$\displaystyle \frac{\partial u'}{\partial t'} - v'_0(1+ Ae^{nt})\frac{\partial u'}{\partial y'}=\frac{dU'}{dt'} + \nu\frac{\partial ^2u'}{\partial y'^2}

$

introducing the non-dimensional qts,

$\displaystyle

y= \frac{y' v'_0}{\nu},

t= \frac{t' v'^2_0}{\nu},

n= \frac{\nu n'}{v'^2_0},

u= \frac{u'}{U'_0},

U= \frac{U'}{U'_0}

$

Wats the non-dimensional form of the eqn?