Flow in a Circular Pipe (Differential Analysis)

Let us analyze the pressure driven flow (simply Poiseuille flow ) through a straight circular pipe of constant cross section. Irrespective of the fact that the flow is laminar or turbulent, the continuity equation in the cylindrical coordinates is written as,

(5.4.8)

The important assumptions involved in the analysis are, fully developed flow so that only and there is no swirl or circumferential variation i.e. as shown in Fig. 5.4.1. So, Eq. (5.4.8) takes the following form;

(5.4.9)

Referring to Fig. 5.4.1, no-sip conditions should be valid at the wall . If Eq. (5.4.9) needs to be satisfied, then , everywhere in the flow field. In other words, there is only one velocity component , in a fully developed flow. Moving further to the differential momentum equation in the cylindrical coordinates,

(5.4.10)

Since, , the LHS of Eq. (5.4.10) vanishes while the RHS of this equation is simplified with reference to the Fig. 5.4.1.

(5.4.11)

It is seen from Eq. (5.4.11) that LHS varies with while RHS is a function of x. It must be satisfied if both sides have same constants. So, it can be integrated to obtain,

(5.4.12)

The constant of integration (c) must be zero to satisfy the condition of no shear stress along the center line . So, the end result becomes,

(5.4.13)

Further, at the wall the shear stress is represented as,

(5.4.14)

It is seen that the shear stress varies linearly from centerline to the wall irrespective of the fact that the flow is laminar or turbulent. Further, when Eqs. (5.4.4 & 5.4.14) are compared, the wall shear stress is same in both the cases.