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Question
Thu May 24, 2012 By:

# Assuming that the velocity of flow of a liquid (v) through a narrow tube depends on the radius of the tube(r) , density of the liquid(D) and coefficient of viscosity(n),find the expression for velocity.

Thu May 24, 2012
In fluid dynamics, the HagenPoiseuille equation is a physical law that gives the pressure drop in a fluid flowing through a long cylindrical pipe. The assumptions of the equation are that the flow is laminar, viscous and incompressible and the flow is through a constant circular cross-section that is substantially longer than its diameter. It is also assumed that there is no acceleration of liquid in the pipe. The equation is also known as the HagenPoiseuille lawPoiseuille law and Poiseuille equation.

The flow of fluid through a pipe of uniform (circular) cross-section is known as HagenPoiseuille flow. The equations governing the HagenPoiseuille flow can be derived directly from the NavierStokes equations in cylindrical coordinates by making the following set of assumptions:

1. The flow is steady (  ).
2. The radial and swirl components of the fluid velocity are zero (  ).
3. The flow is axisymmetric (  ) and fully developed ( ).

Then the second of the three NavierStokes momentum equations and the continuity equation are identically satisfied. The first momentum equation reduces to , i.e., the pressure  is a function of the axial coordinate  only. The third momentum equation reduces to:

 where  is the dynamic viscosity of the fluid.
The solution is


Since  needs to be finite at . The no slip boundary condition at the pipe wall requires that  at  (radius of the pipe), which yields



Thus we have finally the following parabolic velocity profile:



The maximum velocity occurs at the pipe centerline ():



The average velocity can be obtained by integrating over the pipe cross section:



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