A particle in fluid will wander: Its center does a random walk. But the same particle...

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A particle in fluid will wander: Its center does a random walk. But the same particle will also rotate randomly, leading to diffusion in its orientation. Rotational diffusion affects the precision with which a microorganism can swim in a straight line. We can estimate this effect as follows. a) You look up in a book that a sphere of radius R can be twisted in a viscous fluid by applying a torque 7 = Sw, where w is the speed in radians/s and Cr 8πη(??) is the rotational friction coefficient. Unfortunately the dog has chewed your copy of the book and you cant read the last factor. What is it? b)But you didnt want to know about friction, you wanted to know about diffusion. After time t, a sphere will reorient with its axis at an angle to its original direction. Not surprisingly, rotational diffusion obeys <0² >= 4Drt where Dr is the rotational diffusion constant. (This formula is valid as long as t is short enough that this quan- tity is small). Find the dimensions of Dr. c) Get a numerical value for D, for a bacterium, modeled as a sphere of radius 1μm in water at room temperature. d) If this bacterium is swimming, about how long will it take to wander significantly (say 30 deg) off its original direction? A particle in fluid will wander: Its center does a random walk. But the same particle will also rotate randomly, leading to diffusion in its orientation. Rotational diffusion affects the precision with which a microorganism can swim in a straight line. We can estimate this effect as follows. a) You look up in a book that a sphere of radius R can be twisted in a viscous fluid by applying a torque 7 = Sw, where w is the speed in radians/s and Cr 8πη(??) is the rotational friction coefficient. Unfortunately the dog has chewed your copy of the book and you cant read the last factor. What is it? b)But you didnt want to know about friction, you wanted to know about diffusion. After time t, a sphere will reorient with its axis at an angle to its original direction. Not surprisingly, rotational diffusion obeys <0² >= 4Drt where Dr is the rotational diffusion constant. (This formula is valid as long as t is short enough that this quan- tity is small). Find the dimensions of Dr. c) Get a numerical value for D, for a bacterium, modeled as a sphere of radius 1μm in water at room temperature. d) If this bacterium is swimming, about how long will it take to wander significantly (say 30 deg) off its original direction?

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Solution Given data Torque Where W is angular speed in Radsec Rot... View the full answer