laminar flow and shear forces

An inverted parabola, with the area under the curve = the volume of flow moved in a unit time, and zero at the wall.

A laminar profile has the lowest wall shear stress, and has no time- varying component that can "fatigue" attachment methods to failure.

The flow rate will be the only control you have to limit the flow rate. Laminar flow is already the most "quiescent" you can achieve, short of no motion at all.

Now you could form small pockets outside the tube wall, and let the bacteria inhabit the even quieter pockets. Vorticity should be small...

David A. Smith

but the bacteria are about 500 nm wide, so they extend into the tube where they would experience shear stress. i'm guessing i could just figure out the force on the 'inside' surface of the cell if i know that the profile and the dimensions of the channel, right?

thanks, dan

How big is the microchannel, and what is its shape? You'll need to know this to get your "flow thrust".

You'll have to wait for someone that has done CFD on this type of problem.

I would:

- use the bacterial dimensions and quantity to establish an average roughness,

- use one of the empirical relationships for rough pipe flow in the appropriate flow regime, then

- estimate how much power is delivered (essentially) down to each bacteria.

Frankly, I'd just stick some critters in, and see how they fare at different flow rates...

David A. Smith

the channel has a rectangular shape, about 200 microns wide and 50 microns high. the wall area covered by bacteria is rather small relative to the whole wall area, so i'm not sure that the bacteria are disrupting the flow that much.

i'll probably just do that in the end, but these guys take about 1.5 hours to grow, and sometimes you don't realize for a few generations that they're not happy.

dan

You say that the bacteria are about 0.5 microns in diameter, and the channel is 50 x 200 microns. Hence, bacteria are small compared to channel diameter, and channel velocity profile can be approximated by a linear slope next to the wall.

Suggested solution:

1. Figure out velocity profile in channel in the absence of bacteria. For low Reynolds number (Re) flow in your microchannel, the flow will be laminar. Laminar flow in a circular tube is called Poiseuille flow. Many fluid mechanics books that disucss Poiseuille flow follow with a discussion of flow in non-circular channels. ("Fluid Mechanics" by Frank M. White would be a good choice.)

1. Be sure flow is "fully developed". Again, most fluid mechanics books discuss concepts like "fully developed" and "entrance length" within a few pages of the place where they discuss Poiseuille flow. Basically, this involves being sure your channel is long enough for the Poiseuille solution to be valid.

2. Use the laminar flow solution to find the wall shear stress. This may be all you need to know -- if you have data on bacteria viability in flow fields, perhaps this is reported in terms of the maximum shear stress they can withstand.

The wall shear stress is the fluid viscosity times the velocity gradient evaluated at the wall.

1. If you really, really want to know the force on a bacteria attached to the wall:

A. Assume a spherical cow (bacteria).

B. Approximate fluid velocity profile in the neighborhood of the wall by a linear profile. ( Slope = shear stress / viscosity )

C. Find creeping flow solution (Stokes flow, low Re flow, etc.) for force on a sphere attached to a wall in a linear velocity gradient. Fortunately, this has apparently been done already. From "Dynamics of bubbles, drops, and rigid particles," by Z. Zapryanov and S. Tabakova, p. 237:

"An exact solution for a viscous flow around a fixed sphere in contact with a fixed plane wall, when the fluid motion in the absence of the sphere is assumed to be a uniform linear shear flow, is given by O'Neill (1968) using tangent- sphere coordinates."

Olin Perry Norton

thanks to all! clearly there's a lot of reading i need to do...

I reckon you received some marvelous tips. The only thing I don't recall mention of, is the buffeting that small particles get in this miniature world dominated by viscous forces, from Brownian motion. If you can see dust motes dancing about in water through a microscope, then you can guess how bacteria might be affected.

Brian Whatcott Altus OK