Commit efec80a1 authored by Olivier's avatar Olivier

Exercises 6: updated formula sheet, fixed typos in problem 1

parent 15b70349
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\atstartofexercises
\fluidmechexercisestitle
......@@ -10,15 +10,15 @@
\mecafluexboxen
\begin{boiboite}
Continuity equation:
\begin{equation}
\frac{1}{\rho} \totaltimederivative{\rho} + \divergent{\vec V} = 0 \tag{\ref{eq_continuity_der_alt}}
\end{equation}
Continuity equation for incompressible flow:
\begin{IEEEeqnarray}{rCl}
\divergent{\vec V} & = & 0\ztag{\ref{eq_continuity_der_inc}}
\end{IEEEeqnarray}
Navier-Stokes equation for incompressible flow:
\begin{equation}
\rho \totaltimederivative{\vec V} = \rho \vec g - \gradient{p} + \mu \laplacian{\vec V} \tag{\ref{eq_navierstokes}}
\end{equation}
\begin{IEEEeqnarray}{rCl}
\rho \totaltimederivative{\vec V} &=& \rho \vec g - \gradient{p} + \mu \laplacian{\vec V} \ztag{\ref{eq_navierstokes}}
\end{IEEEeqnarray}
\end{boiboite}
......@@ -26,7 +26,7 @@ Navier-Stokes equation for incompressible flow:
\label{exo_ns_revision}
%homemade
For the continuity equation (eq.~\ref{eq_continuity_der_alt}), and then for the incompressible Navier-Stokes equation (eq.~\ref{eq_navierstokes}),
For the continuity equation (eq.~\ref{eq_continuity_der_inc}), and then for the incompressible Navier-Stokes equation (eq.~\ref{eq_navierstokes}),
\begin{enumerate}
\item Write out the equation in its fully-developed form in three Cartesian coordinates;
\item State in which flow conditions the equation applies.
......@@ -35,8 +35,8 @@ Navier-Stokes equation for incompressible flow:
Also, in order to revise the notion of total (or substantial) derivative:
\begin{enumerate}
\shift{2}
\item Describe a situation in which the total time derivative $\text{D}/\text{D}t = 0$ of a property is non-zero, even though the flow is entirely steady ($\partial{}/\partial{t} \neq 0$).
\item Describe a situation in which the the flow is unsteady, although some property of the fluid, when measured from the point of view of the particle, is not changing with time.
\item Describe a situation in which the total time derivative $\text{D}/\text{D}t$ of a property is non-zero, even though the flow is entirely completely ($\partial{}/\partial{t} = 0$).
\item Describe a situation in which the the flow is unsteady, although some property of the fluid, when measured from the point of view of a fluid particle, is not changing with time.
\end{enumerate}
......@@ -131,7 +131,7 @@ Navier-Stokes equation for incompressible flow:
\NumTabs{2}
\begin{description}
\item [\ref{exo_ns_revision}]%
\tab 1) For continuity, use eqs.~\ref{eq_totaltimederivativeheavy} and~\ref{eq_def_gradient} in~equation~\ref{eq_continuity_der_alt}. For Navier-Stokes, see eqs.~\ref{eq_ns_cartone}, \ref{eq_ns_carttwo} and~\ref{eq_ns_cartthree} p.~\pageref{eq_ns_cartone};
\tab 1) Continuity: eq.~\ref{eq_continuity_der_inc_dev} p.\pageref{eq_continuity_der_inc_dev}. Navier-Stokes: see eqs.~\ref{eq_ns_cartone}, \ref{eq_ns_carttwo} and~\ref{eq_ns_cartthree} p.~\pageref{eq_ns_cartone};
\tab 2) Read \S\ref{ch_continuity_der} p.~\pageref{ch_continuity_der} for continuity, and \S\ref{ch_navier-stokes} p.\pageref{ch_navier-stokes} for Navier-Stokes;
\tab 3) and 4) see \S\ref{ch_substantial_derivative} p.~\pageref{ch_substantial_derivative}.
\item [\ref{exo_acceleration_field}]%
......@@ -139,20 +139,20 @@ Navier-Stokes equation for incompressible flow:
\item [\ref{exo_volumetric_dilatation_rate}]%
\tab $\divergent{\vec V} = -x^2 + x - z$; thus at the probe it takes the value $\left(\divergent{\vec V}\right)_\text{probe} = \SI{-4}{\per\second}$.
\item [\ref{exo_incompressiblity}]%
\tab Apply equation~\ref{eq_conti_inc_twod} p.\pageref{eq_conti_inc_twod} to $\vec V$: the answer is yes.
\tab Apply equation~\ref{eq_continuity_der_inc_dev} p.\pageref{eq_continuity_der_inc_dev} to $\vec V$: the answer is yes.
\item [\ref{exo_missing_components}]%
\tab 1) Applying equation~\ref{eq_conti_inc_twod}: $w_1 = \num{-3}xz - \frac{1}{2}z^2 + f_{(x,y,t)}$;
\tab 1) Applying equation~\ref{eq_continuity_der_inc_dev}: $w_1 = \num{-3}xz - \frac{1}{2}z^2 + f_{(x,y,t)}$;
\tab 2) idem, $v_2 = \num{-3}axy - bzy^2 + f_{(x,z,t)}$.
\item [\ref{exo_another_acceleration_field}]%
\tab $\totaltimederivative{\vec V} = (3)\vec i + (3z + y^2x)t\vec j + (y^2 + 2xyzt) \vec k$. At the probe it takes the value $\num{3}\vec{i} + \num{250}\vec{j} + \num{496}\vec{k}$.
\item [\ref{exo_vortex_continuity}]%
\tab Apply equation~\ref{eq_conti_inc_twod} to $\vec V$ to verify incompressibility.
\item [\ref{exo_pressure_fields}]%
\tab Note: the constant (initial) value~$p_0$ is sometimes implicitly written in the unknown functions $f$.
\tab\tab 1) $p = -\rho \left[abx + \frac{1}{2}a^2 x^2 + bcy + \frac{1}{2}a^2 y^2 \right] + p_0 + f_{(t)}$;
\tab 2) $p = -\rho \left(8 x^2 + 8 y^2\right) + p_0 + f_{(t)}$;
\tab Note: the constant (initial) value~$p_{ini}$ is sometimes implicitly written in the unknown functions $f$.
\tab\tab 1) $p = -\rho \left[abx + \frac{1}{2}a^2 x^2 + bcy + \frac{1}{2}a^2 y^2 \right] + p_{ini} + f_{(t)}$;
\tab 2) $p = -\rho \left(8 x^2 + 8 y^2\right) + p_{ini} + f_{(t)}$;
\tab 3) $\partialderivative{}{x}\left(\partialderivative{p}{y}\right) \neq \partialderivative{}{y}\left(\partialderivative{p}{x}\right)$, thus we cannot describe the pressure with a mathematical function;
\tab 4) $p = -\rho \left[\frac{U_0^2}{L} \left(x + \frac{x^2}{2L} + \frac{y^2}{2L}\right) - g_x x - g_y y \right] + p_0 + f_{(t)}$.
\tab 4) $p = -\rho \left[\frac{U_0^2}{L} \left(x + \frac{x^2}{2L} + \frac{y^2}{2L}\right) - g_x x - g_y y \right] + p_{ini} + f_{(t)}$.
\end{description}
\atendofexercises
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