Nomenclature

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

height of thin fluid film

$w\left(t\right)$

half-width of thin fluid film

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

fluid pressure in thin fluid film

Re

Reynolds number of thin fluid film flow

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

fluid injection/leak-off velocity

$V\left(t\right)$

total volume of thin fluid film per unit breadth

X

Lie point symmetry

$\eta$

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

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

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

$f\left(\eta \right)$

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

$g\left(\eta \right)$

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

$\beta$

constant of proportionality in equation $g\left(\eta \right)=\beta f\left(\eta \right)$

$\alpha$

$\frac{c_4}{c_2}$ where $c_4$ and $c_2$ are constants in the Lie point symmetry

$\epsilon \left(t\right)$

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

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

stream function