjacent to the solid surface ... 

In the viscous sublayer, the magnitude of the time-averaged value of the shear stress f is given by Newton s law of viscosity which can be written in this case as... 

The velocity profile for steady, fully developed, laminar flow in a pipe can be determined easily by the same method as that used in Example 1.9 but using the equation of a power law fluid instead of Newton s law of viscosity. The shear stress distribution is given by... 

A law similar to these two diffusional processes is Newton s law of viscosity, which relates the flux (or shear stress) ryx of the x component of momentum due to a gradient in ux this law is written as... 

The equations for one-dimensional momentum and mass flow are directly analogous to Fourier s Law. A velocity gradient, dv /dy, is the driving force for the bulk flow of momentum, or momentum flux, which we call the shear stress (shear force per unit area), Xyx- This leads to Newton s Law of Viscosity ... 

We seek to nnderstand the response of a material to an applied stress. In Chapter 4, we saw how a flnid responds to a shearing stress through the application of Newton s Law of Viscosity [Eq. (4.3)]. In this chapter, we examine other types of stresses, snch as tensile and compressive, and describe the response of solids (primarily) to these stresses. That response usually takes on one of several forms elastic, inelastic, viscoelastic, plastic (ductile), fracture, or time-dependent creep. We will see that Newton s Law will be useful in describing some of these responses and that the concepts of stress (applied force per unit area) and strain (change in dimensions) are universal to these topics. 

Equation (5.5) is known as Hooke s Law and simply states that in the elastic region, the stress and strain are related through a proportionality constant, E. Note the similarity in form to Newton s Law of Viscosity [Eq. (4.3)], where the shear stress, r, is proportional to the strain rate, y. The primary differences are that we are now describing a solid, not a fluid, the response is to a tensile force, not a shear force, and we do not (yet) consider time dependency in our tensile stress or strain. 

Recall also from Section 4.0 that the viscous shear rate, )> , can be related to the viscous shear stress through the viscosity, p, according to Newton s Law of Viscosity, Eq. (4.3) ... 

We begin with a brief discussion of Newton s law of viscosity and follow this with a discussion of Newtonian flow (i.e., the flow of liquids that follow Newton s law) in a few standard configurations (e.g., cone-and-plate geometry, concentric cylinders, and capillaries) under certain specific boundary conditions. These configurations are commonly used in viscometers designed to measure viscosity of fluids. 

Describe the physical significance of Newton s law of viscosity. Is Newton s law always applicable ... 

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