Table 1 Summary of the common models for predicting the maximum spreading factor.

Jones¹⁰

\(\:{\beta\:}_{max}=\sqrt{\frac{4}{3}{\text{R}\text{e}}^{0.25}}\)

(1)

Scheller and Bousfield¹¹

\(\:{\beta\:}_{max}=0.61{\left(\frac{\text{W}\text{e}}{\text{O}\text{h}}\right)}^{0.166}\)

(2)

Asai et al.¹²

\(\:{\beta\:}_{max}=1+0.48{\text{W}\text{e}}^{0.5}\text{exp}\left(-1.48{We}^{0.22}{Re}^{-0.21}\right)\)

(3)

Roisman¹³

\(\:{\beta\:}_{max}=0.87{\text{R}\text{e}}^{0.2}-0.4{\text{R}\text{e}}^{0.4}{\text{W}\text{e}}^{-0.5}\)

(4)

Chandra and Avedisian¹⁴

\(\:\frac{3\text{W}\text{e}}{2\text{R}\text{e}}{\beta\:}_{max}^{4}+\left[1-\text{cos}\left(\theta\:\right)\right]{\beta\:}_{max}^{2}-\left(\frac{\text{W}\text{e}}{3}+4\right)=0\)

(5)

Mao et al.¹⁵

\(\:\left[\frac{1}{4}\left[1-\text{cos}\left(\theta\:\right)\right]+0.2\frac{{\text{W}\text{e}}^{0.83}}{{\text{R}\text{e}}^{0.33}}\right]{\beta\:}_{max}^{3}-\left(\frac{\text{W}\text{e}}{12}+1\right){\beta\:}_{max}+\frac{2}{3}=0\)

(6)

Ukiwe and Kwok¹⁶

\(\:\left(\text{W}\text{e}+12\right){\beta\:}_{max}=8+{\beta\:}_{max}^{3}\left[3\left[1-\text{cos}\left({\theta\:}_{Y}\right)\right]+4\frac{\text{W}\text{e}}{\sqrt{\text{R}\text{e}}}\right]\)

(7)

Aksoy et al.¹⁷

\(\:3.18\frac{{\text{W}\text{e}}^{0.72}}{{\text{R}\text{e}}^{0.86}}{\beta\:}_{max}^{6.5}=\left(\text{W}\text{e}+12\right){\beta\:}_{max}-{\beta\:}_{max}^{3}\left[3\left[1-\text{cos}\left({\theta\:}_{Y}\right)\right]\right]-8\)

(8)

Pasandideh-Fard et al.¹⁸

\(\:{\beta\:}_{max}=\sqrt{\frac{We+12}{3\left[1-\text{cos}\left({\theta\:}_{A}\right)\right]+4\left(\frac{\text{W}\text{e}}{\sqrt{\text{R}\text{e}}}\right)}}\)

(9)