1D viscous flow upwards against gravity Inviscid burgers equation for fluid flowing upwards against gravity:
$$ 
u \frac{du}{dy} = -g  
$$
I can solve for the velocity profile by simple integration and applying Dirichlet b.c. $u(0)=u_0$:
$$
u(y) = \sqrt{u_0^2 - 2gy}
$$
Now how to solve the viscous case
$$
\rho u \frac{du}{dy} = \mu \frac{d^2u}{dy^2} - \rho g 
$$
with B.C. $u(0)=u_0$ and $du/dy(0)=0$ ?

A series solution can be build as follows.
Make $u=\sum_{k=0}^n a_k y^k$ and substitute into the equation resulting the linear system. Here $n=5$
$$
\left(
\begin{array}{cccccc}
 1 & 0 & 0 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 & 0 & 0 \\
 a_1 \rho  & a_0 \rho  & -2 \mu  & 0 & 0 & 0 \\
 2 a_2 \rho  & 2 a_1 \rho  & 2 a_0 \rho  & -6 \mu  & 0 & 0 \\
 3 a_3 \rho  & 3 a_2 \rho  & 3 a_1 \rho  & 3 a_0 \rho  & -12 \mu  & 0 \\
 4 a_4 \rho  & 4 a_3 \rho  & 4 a_2 \rho  & 4 a_1 \rho  & 4 a_0 \rho  & -20 \mu  \\
\end{array}
\right)\left(\begin{array}{c}a_0\\ a_1\\a_2\\a_3\\a_4\\a_5\end{array}\right)=
\left(\begin{array}{c}u_0\\ 0\\-\rho g\\0\\0\\0\end{array}\right)
$$
Follows a MATHEMATICA script which handles the algebra
n = 5;
d[u_, y_] := rho u D[u, y] - mu D[u, y, y] + rho g
U = Sum[Subscript[a, k] y^k, {k, 0, n}]
res = d[U, y]
coefs = CoefficientList[res, y]
A = Table[Subscript[a, k], {k, 0, n}]
equs = Take[coefs, {1, n - 1}]
bcs = {U - u0, D[U, y]} /. {y -> 0}

equstot = Join[bcs, equs]
sols = Solve[equstot == 0, A]

U0 = U /. sols

