Ten QuestionsRRaghavanNov em ber 12, 2014A fool can ask more questions in a minute than the wise can answer in alifetime. Proverb (somewhere)There are some questions here more or less loosely related to what I talkabout. Not all of the questions are sane.1. Read http://geometeer.com/geometeerignorance.html. What do you think?2. Consider a plane wave propagating in the positive region. This is asolution of the wave equation which is a certain limit of the equations offluid dynamics Suppose that the velocity of the fluid is = cos ( − ) (1)the velocity of the wave being = . For example, this could begenerated by an oscillating plate at =0. Now this means (and this istrue) that if there is a particle at the position at time , i.e.,giventhat () is the −coordinate of the particle we have ()=,thenitsvelocity is 0()= of the abo ve equation. Suppose this particle startedoff at =0.(i) Integrate0()= cos ( () − ) (2)with the initial condition (0) = 0.(Hint:first use units such thatposition is measured in units of 1 and times in units of 1 and furtheruse the independent variable ()= () − .)(ii) How does the position behave as a function of ?(iii) Hopefully, your answer is that the position oscillates but keeps in-creasing linearly with over each period. So the fluid particles next tothe vibrating wall vacate from there and move steadily to the right, leavinga vacuum at the wall. What happened?3. Look at the paper http://www.ncbi.nlm.nih.gov/pubmed/21241713 whichdeals with perivascular flow due to the pulsations of an artery, and pro-vides a peristaltic mechanism for it. Note that they assume that theperivascular space is bounded between the pulsating artery and a fixedwall. Assume that the wall also pulsates in phase and amplitude the1same as the artery. When this happens, show that the term independentof the mean pressure drop over a wave length in their equation (12) is zero.4. A plane wave of pressure in( − ) obeying the w av e equation prop-agates from infinitely far to the left to hit an elastic layer beginning at =0and extending to = after which there is an immovable rigidwall. If the pressure at the left edge of the layer is (), then assume thedeformation of the layer is proportional to that, i.e.,itis ()= ()where is a (dimensional) constant.(a) Find the equation determining () as a function of the function inwhich is known, as opposed to which is unkno wn.(b) Find the expression for () in terms of inand ().Hin t : Integrate the Euler equation that relates the vibrational velocity of the layer to the pressure ,= −(3)and that = in( − )+ref( + ) (4)together with ( =0)=0() (5)5. It has been observed in brain tumor and other patients, that the shapeof the pressure—edema curve is roughly shaped like a reversed L. Edemameans excess water in the brain, and thepressureispressuremeasuredin the cerebrospinal fluid. In other words, the pressure increases v erylittle as edema increases until it reaches a certain considerable level, andthen the pressure increases v e ry rapidly so that further edema will quicklyresult in morbidity and death. Headaches are a sign that the “knee” inthis curve has been passed.Usual stress—strain relations in elasticity are linear: the pressure increaseslinearly with expansion. In vent an explanation that explains the pressure—edema relation, and argue why it might pertain in the brain. Can youthink of something at Party City with a somewhat similar behavior?6. Isaac Newton proved that a spherically symmetric mass has a gravitationalforce outside the body exactly as if all its mass was concentrated at thecenter.Imagine: intraparenchymal drug delivery is now a $100 billion dollar busi-ness. Unfortunately, the catheters and needles that deliver these drugsdo not distribute the drugs very far at all. You run into this guy youknew in high school. He has this great idea for sending the infusate farfrom the catheter. Remember that garden hose in your backyard? hesays. When you squeeze the hose you shoot the water off much farther.2Well what we do is narrow the catheter tip by a factor of 10, and I hav ethe technology to do that. For a small investment of $10K, he promisesyou a 100 fold return. You still infuse the fluid at the same low flow rate,but because the flow is going through a much smaller port, it will shootout at a much higher ve locity and whoosh, it will go farther. (The flowrates involved are a few microliters per minute.)(a) What does physics and the physics of tissue say about this idea?(b) Was my remark about Newton’s proof a non sequitur ?Epilog: Your high school buddy is now Chief Engineer at Johnson andJohnson Medical Devices. You are unemployed and living at home withmom and dad. You missed your chance.7. Look at an illustration or image of cerebral vasculature. You can seethat the surface and ev en deeper vasculature follow the sulci (grooves) inthe brain. Do y ou think the vasculature follows the grooves during braindevelopment or are the grooves forced by the vasculature as the brainforms? (Don’t look this up.) Argue for your hypothesis.8. Francis Crick (shame on you if you don t know who he was) and anotherscientist named Christof Koch proposed that the claustrum, a structurein our brain was central to integrating information and to obtaining inte-grated conscious percepts, see h ttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC1569501/.Can you propose an experimental procedure to disprove their silly, I meantto say: test their bold hypothesis?9. Giulio Tononi is a sleep specialist who has made waves because he claimshe knows how to measure consciousness. (If you google Tononi and con-sciousness you will get more references than you should read in a lifetime.)Roughly speaking his measure is the integrative ability of consciousness –given ho w much raw information is presented, what is the relative entropy(whic h is a precise concept in information theory also kno wn as the Kull-back Leibler measure) of the integrated information that a consciousnessmakesofitrelativetotheraw. Themoreconscioustheentity,thehigherthe relative entropy. Withouth knowing much more, try to produce acoun terexample that would help show
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