- Hagen-Poiseuille flow from the Navier-Stokes equations
The flow of fluid through a pipe of uniform (circular) cross-section is known as Hagen-Poiseuille flow. The Hagen-Poiseuille flow is an exact solution of the
Navier-Stokes equations influid mechanics . The equations governing the Hagen-Poiseuille flow can be derived from the Navier-Stokes equation incylindrical coordinates by making the following set of assumptions:# The flow is steady ( partial(...)/partial t = 0 ).
# The radial and swirl components of the fluid velocity are zero ( u_r = u_ heta = 0 ).
# The flow is axisymmetric ( partial(...)/partial heta = 0 ) and fully developed (partial u_z/partial z = 0 ).Then the second of the three Navier-Stokes
momentum equations and thecontinuity equation are identically satisfied. The first momentum equation reduces to partial p/partial r = 0 , i.e., thepressure p is a function of the axial coordinate z only. The third momentum equation reduces to::frac{1}{r}frac{partial}{partial r}left(r frac{partial u_z}{partial r} ight)= frac{1}{mu} frac{partial p}{partial z}:The solution is :u_z = frac{1}{4mu} frac{partial p}{partial z}r^2 + c_1 ln r + c_2 Since u_z needs to be finite at r = 0 , c_1 = 0 . The no slip
boundary condition at the pipe wall requires that u_z = 0 at r = R (radius of the pipe), which yields:c_2 = -frac{1}{4mu} frac{partial p}{partial z}R^2.
Thus we have finally the following
parabolic velocity profile::u_z = -frac{1}{4mu} frac{partial p}{partial z} (R^2 - r^2).
The maximum velocity occurs at the pipe centerline (r=0 ):
:u_z}_{max}=frac{R^2}{4mu} left(-frac{partial p}{partial z} ight).
The average velocity can be obtained by integrating over the pipe
cross-section :: u_z}_mathrm{avg}=frac{1}{pi R^2} int_0^R u_z cdot 2pi r dr = 0.5 {u_z}_mathrm{max}.The Hagen-Poiseuille equation relates the pressure drop Delta p across a circular pipe of length L to theaverage flow velocity in the pipe u_z}_mathrm{avg} and other parameters. Assuming that the pressure decreases linearly across the length of the pipe, we have frac{partial p}{partial z} = frac{Delta p}{L} (constant). Substituting this and the expression for u_z}_mathrm{max} into the expression for u_z}_mathrm{avg} , and noting that the pipe diameter D = 2R , we get: : u_z}_{avg} = frac{D^2}{32 mu} frac{Delta P}{L}. Rearrangement of this gives the Hagen-Poiseuille equation:: Delta P = frac{32 mu L {u_z}_mathrm{avg{D^2}.
ee also
*
Poiseuille's Law
*Couette flow
*Pipe flow
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