View Full Version : Question about the shape of a spider's web

Perikles

2013-Jul-18, 04:41 PM

Actually, I mean a spider's silk, not the spun web. The wiki article on the catenary (http://en.wikipedia.org/wiki/Catenary) shows how all the segments of an actual web are parts of a catenary, but my question is about long single strands of silk which spiders produce overnight for me to walk into in the morning. I see them against the light of the rising sun, spun between plants and/or other objects (the motorbike is a favourite) which can be several metres apart. If they hang in total calm, they form catenaries (assuming the spider is elsewhere), but if there is a gentle breeze, they become (almost) horizontal. I saw a large one this morning which looked very much like the arc of a circle.

Can anyone tell me how to calculate the shape of a silk caused by a constant wind and neglecting gravity? Assuming the silk is of uniform density, the force of the wind is not constant on each element of the web (or is it?) because it must depend on the angle between each element and the wind direction. The wiki article gives clues how to calculate this, but I'm very rusty and the maths has baffled me.

Jeff Root

2013-Jul-18, 08:06 PM

http://en.wikipedia.org/wiki/Fluid_dynamics

http://en.wikipedia.org/wiki/Aerodynamics

The article you cite would appear to suggest that a thread

suspended at the ends in a perfectly uniform wind would

have a shape closer to a parabola than a catenary, but you

are right that the force of the wind must vary with the angle

of incidence. Way beyond my capabilities, especially if the

wind is not symmetrical relative to the suspension points.

My first thoughts were of a spider thread suspended from

a single point, waving in the wind like a flag.

-- Jeff, in Minneapolis

profloater

2013-Jul-19, 02:03 PM

we used to have to do a lot of catenary stuff with hypothetical chains I can look up my notes. However I believe a normal gravity formula will apply to wind loading per unit length because the drag on a unit length will be a constant just like the weight of a unit length. at least to first order, the drag is skin friction drag of the boundary layer.

profloater

2013-Jul-19, 02:14 PM

So I think you have y = L cosh(x/L) for the shape, the defining characteristic is the constant "horizontal" force component. this assumes the drag force is constant per unit length.

Perikles

2013-Jul-19, 03:41 PM

However I believe a normal gravity formula will apply to wind loading per unit length because the drag on a unit length will be a constant just like the weight of a unit length. I really don't see how. Suppose the silk is blown by the wind to make a shape something like 180 degrees of a circle (which is what I saw happening). The silk at the ends, attached to two objects, will suffer very little or no drag from the wind because the wind comes almost at a tangent to those points. That can't be the same drag as in the middle where the wind comes perpendicular to the arc. So I can't see how the shape can be a hyperbolic cosine expression. Surely, the drag per unit length must be proportional to the cosine of the angle of incidence at all points.

profloater

2013-Jul-19, 03:44 PM

I left out a wind force variable something like kv^2 so y = kv^2. L cosh (x/L) giving a series of curves for increasing v wind speed. Unfortunately near the ground the wind speed is unlikely to be constant with height so not a pure catenary at all.

Perikles

2013-Jul-19, 03:46 PM

I left out a wind force variable something like kv^2 so y = kv^2. L cosh (x/L) giving a series of curves for increasing v wind speed. Unfortunately near the ground the wind speed is unlikely to be constant with height so not a pure catenary at all.Let's just assume that the silk is fixed between two points of equal height and the wind speed is constant at all points along the silk, and gravity is ignored.

profloater

2013-Jul-19, 03:53 PM

I really don't see how. Suppose the silk is blown by the wind to make a shape something like 180 degrees of a circle (which is what I saw happening). The silk at the ends, attached to two objects, will suffer very little or no drag from the wind because the wind comes almost at a tangent to those points. That can't be the same drag as in the middle where the wind comes perpendicular to the arc. So I can't see how the shape can be a hyperbolic cosine expression. Surely, the drag per unit length must be proportional to the cosine of the angle of incidence at all points.

Well I took the view that the shape drag would be very small compared to the friction drag on such a fine thread. If the shape lift and drag are significant, then yes it would be very different. They say sails are catenaries although the same objection applies that the wind force must depend on the actual angle and since the sail is modifying the wind direction, it must be an approximation, more so than with a thread. As the area reduces the shape effect would diminish with the skin drag but for the boundary layer that is, I suspect, many times bigger than the thread dimension. However I could be wrong. The shape drag is easy to estimate if the dimension of the thread is known, it is a cylinder. Of course the skin friction drag is always along the wind direction but the shape air forces would have a lift component. That's why I assumed the force mimics gravity.

profloater

2013-Jul-19, 04:15 PM

The switch over occurs at Reynolds number of about 1.5 . 10^5 below that the drag is mainly shape drag and the drag coefficient for a cylinder is about 1.2. If the thread is about 0.025 mm i.e. 0.000025 m for wind speed 1 m/s the RE is huge about 4 10^8 I think so boundary forces dominate and the drag coefficient drops to 0.4.

Re = wind velocity (m/s) x diameter (m) / 1.456.10^-5 in air.

Perikles

2013-Jul-19, 04:21 PM

Er - am I supposed to know what RE is? :confused:

profloater

2013-Jul-19, 04:24 PM

Let's just assume that the silk is fixed between two points of equal height and the wind speed is constant at all points along the silk, and gravity is ignored.

sorry I was typing while you posted, my assumption of 25 microns, one thou inch, may be wrong but still the Re will be very high meaning that viscous effects dominate over inertial, which was my assumption. If so then the drag on a unit length is the same at all angles to the wind. So it's like gravity, only sideways and bigger than gravity. but if that's right it should stay in a plane as wind speed increases from zero. The stretch effect would not change that unless it is very large.

profloater

2013-Jul-19, 04:32 PM

Er - am I supposed to know what RE is? :confused:

Re is reynolds number used to determine the flow regime in all fluids, its a non dimensional number, useful in comparing all sorts of flow situations. Basically it compares inertial and viscous forces, at low Re (less than 10^5) we are in aerodynamics, but at high Re we are in viscous forces, laminar and turbulent boundary layers as in specks of dust etc in a fluid. For a given fluid with its density and viscosity fixed, Re depends on V and characteristic L, in this case the diameter. I have not checked good old Wiki but I bet it's there.

profloater

2013-Jul-19, 04:46 PM

well it seems drag line is only 3 microns and density 1.3 so weight per unit length is 9. 10^-11 N

drag at 1/2 ro v^2 Cd A with Cd 0.4 at 1 m/s 0.6 10^-6 N

of course I may have cobbled that up by a few OOM!

Perikles

2013-Jul-19, 04:52 PM

So if I understand that correctly, with the drag per unit length constant and independent of the actual angle between the unit and the wind direction (which I find very suprising) then the silk will make an ordinary catenary shape.

Thanks for that!

profloater

2013-Jul-19, 05:30 PM

The only way you could get a circle would be equivalent to a hoop stress with constant pressure radially and a fixed tangential tension, That in my view can happen in a billowing sail situation. However a fully developed catenary must be quite close to a semi circle except right at the ends. The horizontal force along the thread is a constant but the "vertical" force, in this case the horizontal at right angles, of course is greatest at the ends where it would break first, and since it is in fact very stretchy the material near the ends is growing longer, with its diameter decreasing (no idea what Poissons ration is for spider silk but for elastomer 0.5) so now the force near the ends will be less (lower diameter,) and the shape is unique. It is likely the atoms align as it stretches becoming stiffer so enough for a good research project.

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