A. Hydrodynamic entrance region of isothermal duct flow

In many practical cases the flow in pipes is not fully developed, but rather in the process of developing and hence eventually reaching the fully developed state.

One typical example of such a developing flow is the flow in the entrance region of a pipe. At the entrance section of the pipe the flow is assumed to have constant velocity over the whole cross-section. In the inlet region of the pipe the velocity profile is gradually transformed and tends, under the influence of viscous forces, to assume the asymptotic velocity profile at the fully developed pipe flow, discussed in the previous sections. The hydrodynamic entrance region of isothermal flow in a horizontal pipe of constant cross-section will be considered here.

According to Equation 15778.37 and Equation 15778.47, the pressure drop between the entrance section (index 1, x = 0) and the section at station x is given by

\[\label{eq1}\frac{p_1-p(x)}{(\rho/2)\bar{u}^2}=\frac{4}{d_h}\int^x_0\frac{\tau_w(x)}{(\rho/2)\bar{u}^2}dx +2\left[\beta(x)-1\right]\tag{1}\]