I am presently working on a problem in fluid dynamics where our group is investigating the behavior of temperature, velocity, and pressure at the leading edge of a flat plate when fluid flows past it. The momentum equation, after introducing stream functions and cross-differentiation, reads
$$\rho\left(\psi_{y}\nabla^{2}\psi_{x} - \psi_{x}\nabla^{2}\psi_{y} \right) = \mu\nabla^{4}\psi$$
I have made use of the self-similar transformation $\eta=\frac{y}{x}$ with $f(\eta,R)=\frac{\psi}{Ux}$ to obtain the momentum equation as the following PDE (It checks out with paper [1] which I have referred (pg 156); note: there is one error in the paper, it has $2\eta ff_{\eta}$ instead of $2\eta ff_{\eta\eta}$ in the $R[...]$ terms)
$$\left[(1+\eta^{2})^{2}f_{\eta\eta\eta\eta}+8(1+\eta^{2})f_{\eta\eta\eta}+4(1+3\eta^{2})f_{\eta\eta}\right]$$
$$+\left[2\eta ff_{\eta\eta}+(1+\eta^{2})(ff_{\eta\eta}+f_{\eta}f_{\eta\eta})- 4(1+3\eta^{2})f_{\eta\eta R}-4\eta(1+\eta^{2})f_{\eta\eta\eta R}\right]R$$
$$+\left[2\eta(f_{R}f_{\eta\eta}-ff_{\eta\eta R})-(1+\eta^{2})(f_{\eta}f_{\eta\eta R}-f_{R}f_{\eta\eta\eta})+2(1+3\eta^{2})f_{\eta\eta RR}\right]R^{2}$$
$$+\left[2\eta(f_{\eta}f_{\eta RR}-f_{R}f_{\eta \eta R})+ff_{\eta RR}-3f_{\eta}f_{RR}+4f_{RRR}-4\eta f_{\eta RRR}\right]R^{3}$$
$$+\left[f_{RRRR}+f_{R}f_{\eta RR}-f_{\eta}f_{RRR}\right]R^{4}=0$$
Boundary conditions: At the plate, $\eta=0$ and we have $f_{\eta}=0$ & $f+f_{R}R=0$. At the leading edge, $\eta \rightarrow \infty$ and we have $f_{\eta}\rightarrow 1$ & $f+f_{R}R\rightarrow \eta$.
Since this problem is at the leading edge, we consider Reynolds numbers of the order $10^{-3}$ and hence, we can neglect all $\mathcal{O}(R^{2})$ and higher order terms of the PDE, but still, solving the resulting PDE is daunting. However, does a closed-form solution exist for the entire PDE?
The paper uses an asymptotic expansion in $R$ and setting $u=f_{\eta\eta}$, we obtain the following second-order ODE
$$(1+η^2 )^2 u_{ηη}+8η(1+η^2 )u_{η}+4(1+3η^2 )u=0, $$
to solve the PDE but I require a solution in $\eta$ and $R$ since I need to use it in finding the heat transfer coefficient via the energy equation. Just to give some context, the energy equation under a self-similar transformation reads
$$R^{2} \left(T_{RR}-PrU\left(f_{\eta}f_{\eta R}-f_{R}f_{\eta\eta} \right) \right)+R\left(PrU-4\eta T_{\eta} \right) +2\eta T_{\eta} + \left(1+\eta^{2} \right)T_{\eta\eta}=0,$$
where $Pr$ is the Prandtl number and $T=T(\eta,R)$. Setting $Pr\equiv1$, we can now define a second-order operator $L$ as follows
$$L:= \frac{1}{R^{2}}\left(1+\eta^{2}\right)\partial_{\eta\eta}^{2}+2\eta \left(1-2R \right)\partial_{\eta},$$
such that
$$L[T]+T_{RR}=H\ \ in\ V_{R},$$
where $H=-U(f_{\eta}f_{\eta R}-f_{R}f_{\eta\eta})$. Thus, with boundary conditions, we define the following IBVP
$$L[T]+T_{RR}=H\ \ in\ V_{R}$$
$$T=T_{\infty}\ \ on\ \partial V\times [0,R]$$
$$T=h,\ T_{R}=g\ \ on\ V\times\{R=0\} $$
where $V$ is an open set of $\Bbb{R}^{2}$, $V_{R}=V\times (0,R]$ for a fixed Reynold’s number $R>0$, $H:V_{R}\rightarrow \Bbb{R}$ and $h,g:V\rightarrow \Bbb{R}$ are given and $T:\bar{V}_{R}\rightarrow \Bbb{R}$ is unknown. Solving this PDE gives $T(\eta,R)$ and later gives $h(x)$ in terms of $f_{\eta},f_{\eta R},f_{R},f_{\eta \eta}$. I did try to solve the momentum PDE neglecting the terms of $R^{2}$ and higher powers and later trying to compare it to a standard form mentioned in Polyanin, but couldn't find a solution.
[1]: https://digitalcommons.lsu.edu/gradschool_dissertations/1087/
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