Poiseuille Flow

(10.23) |

is the uniform effective pressure gradient along the pipe, and

(10.24) |

the time independent velocity profile driven by this gradient. It follows from Section 1.19 that and . Hence, Equation (10.2) reduces to

(10.25) |

Taking the -component of this equation, we obtain

(10.26) |

where use has been made of Equation (1.155). The most general solution of the previous equation is

(10.27) |

where and are arbitrary constants. The physical constraints are that the flow velocity is non-singular at the center of the pipe (which implies that ), and is zero at the edge of the pipe [i.e., ], in accordance with the no slip condition. Thus, we obtain

(10.28) |

The volume flux of fluid down the pipe is

According to the previous analysis, the quantity should be directly proportional to the effective pressure gradient along the pipe. The accuracy with which experimental observations show that this is indeed the case (at relatively low Reynolds number) is strong evidence in favor of the assumptions that there is no slip at the pipe walls, and that the flow is non-turbulent. In fact, the result (10.29), which is known as