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.

Expert Reply
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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