Denny's paradox
Encyclopedia
In the study of animal locomotion on the surface layer of water
, Denny's paradox refers to the apparent impossibility of surface-dwelling animals such as the water strider
generating enough propulsive force to move. It is named after Mark Denny
.
If capillary waves are assumed to generate the momentum transfer to the water, the animal's legs must move faster than the phase speed
of the waves, given by
where
is the acceleration due to gravity
,
is the strength of surface tension
, and
the density
of water
. For standard conditions, this works out to be about 0.23 m/s.
Infant water striders' legs move at speeds much less than this and, according to this physical picture, cannot move.
Writing in the Journal of Fluid Mechanics
, David Hu and John Bush state that Denny's paradox "rested on two flawed assumptions. First, water striders' motion was assumed to rely on the generation of capillary waves, since the propulsive force was thought to be that associated with wave drag
on the driving leg. Second, in order to generate capillary waves, it was assumed that the strider leg speed must exceed the minimum wave
speed,
m/s. We note that this second assumption is strictly true only for steady motions" .
Animal locomotion on the surface layer of water
Animal locomotion on the surface layer of water is the study of animal locomotion in the case of small animals that live on the surface layer of water, relying on surface tension to stay afloat....
, Denny's paradox refers to the apparent impossibility of surface-dwelling animals such as the water strider
Water strider
Gerridae is a family of true bugs in the order Hemiptera, commonly known as water striders, water bugs, magic bugs, pond skaters, skaters, skimmers, water scooters, water skaters, water skeeters, water skimmers, water skippers, water spiders, or Jesus bugs...
generating enough propulsive force to move. It is named after Mark Denny
Mark Denny
Mark W. Denny is a professor of biology at Stanford University.His research on the intertidal zone of wave-swept shores has led to increased understanding of this habitat...
.
If capillary waves are assumed to generate the momentum transfer to the water, the animal's legs must move faster than the phase speed
of the waves, given bywhere
is the acceleration due to gravityStandard gravity
Standard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...
,
is the strength of surface tensionSurface tension
Surface tension is a property of the surface of a liquid that allows it to resist an external force. It is revealed, for example, in floating of some objects on the surface of water, even though they are denser than water, and in the ability of some insects to run on the water surface...
, and
the densityDensity
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
. For standard conditions, this works out to be about 0.23 m/s.
Infant water striders' legs move at speeds much less than this and, according to this physical picture, cannot move.
Writing in the Journal of Fluid Mechanics
Journal of Fluid Mechanics
The Journal of Fluid Mechanics is the leading scientific journal in the field of fluid mechanics. It publishes original work on theoretical, computational and experimental aspects of the subject. The journal is usually referred to by its initials JFM by academics in the field. Within citations,...
, David Hu and John Bush state that Denny's paradox "rested on two flawed assumptions. First, water striders' motion was assumed to rely on the generation of capillary waves, since the propulsive force was thought to be that associated with wave drag
Wave drag
In aeronautics, wave drag is a component of the drag on aircraft, blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects.- Overview :...
on the driving leg. Second, in order to generate capillary waves, it was assumed that the strider leg speed must exceed the minimum wave
speed,
m/s. We note that this second assumption is strictly true only for steady motions" .