3.4 Forces on Materials
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• capillary adhesive forces of water with the walls of the xylem and the elaborated
microtubules at the top of the plant;
• water concentration gradients due to controlled evapotranspiration from foliage
surfaces;
• osmotic pressure in the plant foliage due to the active transport of sugar solution
down the phloem in the branches or trunk and active absorption of water from
dew or mist settling on plant top foliage;
• upward osmotic pressure in the xylem from the roots of the plant.
If air leaks into the microtubules at the top of a tree, fluid cavitation can occur.
The adhesive forces at the top level of the tree is enhanced by any branching a
given xylem tube undergoes, which increases the ratio of water-contact surface area
compared to the volume per length in the xylem tubule.
The tallest trees in the world are the California redwoods, standing as much
as 112 m high. Calculations show that the height limitation does not come from
physical constraints, such as resilience under gravity or wind bending. Rather, the
limitation is likely to be dominated by hydraulic negative pressure causing cavitation
at the top of the xylem fluid column (with negative pressures on the order of
−2 MPa).
The interface between two immiscible fluids (‘1’ and ‘2’) is called a capillary
surface. Under static conditions, the surface is determined by a balancing of pressure
forces and surface tension. Let n be the normal to the surface pointing in the
direction toward fluid ‘2’ from ‘1’. Then
p 2 − p 1 = σ ∇ · n .
(3.14)
This relation can be expressed as an equation for a capillary surface. More details
can be found in Appendix D.
3.4.5 Stress in Materials
Humans, even at rest on the surface of the Earth, have stresses in their bones and
elsewhere brought on by gravity.
Within material substances, such as bones or flesh, the forces on each small
element of mass δm can be separated into two types: those which act throughout the
volume of the mass, and those which act on the external surface areas of the mass.
Newton’s 2nd law can then be written:
f V δV +
f S δA = δma. For example,
gravity generates body forces since it acts on every atom within the mass. In this
case, f V = f G ≡ ρg where ρ is the mass density, ρ ≡ δm/δV , and g is the local
gravitational field. The surface force per unit area, f S , is referred to as the ‘stress’
acting on the surface. Stress is measured in newtons per square meter. One newton
per square meter is defined as a ‘pascal’ (Pa).
51
• capillary adhesive forces of water with the walls of the xylem and the elaborated
microtubules at the top of the plant;
• water concentration gradients due to controlled evapotranspiration from foliage
surfaces;
• osmotic pressure in the plant foliage due to the active transport of sugar solution
down the phloem in the branches or trunk and active absorption of water from
dew or mist settling on plant top foliage;
• upward osmotic pressure in the xylem from the roots of the plant.
If air leaks into the microtubules at the top of a tree, fluid cavitation can occur.
The adhesive forces at the top level of the tree is enhanced by any branching a
given xylem tube undergoes, which increases the ratio of water-contact surface area
compared to the volume per length in the xylem tubule.
The tallest trees in the world are the California redwoods, standing as much
as 112 m high. Calculations show that the height limitation does not come from
physical constraints, such as resilience under gravity or wind bending. Rather, the
limitation is likely to be dominated by hydraulic negative pressure causing cavitation
at the top of the xylem fluid column (with negative pressures on the order of
−2 MPa).
The interface between two immiscible fluids (‘1’ and ‘2’) is called a capillary
surface. Under static conditions, the surface is determined by a balancing of pressure
forces and surface tension. Let n be the normal to the surface pointing in the
direction toward fluid ‘2’ from ‘1’. Then
p 2 − p 1 = σ ∇ · n .
(3.14)
This relation can be expressed as an equation for a capillary surface. More details
can be found in Appendix D.
3.4.5 Stress in Materials
Humans, even at rest on the surface of the Earth, have stresses in their bones and
elsewhere brought on by gravity.
Within material substances, such as bones or flesh, the forces on each small
element of mass δm can be separated into two types: those which act throughout the
volume of the mass, and those which act on the external surface areas of the mass.
Newton’s 2nd law can then be written:
f V δV +
f S δA = δma. For example,
gravity generates body forces since it acts on every atom within the mass. In this
case, f V = f G ≡ ρg where ρ is the mass density, ρ ≡ δm/δV , and g is the local
gravitational field. The surface force per unit area, f S , is referred to as the ‘stress’
acting on the surface. Stress is measured in newtons per square meter. One newton
per square meter is defined as a ‘pascal’ (Pa).