Ten Questions R Raghavan November 12 2014 A fool can ask more questions in a minute than the wise can answer in a lifetime Proverb somewhere There are some questions here more or less loosely related to what I talk about 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 a solution of the wave equation which is a certain limit of the equations of fluid dynamics Suppose that the velocity of the fluid is cos 1 the velocity of the wave being For example this could be generated by an oscillating plate at 0 Now this means and this is true that if there is a particle at the position at time i e given that is the coordinate of the particle we have then its velocity is 0 of the above equation Suppose this particle started o at 0 i Integrate 0 cos 2 with the initial condition 0 0 Hint first use units such that position is measured in units of 1 and times in units of 1 and further use the independent variable ii How does the position behave as a function of iii Hopefully your answer is that the position oscillates but keeps increasing linearly with over each period So the fluid particles next to the vibrating wall vacate from there and move steadily to the right leaving a vacuum at the wall What happened 3 Look at the paper http www ncbi nlm nih gov pubmed 21241713 which deals with perivascular flow due to the pulsations of an artery and provides a peristaltic mechanism for it Note that they assume that the perivascular space is bounded between the pulsating artery and a fixed wall Assume that the wall also pulsates in phase and amplitude the 1 same as the artery When this happens show that the term independent of the mean pressure drop over a wavelength in their equation 12 is zero 4 A plane wave of pressure in obeying the wave equation propagates from infinitely far to the left to hit an elastic layer beginning at 0 and extending to after which there is an immovable rigid wall If the pressure at the left edge of the layer is then assume the deformation of the layer is proportional to that i e it is where is a dimensional constant a Find the equation determining as a function of the function in which is known as opposed to which is unknown b Find the expression for in terms of in and Hint Integrate the Euler equation that relates the vibrational velocity of the layer to the pressure 3 and that in ref 4 0 0 5 together with 5 It has been observed in brain tumor and other patients that the shape of the pressure edema curve is roughly shaped like a reversed L Edema means excess water in the brain and the pressure is pressure measured in the cerebrospinal fluid In other words the pressure increases very little as edema increases until it reaches a certain considerable level and then the pressure increases very rapidly so that further edema will quickly result in morbidity and death Headaches are a sign that the knee in this curve has been passed Usual stress strain relations in elasticity are linear the pressure increases linearly with expansion Invent an explanation that explains the pressure edema relation and argue why it might pertain in the brain Can you think of something at Party City with a somewhat similar behavior 6 Isaac Newton proved that a spherically symmetric mass has a gravitational force outside the body exactly as if all its mass was concentrated at the center Imagine intraparenchymal drug delivery is now a 100 billion dollar business Unfortunately the catheters and needles that deliver these drugs do not distribute the drugs very far at all You run into this guy you knew in high school He has this great idea for sending the infusate far from the catheter Remember that garden hose in your backyard he says When you squeeze the hose you shoot the water o much farther 2 Well what we do is narrow the catheter tip by a factor of 10 and I have the technology to do that For a small investment of 10K he promises you 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 shoot out at a much higher velocity and whoosh it will go farther The flow rates 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 and Johnson Medical Devices You are unemployed and living at home with mom and dad You missed your chance 7 Look at an illustration or image of cerebral vasculature You can see that the surface and even deeper vasculature follow the sulci grooves in the brain Do you think the vasculature follows the grooves during brain development or are the grooves forced by the vasculature as the brain forms Don t look this up Argue for your hypothesis 8 Francis Crick shame on you if you dont know who he was and another scientist named Christof Koch proposed that the claustrum a structure in our brain was central to integrating information and to obtaining integrated conscious percepts see http www ncbi nlm nih gov pmc articles PMC1569501 Can you propose an experimental procedure to disprove their silly I meant to say test their bold hypothesis 9 Giulio Tononi is a sleep specialist who has made waves because he claims he knows how to measure consciousness If you google Tononi and consciousness you will get more references than you should read in a lifetime Roughly speaking his measure is the integrative ability of consciousness given how much raw information is presented what is the relative entropy which is a precise concept in information theory also known as the Kullback Leibler measure of the integrated information that a consciousness makes of it relative to the raw The more conscious the entity the higher the relative entropy Withouth knowing much more try to produce a counterexample that would help show that this measure cannot possibly be correct Feel free to read a little before trying out an answer 10 Many equations we encounter are loosely of the stimulus response kind For example Darcy s law says v K 6 saying that a pressure gradient stimulus drives a fluid velocity in a very resistive medium in proportion to it Newton s law says p f 3 7 If you dont know what the symbols mean hmm equation for gravity reads E T Anyway Einstein s 8 where the right hand is the stress tensor stimulus of all the energy and matter and the left
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