One tube of length L 0 and diameter D 0 splits symmetrically into two tubes of length L 1 and diameter D 1 . The free ends of the three tubes define an isosceles triangle with the base B and side R...

One tube of length L₀
and diameter D₀
splits symmetrically into two tubes of length L₁
and diameter D₁. The free ends of the three tubes define an isosceles triangle with the base B and side R (Fig. P13.1). The aspect ratio B∕R is a known parameter. An incompressible Newtonian fluid flows from the L₀
tube into the L₁
tubes or in the opposite direction. The flow is laminar and fully developed, and the pressure drop due to the Y junction is negligible. According to

the Hess–Murray rule, the global flow resistance is minimum when D₀∕D₁
= 2^1∕3. Minimize the global flow resistance further by selecting the tube lengths. Report the optimal L₀∕R and L₁∕R as functions of B/R. Show that the optimal bifurcation angle 𝛽 formed between the L₁
tubes is 74.94∘, which is independent of the triangle aspect ratio B /R.