Newton s Law of Viscosity

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. 

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,... 

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. 

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 ... 

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

At the phenomenological level, there are enough further relations between the 14 variables to reduce the number to 5 and make the problem determinate. These further relations are the thermodynamic ones and Stokes and Newton s laws of viscosity and heat flow. These lead from the transport equations to the Navier-Stokes equations. It is noted that these are irreversible. 

On the basis of this model, we shall derive Stokes and Newton s laws of viscosity and heat conduction, with expressions for the coefficients of viscosity (q) and heat conduction (k) which are proportional to x and, for the above choice, have the usual range of values. Their ratio turns out to be... 

In deriving his law of fluid viscosity, Newton wrote in the "Principia", published in 1687, "The resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another" (5). This lack of slipperiness is known as viscosity and led to Newton s law of viscosity, i.e.,... 

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