$3$B$.2$ Laminar flow in a triangular duct $($Fig$.3$B$\mathrm{.}2).{?}^2$

One type of compact heat exchanger is shown in Fig.

$3$B$\mathrm{.}2($a$).$ In order to analyze the performance of such an

apparatus, it is necessary to understand the flow in a duct

whose cross section is an equilateral triangle. This is done

most easily by installing a coordinate system as shown in

Fig. $3$B$\mathrm{.}2($b$).$

$($a$)$ Verify that the velocity distribution for the laminar

flow of a Newtonian fluid in a duct of this type is given

by

$v_z=\frac{(P_0-P_L)}{4LH}(y-H)(3x^2-y^2)$

${?}^2$ An alternative formulation of the velocity profile is given

by L$.$ D$.$ Landau and E$.$ M$.$ Lifshitz, Fluid Mechanics, Pergamon,

Oxford, $2$nd edition $(1987),$ p$.54.$