(adsbygoogle = window.adsbygoogle || []).push({}); PROBLEM FORMULATION:

Considering the region [itex] \Omega [/itex] bounded as a square box within [itex] x \in [0,1], y \in [0,1] [/itex]. We wish to solve the 2D, stationary, advection-diffusion equation,

[itex]

0 = D\nabla^2 \rho(x,y) + \vec{V} \cdot \nabla \rho(x,y)

[/itex]

where [itex] D [/itex] is a scalar constant, and [itex] \vec{V} = V_1\hat{e}_x + V_2 \hat{e}_y [/itex] is a constant advection field vector. The problem has the following boundary conditions,

[itex]

\begin{align}

\nabla \rho(1,y) &= \nabla \rho(x,1) =& 0 \\

\rho(0,y) &= \rho(x,0) =& C = \textrm{known constant}

\end{align}

[/itex]

ATTEMPT AT SOLUTION: Method of Separation of Variables

Assuming [itex] \rho(x,y) = X(x) \cdot Y(y) [/itex], then

[itex]

\begin{align}

\nabla \rho &= Y X_{x} \hat{e}_x + X Y_{y} \hat{e}_y \\

\nabla^2 \rho &= Y X_{xx} + X Y_{yy}

\end{align}

[/itex]

where the subscript denotes partial derivation with respect to that particular variable, with the exception of [itex]\hat{e}[/itex] which represents the unit vector in either direction. Decomposing the advection field vector and inserting these results into the original problem, we get..

[itex]

0 = D (Y X_{xx} + X Y_{yy} ) + Y X_{x} v_{x} + X Y_{y} v_{y}

[/itex]

Multiply this eq. with [itex] 1/(XYD) [/itex] and moving all [itex] X [/itex] -expressions to the left hand side result in the separation of [itex] X [/itex] and [itex] Y [/itex] such that they both have to be equal some unknown constant [itex] \lambda [/itex] (we cannot have a small change in [itex] X [/itex], without the corrosponding change in [itex] Y [/itex] and vice versa). Thus,

\begin{align}

\lambda &= -\frac{X_{xx}}{X} - \frac{X_{x}}{X} \frac{V_{1}}{D} \\

\lambda &= +\frac{Y_{yy}}{Y} + \frac{Y_{y}}{Y} \frac{V_{2}}{D}

\end{align}

Which are two independent, linear, second order ODEs, with general solutions

[itex]

\begin{align}

X(x) &= C_1 \exp[-\alpha^{(+)}_{x} x] \\

&+ C_2 \exp[+\alpha^{(-)}_{x} x] \\

& \\

Y(y) &= C_3 \exp[-\alpha^{(+)}_{y} y] \\

&+ C_4 \exp[+\alpha^{(-)}_{y} y]

\end{align}

[/itex]

with

[itex]

\alpha^{(\pm)}_{m} = \frac{1}{2} \left( \sqrt{\frac{v_{m}^2}{D^2} +4\lambda} \pm \frac{v_{m}}{D}\right), \quad m=1 \vee 2

[/itex]

and [itex] C_i, i=1,2,3,4; [/itex] are constant coefficients. So far, so good!! However, here is where my issues begin to pile up as I'm unable to sort out the [itex] C [/itex] -values with the boundary conditions.

Along thenorthandeastboundaries, I'm able to write [itex] C_1 = (\textrm{some const expression}) \cdot C_2 [/itex] and similar for [itex] C_3, C_4 [/itex]. However, for thewestandsouthbounds I end up with

[itex]

(C_1 + C_2)Y = C \\

(C_3 + C_4)X = C

[/itex]

Which will only be valid for [itex]C=0, C_1 = -C_2, C_3=-C_4[/itex]. This, evidently, results in the trivial solution [itex]\rho(x,y)=0[/itex], which obviously is not what we want ..

Any help is appreciated!!!I've also tried to solve it by integral transforms, but due to the stationarity (the problem-equation is homogenious), I fail as [itex] \rho [/itex] vanishes ..

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# 2D Stationary advection-diffusion eq. as a BVP

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