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
ρ(ψy∇2ψx−ψx∇2ψy)=μ∇4ψ
I have made use of the self-similar transformation η=yx with f(η,R)=ψUx 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 2ηffη instead of 2ηffηη in the R[...] terms)
[(1+η2)2fηηηη+8(1+η2)fηηη+4(1+3η2)fηη]
+[2ηffηη+(1+η2)(ffηη+fηfηη)−4(1+3η2)fηηR−4η(1+η2)fηηηR]R
+[2η(fRfηη−ffηηR)−(1+η2)(fηfηηR−fRfηηη)+2(1+3η2)fηηRR]R2
+[2η(fηfηRR−fRfηηR)+ffηRR−3fηfRR+4fRRR−4ηfηRRR]R3
+[fRRRR+fRfηRR−fηfRRR]R4=0
Boundary conditions: At the plate, η=0 and we have fη=0 & f+fRR=0. At the leading edge, η→∞ and we have fη→1 & f+fRR→η.
Since this problem is at the leading edge, we consider Reynolds numbers of the order 10−3 and hence, we can neglect all O(R2) 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=fηη, we obtain the following second-order ODE
(1+η2)2uηη+8η(1+η2)uη+4(1+3η2)u=0,
to solve the PDE but I require a solution in η 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
R2(TRR−PrU(fηfηR−fRfηη))+R(PrU−4ηTη)+2ηTη+(1+η2)Tηη=0,
where Pr is the Prandtl number and T=T(η,R). Setting Pr≡1, we can now define a second-order operator L as follows
L:=1R2(1+η2)∂2ηη+2η(1−2R)∂η,
such that
L[T]+TRR=H in VR,
where H=−U(fηfηR−fRfηη). Thus, with boundary conditions, we define the following IBVP
L[T]+TRR=H in VR
T=T∞ on ∂V×[0,R]
T=h, TR=g on V×{R=0}
where V is an open set of R2, VR=V×(0,R] for a fixed Reynold’s number R>0, H:VR→R and h,g:V→R are given and T:ˉVR→R is unknown. Solving this PDE gives T(η,R) and later gives h(x) in terms of fη,fηR,fR,fηη. I did try to solve the momentum PDE neglecting the terms of R2 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/