Volumetric flux is given by Q = S vdA = S v(2nr)dr. Case A Q = Ja (ar + b)(2r)dr = 2n ((R - R) +(R R) (30 = aR, + b 145 = aR2 + b 15 b = 30 aR1 45 = aR2 + 30 aR1 a = 15R1 (R2- R1) b = 30 - (R2- R1) 5 15R1 , (R - R) + 30 (R2 - R,))(R R}) - Q = 2n (R2 - R,) Case B ectangular Snip Q = S (ar? + b)(2mer)dr = 21((R R$) + (R; R}) %3D S30 = aR; + b 45 = aR; + b 15R; (R R?) 15 b = 30 aR 45 = aR; + 30 aR a = - b = 30 (R R;) 15 15R; Q = 2n (R-R) 30 4(R- R?)(R - R;) + (R-R3) Assuming uniform flow with average radius/velocity 30+45 Yes to both cases, if the average velocity is defined not as but as the average of the velocity 2 R2 SK v(R)dR -. This will give the average velocity of the fluid in the annulus, and by multiplying by function ie R2-R1 the area we shouldget the average volume flow. If instead we take the average radius of the annulus ravg, and define vavg = v(ravg), in case A vavg will R2 equal- v(R)dR and we will get the correct result, but in case B vavg will not be the average velocity, R2-R1 and we will get a wrong result.