Solution for gravity driven flow of liquid inside a circular tube

DSolve[{ μ/r D[ r D[vz[r],r],r] + ρ gz ==0, vz[R] ==0, vz'[R-h] ==0},vz[r],r]

vz[r]/.%[[1]]

vzee=FullSimplify[%2]

Find the average velocity

Integrate[vzee r,{r,R,R-h}, Assumptions->{h>0,h<R}]/Integrate[ r,{r,R,R-h}, Assumptions->{h>0,h<R}]

FullSimplify[%]

Limit[%10,h->R]

DSolve[{ μ/r D[ r D[vz[r],r],r] + ρ gz ==0, vz[R] ==0, vz'[0] ==0},vz[r],r]

vzz=vz[r]/.%[[1]]

Integrate[%r,{r,R,0}, Assumptions->{h>0,h<R}]/Integrate[ r,{r,R,0}, Assumptions->{h>0,h<R}]

Flow in blood vessel with Catheter

DSolve[{μ/r D[ r v'[r],r] - G ==0, v[λ R] ==0, v[R] ==0},v[r],r]

v[r]/.%[[1]]

FullSimplify[%]

Integrate[%3 r,{r, λ R, R}, Assumptions->{λ>0}]

DSolve[{μ/r D[ r v'[r],r] - G ==0, v'[0] ==0, v[λ R] ==0},v[r],r]

v[r]/.%[[1]]

Integrate[% r,{r, 0 , λ R}, Assumptions->{λ>0}]

%12/%6

FullSimplify[%,Assumptions->{0≤λ≤1}]

FindRoot[%==1,{λ,.5}]