Forces on Fluid Elements
Fluid Elements - Definition:
Fluid element can be defined as an infinitesimal region of the fluid continuum in isolation from its surroundings.
Two types of forces exist on fluid elements
(i) Body Force: distributed over the entire mass or volume of the element. It is usually expressed per unit mass of the element or medium upon which the forces act.
Example: Gravitational Force, Electromagnetic force fields etc.
(ii) Surface Force: Forces exerted on the fluid element by its surroundings through direct contact at the surface.
Surface force has two components:
(i) Normal Force: along the normal to the area
(ii) Shear Force: along the plane of the area.
The ratios of these forces and the elemental area in the limit of the area tending to zero are called the normal and shear stresses respectively.
The sheer force is zero for any fluid element at rest and hence the only surface force on a fluid element is the normal component.
Normal Stress in a Stationary Fluid
Consider a stationary fluid element of tetrahedral shape with three of its faces coinciding with the coordinate planes x, y and z.
Since a fluid element at rest can develop neither shear stress nor tensile stress, the normal stresses acting on different faces are compressive in nature.
Suppose, ΣFx, ΣFy and ΣFz are the net forces acting on the fluid element in positive x,y and z directions respectively. The direction cosines of the normal to the inclined plane of an area ΔA are cos α, cos β and cosɤ g .
Considering gravity as the only source of external body force, acting in the -ve z direction, the equations of static equilibrium for the tetrahedron fluid element can be written as
where (Δx Δy Δz)/6 = Volume of tetrahedral fluid element
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