82 Fully Developed Laminar Flow

As is indicated in the previous section, the flow in long, straight, constant diameter sections of a pipe becomes fully developed. That is, the velocity profile is the same at any cross section of the pipe. Although this is true whether the flow is laminar or turbulent, the details of the velocity profile (and other flow properties) are quite different for these two types of flow. As will be seen in the remainder of this chapter, knowledge of the velocity profile can lead directly to other useful information such as pressure drop, head loss, flowrate, and the like. Thus, we begin by developing the equation for the velocity profile in fully developed laminar flow. If the flow is not fully developed, a theoretical analysis becomes much more complex and is outside the scope of this text. If the flow is turbulent, a rigorous theoretical analysis is as yet not possible.

Although most flows are turbulent rather than laminar, and many pipes are not long enough to allow the attainment of fully developed flow, a theoretical treatment and full understanding of fully developed laminar flow is of considerable importance. First, it represents one of the few theoretical viscous analyses that can be carried out "exactly" (within the framework of quite general assumptions) without using other ad hoc assumptions or approximations. An understanding of the method of analysis and the results obtained provides a foundation from which to carry out more complicated analyses. Second, there are many practical situations involving the use of fully developed laminar pipe flow.

Laminar flow char acteristics are different than those for turbulent flow.

There are numerous ways to derive important results pertaining to fully developed laminar flow. Three alternatives include: (1) from F = ma applied directly to a fluid element, (2) from the Navier-Stokes equations of motion, and (3) from dimensional analysis methods.

Steady, fully developed pipe flow experiences no acceleration.

8.2.1 From F = ma Applied Directly to a Fluid Element

We consider the fluid element at time t as is shown in Fig. 8.7. It is a circular cylinder of fluid of length / and radius r centered on the axis of a horizontal pipe of diameter D. Because the velocity is not uniform across the pipe, the initially flat ends of the cylinder of fluid at time t become distorted at time t + St when the fluid element has moved to its new location along the pipe as shown in the figure. If the flow is fully developed and steady, the distortion on each end of the fluid element is the same, and no part of the fluid experiences any acceleration as it flows. The local acceleration is zero (0V/dt = 0) because the flow is steady, and the convective acceleration is zero (V • V V = u 0 m/0x i = 0) because the flow is fully developed. Thus, every part of the fluid merely flows along its pathline parallel to the pipe walls with constant velocity, although neighboring particles have slightly different velocities. The velocity varies from one pathline to the next. This velocity variation, combined with the fluid viscosity, produces the shear stress.

If gravitational effects are neglected, the pressure is constant across any vertical cross section of the pipe, although it varies along the pipe from one section to the next. Thus, if the pressure is p = p1 at section (1), it is p2 = p1 — Ap at section (2). We anticipate the fact that the pressure decreases in the direction of flow so that Ap > 0. A shear stress, t, acts on the surface of the cylinder of fluid. This viscous stress is a function of the radius of the cylinder, t = t( r).

As was done in fluid statics analysis (Chapter 2), we isolate the cylinder of fluid as is shown in Fig. 8.8 and apply Newton's second law, Fx = max. In this case even though the fluid is moving, it is not accelerating, so that ax = 0. Thus, fully developed horizontal pipe flow is merely a balance between pressure and viscous forces—the pressure difference acting on the end of the cylinder of area pr2, and the shear stress acting on the lateral surface of the cylinder of area 2pr/. This force balance can be written as

(p1)pr2

which can be simplified to give

Fluid element at time t

Element at time t + St

Fluid element at time t

Element at time t + St

Motion of a cylindrical fluid element within a pipe.

Pi"

z2n rt

■ FIGURE 8.8 Free-body diagram of a cylinder of fluid.

0 0

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