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 ...

, kinematic similarity is described as “the

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

at any point in the model flow is proportional by a constant scale factor to the

at the same point in the prototype flow, while it is maintaining the flow’s streamline shape.” Kinematic Similarity is one of the three essential conditions ( Geometric Similarity, Dynamic Similarity and Kinematic Similarity) to complete the similarities between a model and a prototype. The kinematic similarity is the similarity of the

motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...

of the fluid. Since motions can be expressed with distance and time, it implies the similarity of lengths (i.e. geometrical similarity) and, in addition, a similarity of the time interval. To achieve kinematic similarity in a scaled model,

dimensionless numbers A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1), ...

fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...

come into consideration. For example, Reynolds number of the model and the prototype must match. There are other

that will also come into consideration, such as

Womersley number The Womersley number (\alpha or \text) is a dimensionless number in biofluid mechanics and biofluid dynamics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscosity, viscous effects. It is named after John R. Wom ...

Lee Waite, Ph.D., P.E.; Jerry Fine, Ph.D.: Applied Biofluid Mechanics, Second Edition. Common Dimensionless Parameters in Fluid Mechanics, Chapter (McGraw-Hill Professional, 2017), AccessEngineering

Example

Assume we need to make a scaled up model of coronary artery with kinematic similarity. Reynolds Number,

Where,

Density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

of the fluid (

SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...

: kg/m³)

Velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

of the fluid (

: m/s)

Characteristic length In physics, a characteristic length is an important dimension that defines the scale of a physical system. Often, such a length is used as an input to a formula in order to predict some characteristics of the system, and it is usually required by ...

or diameter (

: m)

Dynamic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...

(

: N s/m²)

Kinematic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the intern ...

(

: m²/s)
There are few ways to maintain kinematic similarity. To keep the Reynolds number the same, the scaled-up model can use a different fluid with different

viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

. We can also change the

of the

fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shea ...

to maintain the same dynamic characteristics. The above equation can be written for artery as, And for the scaled-up model, At the condition of Kinematic Similarity, That means, or, Substituting variables by provided values will provide important characteristics data for the

and

flow characteristics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

for the scaled-up model. A similar approach can be taken for the scaled-down model (i.e. oil refinery scaled-down model) as well.

References

{{reflist Dimensionless numbers of fluid mechanics Fluid mechanics

Example

See also

References