Please help me answer 12.5 Life at low Re (a) E.coli swims at about 20 mu m/s by

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Please help me answer 12.5

Life at low Re (a) E.coli swims at about 20 mu m/s by rotating a bundle helical flagella. If the motors were to turn 10 times faster than normal, what would their swimming speed be? If their fluid environment was made 10 times more viscous but the motors were to turn at the same rate, what would the swimming speed be? How does the power output of the motor change in these two hypothetical situations? (b) Two micron-sized spheres, one made of silver and the other gold, sediment (that is, fall under gravity) in a viscous fluid. The silver sphere has twice the radius of the gold one. Which sediments faster? (c) The left ventricle of the human heart expels about 50 cc of blood per heartbeat. Assuming a pulse rate of l heartbeat per second and a diameter of the aorta of about 2 cm, what is the mean velocity of blood in the aorta? What is the Reynolds number? (d) What is the Reynolds number of a swimming bacterium? A tadpole? A blue whale? (Adapted from a problem courtesy of H. C. Berg and D. Nelson.) Protein centrifugation Proteins and other macromolecules can be separated by size using centrifugation. The idea is to spin a sample containing proteins of different size in solution. The spinning produces a centrifugal force per unit mass g_c, which leads to diffusion with a drift velocity that depends on the protein size. We assume that a protein in the sample can be approximated as a ball of radius R. (a) Following the discussion in the chapter fill in the steps leading up to the formula for the drift velocity of the protein as a function of its radius (eqn 12.45), nu_drift =^2(rho_protein - rho_solvent)g_c^R^2/9 eta, where rho_protein and rho_soivent are the densities of the protein and the solvent, and eta is the solvent viscosity. (b) Estimate the drift velocity for hemoglobin in water in an ultracentrifuge with g_c ~ 10^5g, where 10m/s is the acceleration of freely falling objects in Earth's gravitational field. Assume a typical protein density of 1.2g/cm^3. (c) We would like to separate two similar proteins, having the same density, rho' = 1.35 g/cm^3. They have diameters of 4 nm and 5 nm respectively. The two protein species start out mixed together in a thin layer at the top of a 1 cm long centrifuge tube. How large should the centrifuge acceleration g_c be so that the two proteins are separated before they drift to the end of the tube?

Explanation / Answer

We describe the the Reynolds number as a way of characterizing the relative importance of the effects of fluid momentum or moment of inertia and fluid viscosity

Bacterial swimming occurs at very low Reynolds numbers (Re 104) such that the fluid motion is governed by Stokes flow, and nonlinearities in the full hydrodynamic equation are irrelevant.

So a swimming tadpole may have a Reynolds numbers around 100 or 1000. Swimming microorganisms may have Reynolds numbers around 0.0001. Swimming blue whale has a Reynolds number of 300,000,000

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