Skip to main content
Engineering LibreTexts

7.5: Summary and further problems

  • Page ID
    24125
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    In large, complex systems, the information is either overwhelming or not available. Then we have to reason with incomplete information. The tool forthis purpose is probabilistic reasoning—in particular, Bayesian probability. Probabilistic reasoning helps us manage incomplete information. Using it, we can estimate the uncertainty in our divide-and-conquer estimates and understand the physics of random walks and thereby viscosity, boundary layers, and heat flow.

    Exercise \(\PageIndex{1}\): Reynolds number as a ratio of two times

    For an object moving through a fluid, the Reynolds number is defined as \(vL/\nu\), where \(v\) is the object’s speed, L is its size (a length), and \(\nu\) is the fluid’s kinematic viscosity. Show that the Reynolds number has the physical interpretation

    \[\frac{\textrm{momentum-diffusion time over a distance comparable to the size } L}{\textrm{fluid-transport time over a distance comparable to the size } L}.\]

    Exercise \(\PageIndex{2}\): Diamond is special

    Diamond has a high thermal conductivity, much higher even than many metals. The speed of sound in diamond is 12 kilometers per second, and diamond’s specific heat cp is 0.63 kilojoules per kilogram kelvin. Use those values to estimate the mean free path of phonons in diamond, as an absolute length and in units of typical interatomic spacings. How does the mean free path in diamond compare to a typical phonon mean free path of a few lattice spacings?

    Exercise \(\PageIndex{3}\): Baking in three dimensions

    Extend the fish-cooking argument of Section 7.3.3.2 to three dimensions to predict the baking time of a 6-kilogram turkey (assumed to be a sphere). How well does the time agree with experience (for example, with the data given in Problem 7.22)?

    Exercise \(\PageIndex{4}\): Resistive networks to analyze random walks

    Random walks are closely connected to infinite resistive networks (this connection is explored deeply in Random Walks and Electric Networks [11]). In particular, the probability of escape pesc—the probability that an n-dimensional random walker escapes to infinity and never returns to the origin—is related to the resistance R to infinity of a n-dimensional electrical network of unit resistors: \(p_{esc} = 1/2 nR\). Use this connection, along with lumping arguments, to estimate R and thereby show that the two-dimensional random walk is recurrent (pesc = 0) but that the three-dimensional walk is transient (pesc > 0)—consistent with Pólya’s theorem (Problem 7.17).

    Exercise \(\PageIndex{5}\): Turning differential equations into algebraic equations

    The cold days of winter arrive, and the ice on a lake starts thickening as heat flows upward through the ice, turning ever more water into ice. Find the scaling exponent \(\beta\) in

    \[\textrm{ice thickness} \propto (\textrm{time})^{\beta}.\]

    clipboard_e1adb0ca9365e1eb5cf22518dd5eec4f2.png

    Exercise \(\PageIndex{6}\): Thermal and electrical conductivities

    Among the metals, the better thermal conductors, such as copper and gold in comparison to aluminum, iron, or mercury, are also the better electrical conductors. (This connection is quantified in the Wiedemann-–Franz law.) What is the reason for this connection?

    Exercise \(\PageIndex{7}\): Teacup spindown

    You stir your afternoon tea to mix the milk (and sugar, if you have a sweet tooth). Once you remove the stirring spoon, the rotation begins to slow. In this problem you’ll estimate the spindown time \(\tau\): the time for the angular velocity of the tea to fall by a significant fraction. To estimate \(\tau\), consider a lumped teacup: a cylinder with height l and diameter l, filled with liquid. Tea near the edge of the teacup—and near the base, but for simplicity neglect the effect of the base—is slowed by the presence of the edge (a result of the no-slip boundary condition).

    clipboard_e733448dc8a820c557625532f93dbdb93.png

    a. In terms of the viscous torque , Tthe initial angular velocity \(\omega\), and \(\rho\) and l, estimate the spindown time \(\tau\). Hint: Consider angular momentum, and drop all dimensionless constants, such as \(\pi\) and 2.

    b. To estimate the viscous torque T, use the result of Problem 7.25:

    \[\textrm{viscous force} = \rho \nu \times \textrm{velocity gradient} \times \textrm{surface area}.\]

    The velocity gradient is determined by the boundary-layer thickness \(\delta\). In terms of \(\delta\), estimate the velocity gradient near the edge and then the torque T.

    c. Put your expression for T into your earlier estimate for \(\tau\), which should now contain only one quantity that you have not yet estimated (the boundary-layer thickness \(\delta\)).

    d. Estimate \(\delta\) in terms of a growth time t, which is the time to rotate 1 radian. After 1 radian, the fluid is moving in a significantly different direction, so the momentum fluxes from different regions no longer add constructively to the growth of the boundary layer.

    e. Put the preceding results together to estimate the spindown time \(\tau\): symbolically in terms of \(\nu\), \(\rho\), \(l\), and \(\omega\); and then numerically.

    f. Stir your tea and estimate \(\tau\) experimentally, and compare with your prediction. Then enjoy a well-deserved cup.


    This page titled 7.5: Summary and further problems is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Sanjoy Mahajan (MIT OpenCourseWare) .

    • Was this article helpful?