1D viscous flow upwards against gravity

Question

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 $du/dy(0)=0$ ?

Answer 1

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