Meaning the net change in mass leaving w per unit

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Unformatted text preview: ing W per unit time. There are six sides to W. 3 pairs of parallel sides. We will look at these 3 pairs. Left-Hand Face of W: The components v1 and v3 are parallel to that face and contribute nothing to the flow through the left hand face. The mass of fluid entering through that face during a short time interval is given by: The mass of fluid leaving the right hand face would be the same: The difference: Let So the difference: This same logic can be used for the other 2 pairs of sides: Adding these three up gives us the total loss of mass in W during the time interval : We will now shrink our box and also our time increment down to zero: Once were down to an instant in time and a point mass we are then looking at the partial derivative of the density function with respect to time. This function tells you the density at every point and we want to know how it changes: (we put the negative in front to note a loss in mass) Above we did a lot of work to calculate the same thing so: This equation is called the Condition for the Conservation of Mass or the Continuity Equation of a compressible fluid flow. If the flow is steady (independent of time) then and the continuity equation turns into: If the fluid is incompressible (density is constant) then the continuity equation turns into: This relation is known as the Condition of Incompressibility. It expresses the BALANCE of outflow and inflow for a given volume element at any time. This makes complete sense since you have a steady flow with constant density so of course what comes in must go out! Our book develops this idea using a sphere. But in order to measure the flux for a sphere is more complicated and we will need the idea of a surface integral developed next….. Section 14.6 Surface Integrals We learned that a line integral was a generalization of a definite integral. Now we will look at surface integrals which is a generalization of a double integral. This is the surface integral where S is your surface This is a double integral where R is the projection of S onto the xy-plane Realize if then the surface integral will give you the surface area of your surface S. Note: There is no reason why z must be taken as...
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