Consider water in a glass being sucked through a straw. The water rises up in the straw because of a pressure gradient introduced by the sucking action. Now, change the liquid from water to something thicker like a thick milkshake (higher viscosity and higher density). It would need a greater amount of suction to raise the level of the slushy/milkshake in the straw to that achieved by using water.

How should I explain this ? Can this be explained with the Bond number (gravity/capillarity ratio) or should I explain this phenomena based on the Capillary number (viscosity/capillarity ratio)?

Can/Should the velocity of the liquid flow in the straw be calculated with the Hagen-Poisseuille's equation?

  • $\begingroup$ My first impression is that for a water/milkshake sucked through a vertical straw, surface-tension/capillarity is negligible. Gravity term should be included. Milkshakes are probably non-Newtonian fluids. Steady-state solutions will probably be quite similar except for magnitude of pressure required to suck up liquids. Transient solutions could be quite different due to non-Newtonian properties of milkshake (if any.) $\endgroup$ – Mark Rovetta Feb 13 '13 at 20:13
  • $\begingroup$ @MarkRovetta I somehow tend to disagree with you but I could be wrong about gravity being not that important considering the volume of the milkshake in the straw? But yes, as compared to say, water, gravity would be important. Yes, milkshakes are probably non-Newtonian but the general physics governing them would be the same as for Newt. liquids. Viscosity will probably have a few added terms. $\endgroup$ – dearN Feb 14 '13 at 12:42

The classic Poisseuille flow approach is a fine approximate solution for situations that satisfy its assumptions. The effect of gravity can be accounted for well by including it in the pressure-drop term. It should work fine for water being sucked through a soda-straw.

Surface tension forces won't be large for water or milkshakes sucked through an ordinary soda-straw (~5mm dia.) Surface tension (and the two non-dimensional numbers you mention) would become useful in problems of fluid flow through porous media where liquid is flowing through capillaries.

I think milkshakes behave differently than water because they are non-Newtonian. When I pull the straw out of the milkshake, a thick layer (2-3 mm) clings to the outside of the straw ...

I can set a cherry on the top of my milkshake and it does not sink into the glass. That's not because it's floating (cherries are more dense than milkshakes) its because the forces in the milkshake beneath the cherry do not exceed the critical shear-stress required to cause flow. Below this critical shear-stress, milkshakes behave as a solid.

Note that the dimension of critical shear-stress (material property of milkshakes) multiplied by the characteristic length of the zone of yielding flow beneath the cherry just happens to have the same dimension as surface-tension. But that doesn't mean you're justified assuming the Bond or Cappillary numbers have physical meaning in this case. Dimensions may agree but the physics are different.

The Poisseuille flow approach assume the fluid behaves as a Newtonian-fluid. That assumption is likely violated for a milkshake. So Poisseuille flow solutions might be a poor approximation for analyzing milkshakes.

  • $\begingroup$ Do the derivation again without assuming non-Newtonian fluid and you'll see what difference it makes. It's actually a good derivation to work through. Since we're talking a straw, make sure it's all done in cylindrical coordinates and not Cartesian. $\endgroup$ – tpg2114 Feb 15 '13 at 15:57
  • $\begingroup$ If the assumptions are satisfied, Poisseuille flow is a fine approximation. If the assumptions are inapplicable to the real case, there's no guarantee that the result is meaningful. $\endgroup$ – Mark Rovetta Feb 15 '13 at 18:53

It has to do with the pressure from the atmosphere. Milkshakes are thicker than water and they are also stickier causing them to stick more to the straw.


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