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3.1.2: Continuity equation

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    One of the fundamentals of physics stays that the matter in the interior of an isolated system is not created nor destroyed, it is only transformed. If one thinks in open systems (not isolated), such as human beings or airplanes in flight, its mass is constantly varying.

    In a fluid is not easy to identify particles or fluid volumes since they are moving and deforming constantly within time. That is way the conservation of mass must be understood in a different way:

    Recall the concept of stream tube. Assuming through its walls there is no flow, and that the flow is stationary across any section area (the velocity is constant), the mass that enters per unit of time in Section \(A_1 (\rho_1 V_1 A_1)\) will be equal to the mass that exits Section \(A_2 (\rho_2 V_2 A_2)\), where \(\rho\) is the density, \(V\) is the velocity, and \(A\) is the area. Therefore, the continuity of mass stays:

    \[\rho_1 V_1 A_1 = \rho_2 V_2 A_2.\]

    Since Section \(A_1\) and Section \(A_2\) are generic, one can claim that the product \(\rho VA\) is constant along the stream tube. The product \(\rho VA\) is referred to as mass flow  \(\dot{m}\) (with dimensions [kg/s]).

    截屏2022-01-14 下午8.50.00.png
    Figure 3.3: Continuity equation.

    Compressible and incompressible flow

    In many occasions occurs that the density of a fluid does not change due to the fact that it is moving. This happens in liquids and, in some circumstances, in gases (think in the air confined in a room). Notice that one can not say that the air is incompressible, but an air flow is incompressible.

    The movement of air in which the velocity is inferior to 100 [m/s] can be considered incompressible. When the air moves faster, as is the case in a jet airplane, the flow is compressible and the studies become more complicated as it will be seen in posterior courses.

    3.1.2: Continuity equation is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Manuel Soler Arnedo via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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