For a certain fluid flow problem it is known that both the Froude number and the Weber number are important dimensionless parameters. If the problem is to be studied by using a 1: 15 scale model, determine the required surface tension scale if the density scale is equal to \(1 .\) The model and prototype operate in the same gravitational field.

The given parameters are: the scale of the model which is 1:15, the Froude number, the Weber number, and the surface tension scale which we need to determine. The density scale is equal to 1 and both the model and prototype operate under the same gravitational field.

The Froude number (F) is defined as \(F = \frac{U}{\sqrt{gL}}\), where \(U\) is the flow velocity, \(g\) is the acceleration due to gravity, and \(L\) is a characteristic length. From the given information, the Froude number for the model (F_m) is the same as that for the prototype (F_p). Hence, \(F_m = F_p\) or \(U_m/\sqrt{g_mL_m} = U_p/\sqrt{g_pL_p}\). As they operate in the same gravitational field \(g_p = g_m = g\), we get the ratio \(U_m/U_p = \sqrt{L_m/L_p}\), therefore, the velocity scales as the square root of the linear dimension.

The Weber number (We) is defined as \(We = \frac{\rho U^2 L}{\sigma}\), where \(U\) is the velocity, \(L\) is a characteristic length, \(\rho\) is the density, and \(\sigma\) is the surface tension. From the given information, the Weber number for the model (We_m) is the same as that for the prototype (We_p). Hence, \(We_m = We_p\) or \(\frac{\rho_m U_m^2 L_m}{\sigma_m} = \frac{\rho_p U_p^2 L_p}{\sigma_p}\). Given that the density scale \(\rho_m/\rho_p = 1\), we can simplify this to \(U_m^2/\sigma_m = U_p^2/\sigma_p\). Substituting the scale from step 2 for \(U_m/U_p=\sqrt{L_m/L_p}\), we get the relation \(\sqrt{L_m/L_p}^2/\sigma_m = \sqrt{L_m/L_p}^2/\sigma_p\), which simplifies to \(\sigma_m = \sigma_p (L_m/L_p)\).

By applying the ratio of the scale model 1:15, we substitute the corresponding values into the formula derived from step 3 to give \(\sigma_m = \sigma_p (1/15)\). This gives us the scale for the surface tension.