question about fluid flow

[amestramgram]

Hi, in the Bernoulli principle it says the smaller the radius of the tube the faster the velocity of the liquid flow

in the Poiseuille principle it says the BIGGER the radius of the tube the greater the velocity of liquid flow

could somebody explain me why there is a contradiction?

thanks

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[dualityim]

You may be a bit confused here - Poiseuille's equation gives laminar flow, which is velocity times cross sectional area. In other words, flow, not velocity, is proportional to radius

 

[aevea]

Hi, in the Bernoulli principle it says the smaller the radius of the tube the faster the velocity of the liquid flow
in the Poiseuille principle it says the BIGGER the radius of the tube the greater the velocity of liquid flow

could somebody explain me why there is a contradiction?
thanks

The key to answering your question is understanding that the flow equation (and Bernoulli's principle) are for comparing different points within the same flowing system, while Poiseuille's principle is for comparing different systems.

The continuity equation (volume rate flow equals cross-sectional area times velocity) shows that since volume rate flow is constant within a system of flowing incompressible fluid, the area of the pipe is inversely proportional to the velocity. So fluid has got to move faster where the area is smaller to get the same amount of stuff through. I always think about putting my thumb on the end of a water hose. So within a particular system the velocity of the moving fluid is inversely proportional to the square of the radius.

Poiseuille's principle is for understanding real fluids and how they are affected by factors that increase or decrease friction. In comparing different systems, viscosity and the length of the pipe increase the energy lost to friction (resistance to flow) and therefore decrease the flow rate and velocity of the fluid. Because a wider pipe has less surface area to "rub" against the fluid, increased radius decreases resistance and therefore increases overall system flow rate and velocity. Poiseuille's just says that a given system will have greater flow if it has less resistance.