I am presently working on a problem in fluid dynamics where our group is investigating the behavior of temperature and velocity 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 following PDE (It checks out with this paper 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. Any help is appreciated.

The paper uses an asymptotic expansion in $R$ to solve the PDE but I require a solution in $\eta$ and $R$ since I need to use it in the energy equation. I did try to solve the 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.

Edit: As per Tomas Kojar’s suggestion in the comment section, consider the following two equations Now, setting $u=f_{\eta\eta}$, we obtain the following second-order PDE $$(1+η^2 )^2 u_{ηη}+8η(1+η^2 )u_{η}+4(1+3η^2 )u=0 $$ Which is solved with boundary conditions to obtain the solution $f_{1}(\eta,R)=\frac{2}{\pi}tan^{-1}(η)+\frac{2η}{π(1+η^2)}$. Now, the second equation reads $$[2ηff_{ηη}+(1+η^2 )(ff_{ηηη}+f_η f_{ηη} )-4(1+3η^2 ) f_{ηηR}-4η(1+η^2 ) f_{ηηηR}]R=0,$$ and since $R\neq 0$, we obtain a non-linear PDE which upon setting $v(\eta,R)=f_{\eta\eta\eta}$ yields the following PDE $$η^6 v^2+η^2 v^3 (1+η^2 )-4η(1+η^2 ) v_R=0.$$ But, this is still complicated to solve.

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