Physics 414: Introduction to Biophysics

Professor Henry Greenside November 6, 2018

1 Too hard to apply Newton’s law of viscosity directly to different kinds of fluid flow, want general quantitative description: Navier-Stokes equations (~1850)

Three-dimensional Navier-Stokes equation for isothermal incompressible flow. This partial differential equation is a so-called initial-boundary-value mathematical problem: to have a well-defined problem one needs to specifiy initial data v(t ,x,y,z) at all points in space at 0 some starting time t , and specify boundary conditions on the velocity field v(t,x,y,z) . 0

2 Vocabulary: advective or material derivative

3 Advective derivative is a scalar that distributes over components of a vector

You try it, evaluate the following expression

4 The Navier-Stokes equations for isothermal incompressible flow

5 Evolution equations for density, velocity, temperature, and concentration (the kitchen sink)

You can see how scalar quantities like T and c are advected by the velocity field v. These equations are useful for simulations of atmosphere, oceans, stellar plasmas but rarely needed for dynamics at the cellular or subcellular level in biophysics.

6 For initial data, need to know v(t ,x,y,z) at all points 0 at some starting time t , one reason why accurate 0 weather forecasting is difficult

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

8 How strong or important is viscosity does not depend just on dynamic viscosity η but on relative magnitude of inertia to dissipation: the Reynolds number

Reynolds number can also be interpreted as the ratio of the kinetic energy of motion for a small region (blob) of fluid to the energy associated with work done against the viscous friction, see Eqs. (12.28) and (12.29) on page 496 of PBOC2.

9 Nonequilibrium systems are often characterized by dimensionless ratios that compare how strong are mechanisms driving system out of equilibrium to dissipative mechanisms restoring equilibrium

For example, a convecting fluid is driven out of equilibrium by a temperature difference that leads to buoyancy forces, and two mechanisms try to restore the fluid to equilibrium, viscous friction (characterized by kinematic viscosity ν) and thermal conductivity. An analysis of the Navier-Stokes equations coupled to the heat-diffusion equation leads to a Reynolds-number-like dimensionless parameter called the Rayleigh number:

Fluid changes from motionless (v=0) to convecting when Ra exceeds a certain critical numerical value called the “critical Rayleigh number”. For fluid convecting between two infinitely wide parallel horizontal plates, the critical Rayleigh number has theoretically shown to be 1708, which implies that a minimum temperature difference is needed for convection to begin.

10 Experiments by Osborne Reynolds showed that kind and complexity of flow depended on magnitude of Reynolds number

1) Re for E. coli swimming (v ~ 30 lengths/s) 2) Re for person walking?

11 Laminar flow Re < 1: time independent, parallel stream lines

Note: streamlines are a widely used way to visualize vector fields of different kinds like a fluid velocity field v(x,y,z,t) or an electric field E(x,y,z,t). Streamlines are useful when one wants to summarize the directions of the vectors filling some region of space without worrying about their magnitudes.

By definition, a stream line of some vector field v is any continuous parametric curve r(s) through space such that the vector field v is tangent to the stream line at any point (x,y,z) on the stream line. That is, the tangent to any point on a streamline points in the direction of the vector field. In the above middle figure, the dotted curves are several different stream lines of the two-dimensional velocity field v(x,y) of a time- independent fluid flowing left to right past a long cylinder whose axis is perpendicular to the page of the figure. The small arrows just above the top of the cylinder and just below the bottom of the cylinder are tangent vectors to those streamlines that point in the direction of the fluid flow, but you cannot tell how big are the velocity vectors from this figure, e.g., you cannot tell that the velocity is decreasing in magnitude as you approach the surface of the cylinder because of the no-slip velocity boundary condition.

Note that a streamline is generally NOT the path traced out by a small particle that is placed at a given location on the streamline and then released to move with the fluid (although a particle will follow a given streamline for some distance that gets longer as the radius of curvature of the streamline becomes larger). A streamline is a geometric construct that ignores physical effects like inertia, which causes a particle to move across curving streamlines as the particle moves. You can find many YouTube videos that explain how to calculate streamlines mathematically from a given flow, the methods generally require numerical methods for integrating differential equations.

Finally, the image in the upper left is from an actual fluid experiment in which small (compared to the radius of the cylinder) neutrally buoyant white trace particles have been injected into the flow on the far left of the experimental apparatus. A time-exposed photograph then reveals the experimental stream lines since a sufficiently small and low-mass particle will follow along part of a stream line. The main insight of the above two figures is that laminar flow has locally parallel stream lines which indicates a simple kind of fluid motion. 12 Drag force (shearing force) on sphere far from side walls at small Reynolds number known analytically

Holds only for relative speed v of sphere with respect to fluid that is so small that flow is laminar, time-independent with locally parallel stream lines. This always holds for motion inside a biological cell.

You can find a derivation of Stokes’ law in the Landau and Lifshitz book “Fluid Mechanics” which is available free online, see the Physics 414 links page http://webhome.phy.duke.edu/~hsg/414/links

13 How far does a spherical bacterium glide when flagellar propulsion suddenly turned off?

At scale of bacterium, because Reynolds number Re is so tiny, viscous friction from shearing is huge compared to kinetic energy so bacterium comes to instant halt, travels negligible distance. (Not like you diving into a swimming pool.) The opposite also holds: when a bacterium starts to move from rest by activating its flagellar motors, it starts moving instantly, a negligible amount of time (0.1 microseconds!) is needed to accelerate up to its terminal speed of say 10 microns/s. 14 Application of Stokes’ law to transport of 1 μm- diameter vesicles along microtubule at ~1 micron/s

From video “Inner Life of a Cell”

F is about 100 times smaller than maximum force ~ 5 pN that kinesin motor can exert so viscous drag on these vesicles is negligible.

15 Stokes’ formula implies that small spherical particles subjected to a constant force (gravity, rotational, electrical) can have a small final terminal speed

For example, small spherical objects pulled downwards by gravity mg reach terminal speed v given by

For cloud water droplet of typical diameter ~10 μm falling through air, get v ~ 5 mm/s, too slow to observe from ground which explains why clouds don’t crash to ground. Stokes’ formula also explains why homogenized milk doesn’t separate into cream, and why ants can fall from the top of the Duke Chapel and reach the ground without harm. 16 Application of Stokes’s drag formula to centrifuges that allow one to separate biomolecules in a solution, need buoyancy correction though

Figure 12.18 page 503 PBOC2

17 One-minute End-of-class Question

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