Physics 414: Introduction to Biophysics Professor Henry Greenside October 24 and October 26, 2017

Remarkable (but challenging) physics reference on fluid dynamics Lev Landau Evgeny Lifshitz

State of a fluid: two thermodynamic variables like T and rho, plus velocity field Relaxation time argument implies that sufficiently small local regions of a fluid will be in thermodynamic equilibrium even though fluid itself is not. So state of fluid will be two thermodynamic variables like temperature T(t,X), local mass density rho(t,X) that vary in time and space, plus a velocity field We will discuss only isothermal (constant temperature) and incompressible (constant density) fluids in this course so a rather restricted class of fluid motion. Incompressible fluid flow is an excellent approximation when all fluid speeds are much smaller than the speed of sound v_s in the fluid (see Landau and Lifshitz “Fluid Mechanics” for justification). This is the case for most cellular, animal motions.

Navier-Stokes equations: nonlinear evolution equation for the fluid velocity field v(t,x,y,z) Three-dimensional Navier-Stokes equation for isothermal incompressible flow

Navier-Stokes equations, as partial differential equations, need initial data and boundary conditions Need to know v field at all points at one time, one reason why weather forecasting and other problems involving fluid dynamics are hard to forecast accurately, only have spatially sparse measurements

Fluid in contact with rigid walls satisfies a “no-slip” boundary condition v = 0

Newton’s law of viscosity shearing flows

Derivation of Newton’s law of viscosity for ideal gas in Reif, Section 12.3 of Chapter 12 Chapter 12 of Reif’s book posted on 414 Sakai webpage, under Resources / Book chapters / reif-fund-stat-thermal-physics-ch-12-pp-461-493.pdf Elementary kinetics theory shows how various dissipation mechanisms arise via molecular collisions, get explicit formulas for dissipation coefficients D, kappa, etc in terms of microscopic details (for gas only) This is optional reading, for your own enjoyment and satisfaction. A surprising result of this calculation is that gas viscosity eta does not depend on gas pressure!

From video “Inner Life of a Cell” Stoke’s drag force (shearing force) on sphere at small Reynolds number calculated analytically From video “Inner Life of a Cell” About 100 times smaller than maximum force ~ 5 pN kinesin motor can exert.

Shearing forces in fluids that do not satisfy Newton’s law of viscosity are called “non-Newtonian” Many biological fluids (blood, cytosol, saliva, mucus) are non-Newtonian. Common reason for non-Newtonian behavior is that fluid is not pure homogeneous fluid but has lots of impurities like ions (at high concentrations), polymers, cells that change its properties.

Blood is a complicated non-Newtonian fluid Complete Blood Count (CBC)

Corn starch mixed with water on loud speaker vibrating at 30 Hz. Dynamics of non-Newtonian flows often complicated, search “oobleck” on YouTube https://youtu.be/3zoTKXXNQIU Corn starch mixed with water on loud speaker vibrating at 30 Hz.

At the blackboard: how pressure and shearing forces arise on small fluid volume in time-independent cylindrical pipe flow See NSF video of Sir Geoffrey Taylor, part 1, at time 5:40 into video. https://youtu.be/hALx7vfmRt4

Hagen-Poiseuille steady laminar Newtonian flow in cylindrical tube of diameter d Poiseuille French scientist, name pronounced (by Americans) as “pwa-zay”

Hagen-Poiseuille plus assumption of minimal metabolic cost implies Murray’s law for bifurcations of vessels transporting air or blood THE PHYSIOLOGICAL PRINCIPLE OF MINIMUM WORK. I. THE VASCULAR SYSTEM AND THE COST OF BLOOD VOLUME Cecil Murray, Proc Nat Acad Sci 12:207-214 (1926).

One-minute End-of-class Question