Viscosity and Thermal Conduction

The basic physical phenomena of viscosity and thermal conduction in a fluid are due to the transport of momentum and energy via random molecular motion. Each molecule in a fluid has momentum and energy, which it carries with it when it moves from one location to another in space before colliding with another molecule. The transport of molecular momentum gives rise to the macroscopic effect we call viscosity, and the transport of molecular energy gives rise to the macroscopic effect we call thermal conduction. This is why viscosity and thermal conduction are labeled as transport phenomena. A study of these transport phenomena at the molecular level is part of kinetic theory, which is beyond the scope of this book. Instead, in this section we simply state the macroscopic results of such molecular motion.

Consider the flow sketched in Figure 15.9. For simplicity, we consider a one­dimensional shear flow, that is, a flow with horizontal streamlines in the x direction but with gradients in the у direction of velocity, du/dy, and temperature, dT/dy.

conduction to velocity and temperature gradients, respectively.

Consider a plane ab perpendicular to the у axis, as shown in Figure 15.9. The shear stress exerted on plane ab by the flow is denoted by xyx and is proportional to the velocity gradient in the у direction, xyx ос 3u/dy. The constant of proportionality is defined as the viscosity coefficient ц. Hence,

Эи

tyx = Iі T~

The subscripts on xyx denote that the shear stress is acting in the x direction and is being exerted on a plane perpendicular to the у axis. The velocity gradient du/dy is also taken perpendicular to this plane (i. e., in the у direction). The dimensions of /і are mass/length x time, as originally stated in Section 1.7 and as can be seen from Equation (15.1). In addition, the time rate of heat conducted per unit area across plane ab in Figure 15.9 is denoted by qY and is proportional to the temperature gradient in the у direction, qy ос ЗT/3y. The constant of proportionality is defined as the thermal conductivity k. Hence,

[15.2]

where the minus sign accounts for the fact that the heat is transferred from a region of high temperature to a region of lower temperature; that is, qy is in the opposite direction of the temperature gradient. The dimensions of к are mass x length/(s2 • K), which can be obtained from Equation (15.2) keeping in mind that qY is energy per second per unit area.

Both д and к are physical properties of the fluid and, for most normal situa­tions, are functions of temperature only. A conventional relation for the temperature variation of ц for air is given by Sutherland’s law,

[15.3]

where T is in kelvin and до is a reference viscosity at a reference temperature, 7q. For example, if we choose reference conditions to be standard sea level values, then до = 1 -7894 x 10-5kg/(m • s) and 7b = 288.16 K. The temperature variation of к is analogous to Equation (15.3) because the results of elementary kinetic theory show that к ос jiCp; for air at standard conditions,

k = l.45 fiCp

where cp = 1000 J/(kg • K).

Equations (15.3) and (15.4) are only approximate and do not hold at high tem­peratures. They are given here as representative expressions which are handy to use. For any detailed viscous flow calculation, you should consult the published literature for more precise values of д and k.

In order to simplify our introduction of the relation between shear stress and viscosity, we considered the case of a one-dimensional shear flow in Figure 15.9. In this picture, the у and z components of velocity, v and w, respectively, are zero. However, in a general three-dimensional flow, u, v, and w are finite, and this requires a generalization of our treatment of stress in the fluid. Consider the fluid element sketched in Figure 15.10. In a three-dimensional flow, each face of the fluid element experiences both tangential and normal stresses. For example, on face abed, xxy and xxz are the tangential stresses, and xxx is the normal stress. As before, the nomenclature rtj denotes a stress in the j direction exerted on a plane perpendicular to the і axis. Similarly, on face abfe, we have the tangential stresses xyx and xyz, and the normal stress xyy. On face adge, we have the tangential stresses x7X and xzy, and the normal stress xzz. Now recall the discussion in the last part of Section 2.12 concerning the strain of a fluid element, that is, the change in the angle к shown in Figure 2.31. What is the force which causes this deformation shown in Figure 2.31? Returning to Figure 15.10, we have to say that the strain is caused by the tangential shear stress.

z

Figure 1 5*10 Shear and normal stresses caused

by viscous action on a fluid element.

However, in contrast to solid mechanics where stress is proportional to strain, in fluid mechanics the stress is proportional to the time rate of strain. The time rate of strain in the xy plane was given in Section 2.12 as Equation (2.135a):

Examining Figure 15.10, the strain in the xy plane must be carried out by тху and ryx. Moreover, we assume that moments on the fluid element in Figure 15.10 are zero; hence, xxy = xyx. Finally, from the above, we know that xxy – xyx ос єху. The proportionality constant is the viscosity coefficient д. Hence, from Equation (2.135a), we have

f dv du

tXy — tyx — Д ( ~dy ) ^ ®-®]

which is a generalization of Equation (15.1), extended to the case of multidimensional flow. For the shear stresses in the other planes, Equations (2.135i> and c) yield

The normal stresses xxx, xyy, and xzz shown in Figure 15.10 may at first seem strange. In our previous treatments of inviscid flow, the only force normal to a surface in a fluid is the pressure force. However, if the gradients in velocity Эм/dx, dv/dy, and dw/dz are extremely large on the faces of the fluid element, there can be a meaningful viscous-induced normal force on each face which acts in addition to the pressure. These normal stresses act to compress or expand the fluid element, hence changing its volume. Recall from Section 2.12 that the derivatives du/dx, dv/dy, and dw/dz are related to the dilatation of a fluid element, that is, to V • V. Hence, the normal stresses should in turn be related to these derivatives. Indeed, it can be shown that

du = A(V • V) + 2д — dx	[15.8]
dv = A(V-V) + 2 pt- dy	[15.9]
dw = A(V • V) + 2д—— 3 z	[15.10]

In Equations (15.8) to (15.10), A. is called the bulk viscosity coefficient. In 1845, the Englishman George Stokes hypothesized that

A = -| д [15.11]

To this day, the correct expression for the bulk viscosity is still somewhat controversial, and so we continue to use the above expression given by Stokes. Once again, the

normal stresses are important only where the derivatives du/dx, dv/dy, and 3w/dz are very large. For most practical flow problems, xxx, xyy, and rzz are small, and hence the uncertainty regarding X is essentially an academic question. An example where the normal stress is important is inside the internal structure of a shock wave. Recall that, in real life, shock waves have a finite but small thickness. If we consider a normal shock wave across which large changes in velocity occur over a small distance (typically 10-5 cm), then clearly du/dx will be very large, and rxx becomes important inside the shock wave.

To this point in our discussion, the transport coefficients /x and к have been considered molecular phenomena, involving the transport of momentum and energy by random molecular motion. This molecular picture prevails in a laminar flow. The values of /x and к are physical properties of the fluid; that is, their values for different gases can be found in standard reference sources, such as the Handbook of Chemistry and Physics (The Chemical Rubber Co.). In contrast, for a turbulent flow the transport of momentum and energy can also take place by random motion of large turbulent eddies, or globs of fluid. This turbulent transport gives rise to effective values of viscosity and thermal conductivity defined as eddy viscosity є and eddy thermal conductivity k, respectively. (Please do not confuse this use of the symbols £ and к with the time rate of strain and strain itself, as used earlier.) These turbulent transport coefficients є and к can be much larger (typically 10 to 100 times larger) than the respective molecular values /x and k. Moreover, є and к predominantly depend on characteristics of the flow field, such as velocity gradients; they are not just a molecular property of the fluid such as ji and k. The proper calculation of є and к for a given flow has remained a state-of-the-art research question for the past 80 years; indeed, the attempt to model the complexities of turbulence by defining an eddy viscosity and thermal conductivity is even questionable. The details and basic understanding of turbulence remain one of the greatest unsolved problems in physics today. For our purpose here, we simply adopt the ideas of eddy viscosity and eddy thermal conductivity, and for the transport of momentum and energy in a turbulent flow, we replace /г and k in Equations (15.1) to (15.10) by the combination /x + £ and k + к; that is,

An example of the calculation of є and к is as follows. In 1925, Prandtl suggested

that

for a flow where the dominant velocity gradient is in the у direction. In Equa­tion (15.12),/ is called the mixing length, which is different for different applications; it is an empirical constant which must be obtained from experiment. Indeed, all turbulence models require the input of empirical data; no self-contained purely the-

oretical turbulence model exists today. Prandtl’s mixing length theory, embodied in Equation (15.12), is a simple relation which appears to be adequate for a number of engineering problems. For these reasons, the mixing length model for є has been used extensively since 1925. In regard to к, a relation similar to Equation (15.4) can be assumed (using 1.0 for the constant); that is,

к = єср [15.13]

The comments on eddy viscosity and thermal conductivity are purely introduc­tory. The modern aerodynamicist has a whole stable of turbulence models to choose from, and before tackling the analysis of a turbulent flow, you should be familiar with the modern approaches described in such books as References 42 to 45.