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# Conservation Laws

Suppose that is the density of some bulk fluid property (e.g., mass, momentum, or energy) at position and time . In other words, suppose that, at time , an infinitesimal fluid element of volume , located at position , contains an amount of the property in question. Note, incidentally, that can be either a scalar, a component of a vector, or even a component of a tensor. The total amount of the property contained within some fixed volume is

 (1.28)

where the integral is taken over all elements of . Let be an outward directed element of the bounding surface of . Suppose that this element is located at point . The volume of fluid that flows per second across the element, and so out of , is . Thus, the amount of the fluid property under consideration that is convected across the element per second is . It follows that the net amount of the property that is convected out of volume by fluid flow across its bounding surface is

 (1.29)

where the integral is taken over all outward directed elements of . Suppose, finally, that the property in question is created within the volume at the rate per second. The conservation equation for the fluid property takes the form

 (1.30)

In other words, the rate of increase in the amount of the property contained within is the difference between the creation rate of the property inside , and the rate at which the property is convected out of by fluid flow. The previous conservation law can also be written

 (1.31)

Here, is termed the flux of the property out of , whereas is called the net generation rate of the property within .

Next: Mass Conservation Up: Mathematical Models of Fluid Previous: Viscosity
Richard Fitzpatrick 2016-03-31