91Ó°ÊÓ

For conduction of thermal energy, the heat flux vector is \(\phi=-K_{0} \nabla u\). If in addition the molecules move at an average velocity \(\mathbf{V}\), a process called convection, then briefly explain why \(\boldsymbol{\phi}=-\boldsymbol{K}_{0} \nabla \boldsymbol{\nabla}+c \rho u \mathbf{V}\). Derive the corresponding equation for heat flow, including both conduction and convection of thermal energy (assuming constant thermal properties with no sources).

Consider the equilibrium temperature distribution for a uniform one- dimensional rod with sources \(Q / K_{0}=x\) of thermal energy, subject to the boundary conditions \(u(0)=0\) and \(u(L)=0\). (a) Determine the heat energy generated per unit time inside the entire rod. (b) Determine the heat energy flowing out of the rod per unit time at \(x=0\) and at \(x=L\). (c) What relationships should exist between the answers in parts (a) and (b)?

Consider the polar coordinates $$ \begin{aligned} &x=r \cos \theta \\ &y=r \sin \theta \end{aligned} $$ (a) Since \(r^{2}=x^{2}+y^{2}\), show that $$ \frac{\partial r}{\partial x}=\cos \theta, \quad \frac{\partial r}{\partial y}=\sin \theta, \quad \frac{\partial \theta}{\partial y}=\frac{\cos \theta}{r}, \quad \text { and } \quad \frac{\partial \theta}{\partial x}=\frac{-\sin \theta}{r} $$ (h) Show that \(\hat{\mathbf{r}}=\cos \theta \hat{\mathbf{i}}+\sin \theta \hat{\mathbf{j}} \quad\) and \(\quad \hat{\boldsymbol{\theta}}=-\sin \theta \hat{\mathbf{i}}+\cos \theta \hat{\mathbf{j}} .\) (c) Using the chain rule, show that $$ \nabla u=\frac{\partial u}{\partial r} \hat{\mathbf{r}}+\frac{1}{r} \frac{\partial u}{\partial \theta} \widehat{\boldsymbol{\theta}} \text {. } $$ (d) If \(\mathbf{A}=A, \hat{\mathbf{r}}+A_{g} \hat{\boldsymbol{\theta}}\), show that $$ \nabla \cdot \mathrm{A}=\frac{1}{r} \frac{\partial}{\partial r}\left(r A_{r}\right)+\frac{1}{r} \frac{\partial}{\partial \theta}\left(A_{\theta}\right) $$ since \(\partial \hat{\mathbf{r}} / \partial \theta=\hat{\boldsymbol{\theta}}\) and \(\partial \hat{\boldsymbol{\theta}} / \partial \theta=-\hat{\mathbf{r}}\) follows from part (b). (e) Show that $$ \nabla^{2} u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}} $$

If both ends of a rod are insulated, derive from the partial differential equation that the total thermal energy in the rod is constant.

Consider a thin one-dimensional rod without sources of thermal energy whose lateral surface area is not insulated. (a) Assume that the heat energy flowing out of the lateral sides per unit surface area per unit time is \(w(x, t)\). Derive the partial differential equation for the temperaturc \(u(x, t)\). (b) Assume that \(w(x, t)\) is proportional to the temperature difference between the rod \(u(x, t)\) and a known outside temperature \(\gamma(x, t)\). Derive that $$ c \rho \frac{\partial u}{\partial t}=\frac{\partial}{\partial x}\left(K_{0} \frac{\partial u}{\partial x}\right)-\frac{P}{A}[u(x, t)-\gamma(x, t)] h(x) $$ where \(h(x)\) is a positive \(x\)-dependent proportionality, \(P\) is the lateral perimeter, and \(A\) is the cross-sectional area. (c) Compare (1.2.11) to the equation for a one-dimensional rod whose lateral surfaces are insulated, but with heat sources. (d) Specialize (1.2.11) to a rod of circular cross section with constant thermal properties and \(0^{\circ}\) outside temperature. (e) Consider the assumptions in part (d). Suppose that the temperature in the rod is uniform [i.e., \(u(x, t)=u(t)\) ]. Determine \(u(t)\) if initially \(u(0)=u_{0} .\)

Consider a one-dimensional rod \(0 \leqslant x \leqslant L\) of known length and known constant thermal properties without sources. Suppose that the temperature is an unknown constant \(T\) at \(x=L\). Determine \(T\) if we know (in the steady state) both the temperature and the heat flow at \(x=0\).

Derive the heat equation in two dimensions by using Green's theorem, (1.5.16), the two-dimensional form of the divergence theorem.

If Laplace's equation is satisfied in three dimensions, show that $$ \oiint \nabla u \cdot \hat{\mathbf{n}} d S=0 $$ for any closed surface. (Hint: Use the divergence theorem.) Give a physical interpretation of this result (in the context of heat flow).

Determine the equilibrium temperature distribution inside a circular annulus \(\left(r_{1} \leqslant r \leqslant r_{2}\right):\) (a) if the outer radius is at temperature \(T_{2}\) and the inner at \(T_{1}\). (b) if the outer radius is insulated and the inner radius is at temperature \(T_{1}\).