Summary
In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero (see the derivation below, which illustrates why these conditions are equivalent). Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that (under the right conditions) even compressible fluids can – to a good approximation – be modelled as an incompressible flow. The fundamental requirement for incompressible flow is that the density, , is constant within a small element volume, dV, which moves at the flow velocity u. Mathematically, this constraint implies that the material derivative (discussed below) of the density must vanish to ensure incompressible flow. Before introducing this constraint, we must apply the conservation of mass to generate the necessary relations. The mass is calculated by a volume integral of the density, : The conservation of mass requires that the time derivative of the mass inside a control volume be equal to the mass flux, J, across its boundaries. Mathematically, we can represent this constraint in terms of a surface integral: The negative sign in the above expression ensures that outward flow results in a decrease in the mass with respect to time, using the convention that the surface area vector points outward. Now, using the divergence theorem we can derive the relationship between the flux and the partial time derivative of the density: therefore: The partial derivative of the density with respect to time need not vanish to ensure incompressible flow. When we speak of the partial derivative of the density with respect to time, we refer to this rate of change within a control volume of fixed position.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (32)
ME-445: Aerodynamics
This course will provide the fluid dynamic background to understand how air flows around two- and three-dimensional wings and bodies and to understand and calculate the aerodynamics forces and moments
ME-280: Fluid mechanics (for GM)
Basic lecture in fluid mechanics
BIOENG-312: Fluid mechanics (for SV)
This introductory course on fluids mechanics presents the basics concepts in fluids statics, dynamics and kinematics.
Show more
Related concepts (18)
Fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical, and biomedical engineering, as well as geophysics, oceanography, meteorology, astrophysics, and biology. It can be divided into fluid statics, the study of fluids at rest; and fluid dynamics, the study of the effect of forces on fluid motion.
Flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall).
Material derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation. For example, in fluid dynamics, the velocity field is the flow velocity, and the quantity of interest might be the temperature of the fluid.
Show more
Related MOOCs (5)
Plasma Physics: Introduction
Learn the basics of plasma, one of the fundamental states of matter, and the different types of models used to describe it, including fluid and kinetic.
Plasma Physics: Introduction
Learn the basics of plasma, one of the fundamental states of matter, and the different types of models used to describe it, including fluid and kinetic.
Plasma Physics: Applications
Learn about plasma applications from nuclear fusion powering the sun, to making integrated circuits, to generating electricity.
Show more