#LyX 2.3 created this file. For more info see http://www.lyx.org/ \lyxformat 544 \begin_document \begin_header \save_transient_properties true \origin unavailable \textclass article \begin_preamble \input{tex/preamble_article.tex} \end_preamble \options twoside,final \use_default_options true \maintain_unincluded_children false \language american \language_package babel \inputencoding utf8 \fontencoding global \font_roman "libertine" "Linux Libertine" \font_sans "default" "Courier New" \font_typewriter "default" "default" \font_math "libertine-ntxm" "auto" \font_default_family default \use_non_tex_fonts true \font_sc false \font_osf false \font_sf_scale 100 100 \font_tt_scale 100 100 \use_microtype false \use_dash_ligatures false \graphics default \default_output_format default \output_sync 1 \output_sync_macro "\synctex=1" \bibtex_command default \index_command default \paperfontsize 10 \spacing single \use_hyperref true \pdf_author "Dr.ir. J.A. de Jong - ASCEE" \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks true \pdf_pdfborder true \pdf_colorlinks true \pdf_backref false \pdf_pdfusetitle true \papersize a4paper \use_geometry true \use_package amsmath 1 \use_package amssymb 1 \use_package cancel 1 \use_package esint 1 \use_package mathdots 1 \use_package mathtools 1 \use_package mhchem 1 \use_package stackrel 1 \use_package stmaryrd 1 \use_package undertilde 1 \cite_engine basic \cite_engine_type default \biblio_style plain \use_bibtopic false \use_indices false \paperorientation portrait \suppress_date false \justification true \use_refstyle 0 \use_minted 0 \index Index \shortcut idx \color #008000 \end_index \leftmargin 3cm \topmargin 3cm \rightmargin 2.5cm \bottommargin 3.5cm \headsep 1cm \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \paragraph_indentation default \is_math_indent 0 \math_numbering_side default \quotes_style english \dynamic_quotes 0 \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \html_math_output 0 \html_css_as_file 0 \html_be_strict false \end_header \begin_body \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash newcommand{ \backslash articletitle}{Numerical solution of the (Stokes) viscous boundary layer} \end_layout \begin_layout Plain Layout \backslash date{ \backslash today} \end_layout \begin_layout Plain Layout % \backslash begin{center} \end_layout \begin_layout Plain Layout % \backslash includegraphics{/home/anne/bin/doctemplates/%ascee_beeldmerk_withacr.eps} \end_layout \begin_layout Plain Layout % \backslash end{center} \end_layout \end_inset \end_layout \begin_layout Title \series bold \size giant \begin_inset ERT status open \begin_layout Plain Layout \backslash articletitle \end_layout \end_inset \begin_inset Newline newline \end_inset \size large \begin_inset Note Note status open \begin_layout Plain Layout \series bold \size giant Subtitle \end_layout \end_inset \end_layout \begin_layout Author \series bold J.A. de Jong \begin_inset Formula $^{1}$ \end_inset \end_layout \begin_layout Standard \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Standard \align center \begin_inset Graphics filename /home/anne/bin/doctemplates/ascee_beeldmerk_withacr.eps width 45text% \end_inset \end_layout \begin_layout Standard \align center \size small \begin_inset Formula $^{1}$ \end_inset ASCEE, Máximastraat 1, 7442 NW Nijverdal, info@ascee.nl \end_layout \begin_layout Standard \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Standard \align left \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash arrayrulecolor{asceelightblue} \end_layout \begin_layout Plain Layout \backslash midrule[2pt] \end_layout \end_inset Internal document ID: \end_layout \end_inset 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\begin_layout Plain Layout External document ID: \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Document status: \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Draft \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Note Note status open \begin_layout Plain Layout \lang dutch Revisiehistorie: \end_layout \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Note Note status open \begin_layout Plain Layout \lang dutch \begin_inset ERT status open \begin_layout Plain Layout \backslash today \end_layout \end_inset : rev. 1 \begin_inset Newline newline \end_inset \begin_inset ERT status open \begin_layout Plain Layout \backslash fourteendaysahead \end_layout \end_inset : rev. 2 \end_layout \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset ERT status collapsed \begin_layout Plain Layout \backslash \backslash \end_layout \begin_layout Plain Layout \backslash bottomrule[2pt]% \end_layout \end_inset \end_layout \end_inset \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset toc LatexCommand tableofcontents \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash thispagestyle{empty} \end_layout \begin_layout Plain Layout % Optionally: set this document to confidential \end_layout \begin_layout Plain Layout % \backslash confidential \end_layout \end_inset \end_layout \begin_layout Section Differential equation \end_layout \begin_layout Standard 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} \end_inset where \end_layout \begin_layout Itemize \begin_inset Formula $t$ \end_inset is time, \end_layout \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 \end_layout \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 \end_layout \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 \end_layout \end_body \end_document