# Why is pressure gradient assumed to be constant with respect to radius in the derivation of Poiseuille's Law?

Poiseuille's Law relies on the fact that velocity is not constant throughout a cross-section of the pipe (it is zero at the boundary due to the no-slip condition and maximum in the center). By Bernoulli's Law, this means that pressure is maximum at the boundary and minimum at the center. But in the book I have it is assumed that the pressure gradient is independent of radius (distance from the center of the pipe), and the pressure gradient is thus extricated from a radius-integral. Can anyone justify this?

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First of all, Bernoulli's law is applicable only to inviscid flow, while Poiseuille's flow is for the viscous fluid. The fact that pressure is constant along the orthogonal cross section of the pipe could be derived from the assumption that the flow is parallel, that is everywhere inside the pipe the velocity field has only z-component (assuming the cylindrical coordinate system, with pipe oriented along the z-axis). Then the r-component of Navier-Stokes equation is then reduced to $0 =- \frac1{\rho}\frac{\partial p}{\partial r}$ (all terms containing velocity components are equal to zero here), which gives the pressure independent of radial coordinate.