Chapter VI laminar flow

22.Laminar flow in circular pipes

As stated in Sec. 19, laminar flow takes place in stratified layers without any mixing of the fluid; it obeys Newton's friction law (see Sec. 4) and is completely determined by the latter. Therefore the theory of laminar flow is based on Newton's law of viscous shear.

Consider a section of a straight horizontal pipe of diameter d = 2r₀ between cross-sections 1-1 and 2-2 at distance l apart (Fig. 44) taken sufficiently far away from the entrance to the pipe. The pres sure is p_x at section 1-1 and p₂ at section 2-2. In view of the uniform­ity of the pipe diameter, the velocity and a coefficient are constarc along the pipe, and Bernoulli's equation for the two sections takes the form

where h_f = loss of bead due to friction. Hence,

which is shown by piezometers tapped at the crossrsections.

Now consider a cylindrical control volume of radius г inside Jie stream, such that it is coaxial with the pipe and its two bases lie in the cross-sections.

The equation of uniform flow for the control volume is given by the equality to zero of the sum of the pressure forces and fricion forces acting on the volume. Expressing the shear stress at the bcund- ary of the cylinder by t, we have

whence,

The equation shows that shear stress across the pipe section varies according to a linear law as a function of the radius. The shear stress profile is shown on the left side of Fig. 44.

Now, introducing Newton's friction law, express the shear stress t in terms of the dynamic viscosity and velocity gradient [Eq. (1.11)]

and substitute for the variable у (the distance from the pipe wall) the radius r:

The minus sign is due to the fact that r is measured in the opposite direction of y.

Substituting the expression for т into the previous equation, we obtain

from which we find the velocity increment dv:

To the positive radius increment there corresponds a negative ve­locity increment (i. e., a retardation), which is demonstrated by the velocity profile in Fig. 44.

Integrating, we have

The integration constant С can be found from the boundary conditions where, at r_o~ r , и = 0, whence

The velocity at any point on a cir­cle of radius г is

(6.1)

This is the law of velocity distribution across a circular, pipe section when the flow is laminar. The velocity distribution curve is a para­bola of the second order. The maximum velocity is at the centre of the cross-section, where г = 0, and -is equal to

(6.2)

The ratio in Eq. (6.1), as is apparent from Fig. 44, represents the hydraulic gradient multiplied by y. This quantity is constant along a straight pipe of uniform diameter.

The law of velocity distribution (6.1) can be used to compute the rate of discharge of a pipe. First express the elementary discharge across a differential area dS:

dQ = vdS.

Here у is a function of the radius determined by Eq. (6.1); the area dS is conveniently taken as a ring of radius r and width dr (Fig. 45). Then,

Integrating over the cross-section, i. e., from r = 0 to г = r₀, we have

(6.3)

The mean velocity at the section is found by dividing the discharge by the area:

(6.4)

Comparing this expression with Eq. (6.2), we arrive at the conclu­sion that in laminar flow the mean velocity is just one-half of the maximum velocity:

v_m = 1/2 v_max.

To obtain the friction law, i. e., to express the loss of head due to friction h_f in terms of rate of discharge and pipe dimensions, deter­mine p_f trom Eq. (6.3):

Dividing through by y, we obtain the loss of head:

Substituting vq for fi, and gQ for y, and going over from r₀ to d == 2r₀, we finally obtain

(6.5)

The friction law shows that in laminar flow through a circular pipe the loss of head due to friction varies as the rate of discharge and the viscosity and inversely as the fourth power of the diameter. This law, commonly known as the Hagen-Poiseuille equation, is used to calculate pipelines with laminar flow.

In Sec. 17 we agreed to express friction losses in terms of the mean velocity by Eq. (4.18). Let us develop Eq. (6.5) into the form

For this, in Eq. (6.5) substitute the expression for the rato of discharge. After the necessary cancellations, we have

Multiplying and dividing through by v_m and rearranging, we obtain

or, finally, bringing to the form of Eq. (4.18'),

(6.6)

where

(6.7)

the subscript at X denoting that the flow is laminar.

Friction loss in laminar flow, it should be borne in mind, is pro­portional to the first power of the velocity. The square of the veloc­ity in Eq. (6.6) was obtained artificially, as a result of multiplica­tion and division by v_m, and the coefficient 'k_l is inversely propor­tional to Re, and consequently, to u_m.

The theory of laminar flow through circular pipes set forth here is generally confirmed by experience and the friction law does not require any correction factors, with two exceptions:

1. At the beginning of a pipe, where the parabolic velocity profile has not yet developed. Resistance along the entrance portion of the pipe is greater than further on. This consideration, however, need be taken into account only in short pipelines. It will be analysed in the following section.

1. When considerable heat exchange is taking place, that is when the flowing fluid is heated or cooled. The resistance in this case is usually above the average.