Extended Bernoulli equation, friction loss, and friction coefficient for microscopic Jeffery-Hamel flow with small Reynolds number up to O(1)

The extended Bernoulli equation is formulated in an exact form for a microscopic and small Reynolds number Jeffery-Hamel flow in a two-dimensional convergent or divergent channel. The friction loss and the friction coefficient derived from the extended Bernoulli equation are also obtained for the purpose of engineering applications. The assumption of microscopic and low Reynolds number flow enables us to make the analysis simple, and the results obtained are expressed in forms easy to use. The zeroth- and first-order approximate solutions of velocity distribution in the channel are obtained by solving the nonlinear ordinary differential equation with the optimal homotopy asymptotic method. The zeroth-order solution is shown to be the same function form as that in the two-dimensional parallel flow, i.e., the two-dimensional Poiseuille flow. The extended Bernoulli equation, the friction loss, and the friction coefficient in a finite region of the channel, which are indispensable for applications, are reasonably derived along a stream line and also expressed by cross-sectional average quantities. The cross-sectional average formulae of the friction loss and the friction coefficient are expressed by the geometry of the channel, i.e., the convergent or divergent angle, the channel length, channel widths at inlet and exit. These formulae include the corresponding well-known ones for the two-dimensional parallel flow as a special case where the angle is zero. The friction coefficient drastically increases according to the increase in the angle, especially in a narrow channel region, and attains more than ten times of the friction coefficient for the parallel flow.