Newton’s law of viscous flow

Viscosity — A measure of the frictional resistance a fluid offers to an applied shear force under the conditions of - laminar flow. According to Newton s law of viscous flow... 

The viscosity of a fluid was defined on p. 102, where it was seen to be a measure of the resistance to flow of the fluid. According to Newton s law of viscous flow, the frictional force F/, resisting the relative motion of two adjacent layers in the liquid, is proportional to the area A and to the velocity gradient dvfdx (see Figure 3.1, p. 102) ... 

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 Newton s Law is analogous to Hooke s Law, except shear strain has been replaced by shear strain rate and the shear modulus by shear viscosity.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)]). 

Newton s law of viscous flow and Hooke s law for solids describe the perfect state for each. In practice however, few if any materials show this ideal behaviour and are more appropriately described as viscoelastic. That is to say, they exhibit both viscous and elastic behaviour. More importantly, the relative contribution of each with regard to a materials response will depend on the time scale of the experiment. If the experiment is relatively slow the material will appear viscous. Conversely, if the experiment is relatively fast the material will appear more elastic. 

Newton s law of viscous flow relates the shear stress (t) and rate of shear (dVIdy) for a fluid particle as follows ... 

Newton s law of viscous flow is named for Sir Isaac Newton, 1642-1727, the great British mathematician and physicist who is famous for Newton s laws of motion and for being one of the inventors of calculus. 

When a steady state has been reached there is no net force on the fluid since it does not accelerate. The frictional force due to viscosity at the surface of the cylinder balances the hydrostatic force pushing the liquid through the cylinder. From Newton s law of viscous flow,... 

In viscous (shearing) flow in a liquid, one layer of molecules flows past an adjacent layer. Newton s law of viscous flow is... 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

This corresponds to a Hamiltonian system which is characterized by a weak oscillatory perturbation of the SHV streamfunction T r, ) —> Tfr, Q + HP, (r, ( ) x sin(fEt). The equations of fluid motion (4.4.4) are used to compute the inertial and viscous forces on particles placed in the flow. Newton s law of motion is then... 

Familiar examples of the relation between generalized fluxes and forces are Fick s first law of diffusion, Fourier s law of heat transfer, Ohm s law of electricity conduction, and Newton s law of momentum transfer in a viscous flow. 

In order to use these general momentum conservation equations to calculate the velocity field, it is necessary to express viscous stress terms in terms of the velocity field. The equations which relate the stress tensor to the motion of the continuous fluid are called constitutive equations or rheological equations of state. Although the governing momentum conservation equations are valid for all fluids, the constitutive equations, in general, vary from one fluid material to another and possibly also from one type of flow to another. Fluids, which follow Newton s law of viscosity (although it is referred to as a law, it is just an empirical proposition) are called Newtonian fluids. For such fluids, the viscous stress at a point is linearly dependent on the rates of strain (deformation) of the fluid. With this assumption, a general deformation law which relates stress tensor and velocity components can be written ... 

The effect of applying a similar loading programme to a linear viscoelastic solid has several similarities (Figure 5.2(b)). In the most general case, the total strain e is the sum of three separate parts e, and ez. e and are often termed the immediate elastic deformation and the delayed elastic deformation respectively, ez is the Newtonian flow, that is that part of the deformation, which is identical with the deformation of a viscous liquid obeying Newton s law of viscosity. 

Because of their complex structure the mechanical behavior of polymeric materials is not well described by the classical constitutive equations Hooke s law (for elastic solids) or Newton s law (for viscous liquids). Polymeric materials are said to be viscoelastic inasmuch as they exhibit both viscous and elastic responses. This viscoelastic behavior has played a key role in the development of the understanding of polymer structure. Viscoelasticity is also important in the understanding of various measuring devices needed for rheometric measurements. In the fluid dynamics of polymeric liquids, viscoelasticity also plays a crucial role. " Also in the polymer-processing industry it is necessary to include the role of viscoelastic behavior in careful analysis and design. Finally there are important connections between viscoelasticity and flow birefringence. ... 

The flow behavior of most thermoplastics does not follow Newton s law of viscosity. To quantitatively describe the viscous behavior of polymeric fluids, Newton s law of viscosity is generalized as follows ... 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

Figure 3.30 The viscous flow (a) streamline motion of gases and liquids at a laminar flow, (b) friction between adjacent layers the Newton s law of internal friction.

The viscous component is dominant in liquids hence their flow properties may be described by Newton s law (Equation 14.3) where 17 is the viscosity, which states that the applied stress 5 is proportional to the rate of strain Ay/At, but is independent of the strain y or applied velocity gradient. 

As the temperature is increased above the rubbery plateau, the linear amorphous polymer assumes a viscous state and may undergo irreversible flow, i.e., flows such that the original shape is lost. The flow of the viscous liquid may approach a Newtonian flow, i.e., its flow properties may be estimated from Newton s law for ideal liquids. 

---