Tube of Arbitrary Cross-Section

By using dimensional considerations, one can obtain the following formulas for the volume rate of flow and the maximum velocity in a rectilinear tube of constant cross-section of arbitrary shape  

The most important dimensionless geometric parameter characterizing the cross-section shape is the ratio y/S /V, where V is the cross-section perimeter. In calculations it is convenient to use the shape parameter 

For tubes of convex cross-section that is nearly circular or at least does not deviate very much from the circular shape, it is natural to assume that the coefficients K. and fCv in (1.5.24) depend only on  

Then in many cases the function K, - /C( ) (which determines the volume rate of flow Q in (1.5.24)) is well approximated by the linear function 

For = 1, the approximate formula (1.5.27) gives the exact value of 1Co corresponding to a circular tube. For example, formula (1.5.27) can be used for tubes whose cross-section is a regular JV-gon (N = 4,5.). In particular, for a tube of square cross-section, Eq. (1.5.25) gives = V 0.886. Substituting this into (1.5.27), we obtain K, - 0.0353. This differs from the exact value K, = 0.0351 only by 0.6% (see (1.5.23a)). For a tube of elliptic cross-section with axial ratio a/b - 1.5, the error in (1.5.27) is about 5%. 

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The general case of a tube of arbitrary cross-section, flow profile and variation of diffusion coefficient, for which the equations were set up in 1, may be considered in a similar way. Again, m0 must be a constant, which may be taken as unity, and for c0 we have the equations... 

VISCOUS FLOW IN A TUBE OF ARBITRARY CROSS-SECTION... 

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