Time Rate of Change of Momentum As was the case in deriving the differential

# Time rate of change of momentum as was the case in

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Time-Rate of Change of Momentum As was the case in deriving the differential equation representing conservation of mass, it will again be convenient here to choose a fluid region corresponding to a fluid element. In contrast to what was done earlier, we will restrict our region R ( t ) to be a fluid element from the start. If, in addition, we utilize an Eulerian view of the fluid flow we recognize that the substantial derivative should be employed to represent acceleration or, in our present case, to calculate the time-rate of change of momentum. As noted above, it is convenient for later purposes to consider the momentum per unit volume, rather than the momentum itself; so for the x component of this we would have D Dt integraldisplay R ( t ) ρu dV , the equivalent of mass × acceleration. Then for the complete velocity vector U we can write D Dt integraldisplay R ( t ) ρ U dV braceleftbigg time rate of change of momentum vector bracerightbigg . (3.28) We remind the reader that application of the substantial derivative operator to a vector is accom- plished by applying it to each component individually, so the above expression actually contains three components, each of the form of that for x momentum. Sum of Forces We next consider the general form of the right-hand side of the word equation given earlier, viz. , the sum of forces acting on the material region (fluid element in the present case). There are two main types of forces to analyze: i ) body forces acting on the entire region R ( t ), denoted integraldisplay R ( t ) F B dV , and ii ) surface forces , integraldisplay S ( t ) F S dA , acting only on the surface S ( t ) of R ( t ). It is useful to view the surface S ( t ) as dividing the fluid into two distinct regions: one that is interior to S ( t ), i.e. , R ( t ), and one that is on the outside of S ( t ). This implies that when we focus attention on R ( t ) alone, as it will be convenient to do, we must somehow account for the fact that we have discarded the outside—which interacts with R ( t ). We do this by representing these effects as surface forces acting on S ( t ). We will treat F B and F S , especially the latter, in more detail later.
3.4. MOMENTUM BALANCE—THE NAVIER–STOKES EQUATIONS 71 The Momentum Equations of Fluid Flow We can now produce a preliminary version of the momentum equations using Eq. (3.28) and the body and surface force integrals; this will ultimately lead to the equations of fluid motion, the Navier–Stokes equations. From the word equation given at the start we have D Dt integraldisplay R ( t ) ρ U dV = integraldisplay R ( t ) F B dV + integraldisplay S ( t ) F S dA . (3.29) Just as was the case for the continuity equation treated earlier, this general form is not very convenient for practical use; it contains both volume and surface integrals, and it is not expressed entirely in terms of the typical dependent variables usually needed for engineering calculations.

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