root/trunk/matml/transport/pedagogy/TMS2004ACP/equations.tex

\documentclass{article}
\usepackage{pstricks}
\begin{document}

Conservation example: heatbal.png, heatcons.png, heatgen.png
$$V\cdot\frac{\partial H}{\partial t} =
A\cdot q_x|_x - A\cdot q_x|_{x+\Delta x} + V\cdot\dot{q}$$
$$\frac{\partial H}{\partial t} = -\frac{\partial q_x}{\partial x}
+ \dot{q}$$
$$\rho c_p\frac{\partial T}{\partial t} =
\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right)
+ \dot{q}$$

Diffusion equation and timescale: diffeq.png, difftime.png
$$\frac{\partial C}{\partial t} = D\nabla^2C + G$$
$$L^2/D$$

Heat conduction and timescale: heateq.png, heatime.png
$$\rho c_p\frac{\partial T}{\partial t} = k\nabla^2 T + \dot{q}$$
$$L^2/\alpha$$

Navier-Stokes and timescale: masseq.png, momeq.png, flowtime.png
$$\frac{D\rho}{Dt} + \rho\nabla\cdot\vec{u} = 0$$
$$\rho\frac{D\vec{u}}{Dt} = -\nabla P + \mu\nabla^2\vec{u} + \vec{F}$$
$$L^2/\nu$$

Boundary layers: solid, liquid
\input{blsolid}

\input{blfluid}

Solid boundary layer heat equation, solution, boundary layer for $\delta_T<<x$:
convcond.png, blssolution.png, blsoneper.png
$$u_x\frac{\partial T}{\partial x} = \alpha\frac{\partial^2T}{\partial y^2}$$
$$T = A + B{\rm erf}\left(\frac{y}{2\sqrt{\alpha x/U_\infty}}\right)$$
$$\delta_T = 3.6\sqrt{\alpha x/U_\infty}$$

Fluid boundary layer extent: blfoneper.png
$$\delta_u = 5.0\sqrt{\nu x/U_\infty}$$

Flow down inclined plane and its Reynolds number: incline.png, inclinere.png
$$u_x = \frac{g\sin\theta}{2\nu}(2Lz-z^2)$$
$${\rm Re} = \frac{L}{\nu}\frac{g\sin\theta L^2}{3\nu}$$

Grashof number: grashof.png
$${\rm Gr} = \frac{g\beta\Delta T L^3}{\nu^2}$$

Nusselt number: nusselt.png
$${\rm Nu} = \frac{h L}{k_{fl}}$$

Heat flux: heatflux.png, deltat.png, nuphysical.png
$$q_y = h\Delta T \simeq k_{fl}\frac{\Delta T}{\delta_T}$$
$$\frac{k_{fl}}{h} \simeq \delta_T$$
$${\rm Nu} \simeq \frac{L}{\delta_T}$$
\end{document}