Newton’s law of viscosity, equation

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. 

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 ... 

As was shown earlier, each of the three transport processes is a function of a driving force and a transport coefficient. It is also possible to make the equations even more similar by converting the transport coefficients to the forms of dif-fusivities. Pick s First Law [equation (1-9)] already has its transport coefficient (Dab) in this form. The forms for Fourier s Law [equation (1-7)] and Newton s Law of Viscosity [equation (1-8)] are... 

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

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... 

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. 

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... 

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 ... 

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. 

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. 

A further complication of determination of the viscosity of a macromol-ecular solution is that the viscosity depends on the concentration of macromolecules. Newton developed a formula for predicting the viscosity of a solution of macromolecules and solvent q, knowing the solvent viscosity r, shape factor for the macromolecule v, and the volume fraction of macromolecules 9. Newton s law of viscosity is given in Equation (4.2) and provides a relationship among viscosity, shape factor, and volume fraction of polymer. 

Eq. 3. 85 is Newton s law of viscosity. It is experimentally observed by several liquids provided their rate of flow is not very high. The fluid flow to which this equation applies is called laminar (or streamlined) flow. The equation does not apply to turbulent flow. Liquids which do not obey Eq. 3.85 are called non-Newtonian liquids. For Newtonian liquids, 77 is independent of dv/dxwhereas for non-Newtonian liquids, 77 changes as dv/dz changes. 

From Newton s law of viscosity, the shear stress = fxiduJdy), where is the absolute viscosity. Substituting, Equation (6.12) becomes... 

Three equations are basic to viscoelasticity (1) Newton s law of viscosity, a = ijy, (2) Hooke s law of elasticity. Equation 1.15, and (3) Newton s second law of motion, F = ma, where m is the mass and a is the acceleration. One can combine the three equations to obtain a basic differential equation. In linear viscoelasticity, the conditions are such that the contributions of the viscous, elastic, and the inertial elements are additive. The Maxwell model is ... 

To be more precise, the general tensor equation of Newton s law of viscosity should be obeyed by a Newtonian fluid (2) however, for onedimensional flow, the applicability of eq 1 is sufficient. For a Newtonian fluid, a linear plot of t versus 7 gives a straight line whose slope gives the fluid viscosity. Also, a log-log plot of t versus 7 is linear with a slope of unity. Both types of plots are useful in characterizing a Newtonian fluid. For a Newtonian fluid, the viscosity is independent of both t and 7, and it may be a function of temperature, pressure, and composition. Moreover, the viscosity of a Newtonian fluid is not a function of the duration of shear nor of the time lapse between consecutive applications of shear stress (3). 

Thus, we have a set of equations of motion given by Eqs. [91] that describe a general coupling to an external field. Our objective is to compute averages of functions of the phase space when the system coupled to the external field has reached a steady state. This is the procedure of nonequilibrium molecular dynamics (NEMD) simulations. An illustrative example to consider is the computation of the shear viscosity from Newton s law of viscosity, which reads ... 

We have thus demonstrated that Newton s law of viscosity, an inherently macroscopic result, can be obtained via linear response theory as the nonequilibrium average in the steady state. Furthermore, the distribution function for the steady state average is determined by microscopic equations of motion. Hence, the SLLOD equations, in the linear regime, reduce to the linear phenomenological law proposed by Newton. Moreover, all the quantities that are needed to compute the shear viscosity can be obtained from a molecular dynamics simulation. 

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 last relation was obtained using the variable transformation s=t — t, which implies ds= — dt The integration limits change with this transformation because when F = —00, s = oc, and at t — t, s — 0. For any liquid, the relaxation modulus G(t) eventually decays to zero fast enough that the integral in the above equation is simply a number with units of stress time. Thus, the stress at long times in the steady simple shear experiment is constant, and proportional to the shear rate 7. Newton s law of viscosity [Eq. (7.100)] already defined the viscosity in steady shear as the ratio of shear stress and shear rate. Therefore, the viscosity of any liquid is the time integral of its stress relaxation modulus ... 

The latter equation is usually referred as the Newtonian model or Newton s law of viscosity. In Equation 5.246, is the dilatational bulk viscosity and t) is the shear bulk viscosity. The usual liquids comply well with the Newtonian model. On the other hand, some concentrated macromolecular... 

In these equations T is the temperature, p the mass density, iua the mass fraction of species A. and o,v the. r-component of the fluid velocity vector. The parameter k is the thermal conductivity, D the diffusion coefficient for species A. and / the fluid viscosity from experiment the values of these parameters are all greater than or equal to zero (this is. in fact, a requirement for the system to evolve toward equilibrium). Equation 1.7-2 is known as Fourier s law of heat conduction, Eq. 1.7-." is called Pick s first law of diffusion, and Eq-. 1.7-4 is Newton s law of viscosity. 

---