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I have heard that the Neumann boundary condition $$ \frac{\partial p}{\partial n} = 0, $$ for the acoustic pressure field in the Helmholtz equation in acoustic wave is related to flux. But we normally associate flux with motion, i.e. velocity, which is not present in the above boundary condition. So why does the Neumann boundary condition represent zero flux at the boundary? What is the relationship between the pressure and displacement/velocity fields that somehow makes this a no flux condition?

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    $\begingroup$ I am not very sure, but perhaps the reason why it is called no flux boundary condition is that since there is no pressure gradient normal to boundary there can be no flow in that direction. $\endgroup$ – Deep Jul 24 '17 at 5:19
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What is the relationship between the pressure and displacement/velocity fields that somehow makes this a no flux condition?

The Neumann boundary condition is jsut a condition/constraint placed on the gradients of some parameter, $Q$, normal to the boundary surface, or: $$ \mathbf{n} \cdot \nabla Q = f\left(\mathbf{r},t\right) \tag{1} $$ where $\mathbf{n}$ is the outward unit normal vector to the surface boundary and $f\left(\mathbf{r},t\right)$ is some known/given scalar function of position and/or time.

In the specific example you show, there is no pressure gradient along the outward unit normal vector. From the Euler equations, we know that: $$ \rho \left( \partial_{t} \ \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = - \nabla \ P + \mathbf{F}_{ext} \tag{2} $$ where $\rho$ is the mass density of the fluid, $\mathbf{u}$ is the fluid element velocity, $P$ is the scalar pressure, $\mathbf{F}_{ext}$ is some external force (usually assumed to be gravity), and $\partial_{j}$ is just the partial derivative with respect to parameter $j$. In the absence of gravity or an external force and spherical symmetry, then Equations 1 and 2 show that: $$ \left( \partial_{t} \ \mathbf{u} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = 0 \tag{3} $$

We can further reduce this using the continuity equation which is given by: $$ \partial_{t} \ \rho + \nabla \cdot \left( \rho \ \mathbf{u} \right) = 0 \tag{4} $$ and a steady state assumption to find that $\nabla \cdot \mathbf{u} = 0$. Generally a divergenceless velocity is interpreted as an incompressible flow but in a 1D spherically symmetric system (i.e., only radial direction matters) this also corresponds to no flow.

So why does the Neumann boundary condition represent zero flux at the boundary?

In a generic dimensional analysis, a flux is just a density multiplied by a velocity. This is often shown in various forms of the continuity equation (e.g., see Equation 4 above for fluid flow), where the first term is the time rate of change of a density and the second is the divergence of a flux. Pressure is a type of momentum flux. Thus, the condition that $\mathbf{n} \cdot \nabla P = 0$ means there is no change in momentum flux along the outward unit normal of the boundary. The general form of pressure is a rank-2 tensor, not a scalar. It reduces to a scalar when the system is symmetric and one-dimensional.

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