On Closed-form Solutions to the Fourth-Order Non-Linear Momentum Equation

Question

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/

Comments

Highly improbable that you will find a closed form solution, you should try numerical schemes.

I mean you can obviously check special functions handbooks available online and see if there's something similiar to your pde.

 

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@MathematicalPhysicist I did solve it numerically but I require some sort of an approximate solution in the independent variables (at least up to order $\mathcal{O}(R)$) so that I can find the heat-transfer coefficient as mentioned in the question and I did check up some handbooks available online but could not find any PDE similar to mine.  

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In that case, I would look at a series solution i.e. $f(\eta , R) = \sum_{m,n} a_n(\eta)b_m(R)$, where $b_m(R) = \sum_k c_{m,k}\cdot R^k$ and $a_n(\eta)=\sum_j d_{n,j} \eta^j$, where $c_{m,k}$ and $d_{n,j}$ are some coefficients.

 

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@MathematicalPhysicist I went with the series method and used Mathlab's algebraic solver to solve the series multiplication until reaching the $p=n+m=4$ power degree, and it's taking forever to solve. Moreover, it's immensely complicated. Is there any other method to approach this problem?

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Not that I am aware of, perhaps there's some symmetry here that I don't see, sorry.

There are some books that deal with symmetry groups and their uses in PDE and ODE, but I haven't yet read any of them.

I can recommend the books that I purchased and are in my shelf that you can try to consult:

Olver's "Applications of Lie Groups to Differential Equations".

Bluman's "Applications of Symmetry Methods to PDEs".

Maybe the answer lies there.