The Navier-Stokes Equations
In Chapter 2, Newton’s second law was applied to obtain the fluid-flow momentum equation in both integral and differential forms. In particular, recall Equations (2.13a to c), where the influence of viscous forces was expressed simply by the generic terms (•Tudviscous, (dd'(viscousi and (d%)viSCous – The purpose of this section is to obtain the analogous forms of Equations (2.13a to c) where the viscous forces are expressed explicitly in terms of the appropriate flow-field variables. The resulting equations are called the Navier-Stokes equations—probably the most pivotal equations in all of theoretical fluid dynamics.
In Section 2.3, we discussed the philosophy behind the derivation of the governing equations, namely, certain physical principles are applied to a suitable model of the fluid flow. Moreover, we saw that such a model could be either a finite control volume (moving or fixed in space) or an infinitesimally small element (moving or fixed in space). In Chapter 2, we chose the fixed, finite control volume for our model and obtained integral forms of the continuity, momentum, and energy equations directly from this model. Then, indirectly, we went on to extract partial differential equations from the integral forms. Before progressing further, it would be wise for you to review these matters from Chapter 2.
For the sake of variety, let us not use the fixed, finite control volume employed in Chapter 2; rather, in this section, let us adopt an infinitesimally small moving fluid element of fixed mass as our model of the flow, as sketched in Figure 15.11. To this model let us apply Newton’s second law in the form F = ma. Moreover, for the time being consider only the jc component of Newton’s second law:
Fx = max [15.14]
In Equation (15.14), Fx is the sum of all the body and surface forces acting on the fluid element in the x direction. Let us ignore body forces; hence, the net force acting on the element in Figure 15.11 is simply due to the pressure and viscous stress distributions over the surface of the element. For example, on face abed, the only force in the jc direction is that due to shear stress, rvv dx dz. Face efgh is a
Figure 1 5.1 1 Infinitesimally small, moving fluid element. Only the forces in the x direction are shown.
distance dy above face abed; hence, the shear force in the x direction on face efgh is [xyx + (дтух/ду) dy] dx dz. Note the directions of the shear stress on faces abed and efgh; on the bottom face, zyx is to the left (the negative x direction), whereas on the top face, zyx + (dzyx/dy) dy is to the right (the positive x direction). These directions are due to the convention that positive increases in all three components of velocity, u, v, and w, occur in the positive directions of the axes. For example, in Figure 15.11, и increases in the positive у direction. Therefore, concentrating on face efgh, и is higher just above the face than on the face; this causes a “tugging” action which tries to pull the fluid element in the positive x direction (to the right) as shown in Figure 15.11. In turn, concentrating on face abed, и is lower just beneath the face than on the face; this causes a retarding or dragging action on the fluid element, which acts in the negative x direction (to the left), as shown in Figure 15.11. The directions of all the other viscous stresses shown in Figure 15.11, including zxx, can be justified in a like fashion. Specifically, on face degh, zzx acts in the negative x direction, whereas on face abfe, zzx + (dzzx/8z)dz acts in the positive x direction. On face adhe, which is perpendicular to the x axis, the only forces in the x direction are the pressure force p dy dz, which always acts in the direction into the fluid element, and zxx dy dz, which is in the negative x direction. In Figure 15.11, the reason why zxx on face adhe is to the left hinges on the convention mentioned earlier for the direction of increasing velocity. Here, by convention, a positive increase in и takes place in the positive x direction. Hence, the value of и just to the left of face adhe is smaller than the value of и on the face itself. As a result, the viscous action of the normal stress
acts as a “suction” on face adhe: that is, there is a dragging action toward the left that wants to retard the motion of the fluid element. In contrast, on face bcgf, the pressure force [p + (dp/dx) dx] dy dz presses inward on the fluid element (in the negative x direction), and because the value of и just to the right of face bcgf is larger than the value of и on the face, there is a “suction” due to the viscous normal stress which tries to pull the element to the right (in the positive x direction) with a force equal to [txx + (dzxx/dx)dx]dydz.
Return to Equation (15.14). Examining Figure 15.11 in light of our previous discussion, we can write for the net force in the л direction acting on the fluid element:
Equation (15.15) represents the left-hand side of Equation (15.14). Considering the right-hand side of Equation (15.14), recall that the mass of the fluid element is fixed and is equal to
m = p dx dy dz
Also, recall that the acceleration of the fluid element is the time rate of change of its velocity. Hence, the component of acceleration in the x direction, denoted by ax, is simply the time rate of change of n; since we are following a moving fluid element, this time rate of change is given by the substantial derivative (see Section 2.9 for a review of the meaning of substantial derivative). Thus,
Du _ .
ax = — [15.17]
Combining Equations (15.14) to (15.17), we obtain
which is the x component of the momentum equation for a viscous flow. In a similar fashion, the у and z components can be obtained as
Dw dp drxz dzyz drzz
Dt dz dx dy dz
Equations (15.18a to c) are the momentum equations in the x, y, and z directions, respectively. They are scalar equations and are called the Navier-Stokes equations in
honor of two men—the Frenchman M. Navier and the Englishman G. Stokes—who independently obtained the equations in the first half of the nineteenth century.
With the expressions for rxy = xyx, xyi = xzy, rzx = rxz, rxx, xyy, and xzz from Equations (15.5) to (15.10), the Navier-Stokes equations, Equations (15.18a to c), can be written as
dи Эи Эи Зи dp Э / Эи
РЧ7 + риТ + pv^~ + pwV~ = UV • V + 2/х —
dt dx dy dz dx dx dx J
Equations (15.19a to c) represent the complete Navier-Stokes equations for an unsteady, compressible, three-dimensional viscous flow. To analyze incompressible viscous flow, Equations (15.19a to c) and the continuity equation [say, Equation (2.52)] are sufficient. However, for a compressible flow, we need an additional equation, namely, the energy equation to be discussed in the next section.
In the above form, the Navier-Stokes equations are suitable for the analysis of laminar flow. For a turbulent flow, the flow variables in Equations (15.19a to c) can be assumed to be time-mean values over the turbulent fluctuations, and p can be replaced by p + є, as discussed in Section 15.3. For more details, see References 42 and 43.