Nomenclature

$v_n\left(t\mathrm{,}x\right)$

fluid injection/leak-off velocity

$h\left(t\mathrm{,}x\right)$

height of thin film

$w\left(t\right)$

half-width of thin fluid film

$\eta$

$x/w\left(t\right)$ scaled similarity variable

$f\left(\eta \right)$

invariant solution for $h\left(t\mathrm{,}x\right)$

$g\left(\eta \right)$

invariant solution for $v_n\left(t\mathrm{,}x\right)$

$\psi \left(t\mathrm{,}x\mathrm{,}z\right)$

stream function

$\alpha$

$c_4/c_2$ where $c_4$ and $c_2$ are constants in the Lie point symmetry

${\beta }^{\mathrm{<mo>\; *\; </mo>}}$

constant in equation $g\left(\eta \right)=-{\beta }^{\mathrm{<mo>\; *\; </mo>}}\eta \frac{\partial f}{\partial \eta }$

$\beta$

$2{\beta }^{\mathrm{<mo>\; *\; </mo>}}/5\left(1-{\beta }^{\mathrm{<mo>\; *\; </mo>}}\right)$

$\epsilon \left(t\right)$

$h\left(t\mathrm{,0}\right)/w\left(t\right)$ thin fluid film ratio

$z^{\mathrm{<mo>\; *\; </mo>}}$

$z/h\left(t\mathrm{,0}\right)$ scaled variable

$\left({\eta }_P\mathrm{,}{z}_P^{\mathrm{<mo>\; *\; </mo>}}\right)$

point of intersection of dividing streamline with free surface in $\left(\eta \mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)$ -plane

$\left(x_P\mathrm{,}{z}_P\right)$

point of intersection of dividing streamline with free surface in $\left(x\mathrm{,}z\right)$ -plane

$\theta \left(\beta \right)$

angle dividing streamline makes with positive $\eta$ -axis

$\theta \left(t\right)$

angle dividing streamline makes with positive x-axis