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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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J.A.
de Jong
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ASCEE, Máximastraat 1, 7442 NW Nijverdal, info@ascee.nl
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\begin_layout Section
Differential equation
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The Stokes layer of the velocity satisfies the following differential equation:
\begin_inset Formula
\begin{equation}
\frac{\partial\hat{u}}{\partial t}-\frac{\mu}{\rho_{0}}\nabla_{\mathrm{T}}^{2}\hat{u}=\hat{K}(t),\label{eq:diffeq}
\end{equation}
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where
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\begin_layout Itemize
\begin_inset Formula $t$
\end_inset
is time,
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\begin_layout Itemize
\begin_inset Formula $\nabla_{\mathrm{T}}^{2}$
\end_inset
is the Laplacian in
\emph on
transverse
\emph default
direction.
\end_layout
\begin_layout Itemize
\begin_inset Formula $\hat{u}$
\end_inset
is the
\emph on
axial
\emph default
velocity,
\end_layout
\begin_layout Itemize
\begin_inset Formula $\hat{K}(t)$
\end_inset
is the forcing function as a function of time, which in acoustics equals
minus the gradient of the pressure,
\end_layout
\begin_layout Itemize
\begin_inset Formula $\mu$
\end_inset
is the dynamic viscosity,
\end_layout
\begin_layout Itemize
\begin_inset Formula $\rho_{0}$
\end_inset
the density.
\end_layout
\begin_layout Section
Harmonic solution
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\begin_layout Standard
If we assume harmonic motion, we may write
\begin_inset Formula
\begin{equation}
\hat{u}=\Re\left(ue^{i\omega t}\right)\quad;\quad K=\Re\left(\hat{K}e^{i\omega t}\right)\quad\mathrm{etc}
\end{equation}
\end_inset
where
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\begin_layout Itemize
\begin_inset Formula $\Re$
\end_inset
is the operator taking the real part of its argument,
\end_layout
\begin_layout Itemize
\begin_inset Formula $i=\sqrt{-1}$
\end_inset
,
\end_layout
\begin_layout Itemize
\begin_inset Formula $\omega$
\end_inset
is the frequency in rad/s.
\end_layout
\begin_layout Standard
For flow close to a plate and unbounded in the
\begin_inset Formula $+y$
\end_inset
-direction, the solution for
\begin_inset Formula $u$
\end_inset
yields:
\begin_inset Formula
\begin{equation}
u=\frac{1}{i\omega\rho_{0}}K\left(1-\exp\left(-\left(1+i\right)y/\delta_{\nu}\right)\right),
\end{equation}
\end_inset
where
\end_layout
\begin_layout Itemize
\begin_inset Formula $\delta_{\nu}=\sqrt{\frac{2\mu}{\rho_{0}\omega}}$
\end_inset
\end_layout
\begin_layout Standard
For oscillating flow between two parallel plates, separated at distances
\begin_inset Formula $2y_{0}$
\end_inset
, and
\begin_inset Formula $y=0$
\end_inset
at the center between the two plates:
\begin_inset Formula
\begin{equation}
u=\frac{1}{i\omega\rho_{0}}\frac{1-h_{\nu}}{1-f_{\nu}}K,
\end{equation}
\end_inset
where
\end_layout
\begin_layout Itemize
\begin_inset Formula $h_{\nu}=\frac{\cosh\left(\left(1+i\right)y/\delta_{\nu}\right)}{\cosh\left(\left(1+i\right)y_{0}/\delta_{\nu}\right)},$
\end_inset
\end_layout
\begin_layout Itemize
\begin_inset Formula $f_{\nu}=\frac{\tanh\left(\left(1+i\right)y_{0}/\delta_{\nu}\right)}{\left(1+i\right)y_{0}/\delta_{\nu}}$
\end_inset
.
\end_layout
\begin_layout Section
Forward Euler time, central in space
\end_layout
\begin_layout Standard
For
\begin_inset Formula $\nabla_{\mathrm{T}}^{2}\equiv\frac{\partial^{2}}{\partial y^{2}}$
\end_inset
, the forward Euler time, central in space (uniform grid) formulation of
Eq.
\begin_inset space ~
\end_inset
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:diffeq"
\end_inset
is
\begin_inset Formula
\begin{equation}
\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}-\frac{\mu}{\rho_{0}}\frac{u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}}{\Delta y^{2}}=K^{n},
\end{equation}
\end_inset
where upper index
\begin_inset Formula $n$
\end_inset
denotes a discrete time instance and lower index
\begin_inset Formula $i$
\end_inset
denotes a discrete position index.
This is an explicit form for the velocity at the next time index
\begin_inset Formula $n+1$
\end_inset
.
Solving for
\begin_inset Formula $u_{i}^{n+1}$
\end_inset
yields:
\begin_inset Formula
\begin{equation}
u_{i}^{n+1}=u_{i}^{n}+\Delta tK^{n}+\frac{\mu\Delta t}{\rho_{0}\Delta y^{2}}\left(u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}\right)
\end{equation}
\end_inset
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\end_body
\end_document
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