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I'm working on a question here where I have solved these boundary conditions;

$$\vec{n} \cdot \vec{j}= -\lambda\vec{n} \cdot \nabla u + k(\rho - \rho_0)u\vec{n}\cdot\vec{g}=0$$

So my question is, how do I know which of these 2 terms is a Dirichlet/Neumann boundary condition?

Edit:

$\rho$ - density of substance, $\rho_0$ - density of the medium, $\vec{g}$ - gravitational field, $u$ - concentration, $k$ - constant.

Question: Find the boundary conditions which must be fulfilled in order for no substance to leave or enter the volume V .

What I did to solve this was I calculated $\vec{n} \cdot \vec{j}=0$ (since no substance can leave the volume, this condition has to be true), where $\vec{j}=-\lambda\nabla u+ku\vec{g}(\rho-\rho_0)$.

This finally gave me the expression that I first posted here (at the top). Now my question is, with the information that is given, how do I know if I have a Neumann Boundary condition or a Dirichlet Boundary condition?

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  • $\begingroup$ As it stands now, it's just a bunch of symbols with some addition, subtraction and equality operators. It doesn't even mean anything. So please do provide some context. $\endgroup$ – Ruslan May 31 '17 at 16:33
  • $\begingroup$ Thank you, I have edited my post. If there is anything else that is unclear please feel free to ask, I really want to understand this kind of problem where I'm supposed to know what kind of boundary condition I'm working with, and why. $\endgroup$ – armara May 31 '17 at 16:59
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If we rewrite your equation

$$-\lambda\vec{n} \cdot \nabla u + k(\rho - \rho_0)u\vec{n}\cdot\vec{g}=0$$

as

$$-\lambda\frac{\partial u}{\partial\vec n} + \left(k(\rho - \rho_0)\vec{n}\cdot\vec{g}\right)u=0,$$

where $\frac{\partial u}{\partial\vec n}$ is the normal derivative, then it's immediately apparent that the boundary condition expressed here is the third type boundary condition, also known as Robin boundary condition, which is a combination of Dirichlet boundary condition (first type) and Neumann one (second type).

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  • $\begingroup$ Hmm, can you really write $n \nabla u = \frac{\partial u}{\partial \vec{n}}$? I thought the gradient was a derivation with respect to the spatial coordinates (x,y,z). $\endgroup$ – armara May 31 '17 at 19:49
  • $\begingroup$ Also, how do you recognize it as a third type boundary condition? I thought that the part that has a derivative was the Neumann-condition, and the part without a derivative was a Dirichlet-condition. $\endgroup$ – armara May 31 '17 at 19:52
  • $\begingroup$ @armara for the first question please follow the link about normal derivative, for the second one note that there's both a derivative and a value of $u$ in the equation, so it's neither first-, nor second-type boundary condition — but a weighted combination thereof. $\endgroup$ – Ruslan Jun 1 '17 at 4:41

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