Thus it will be necessary to introduce physical descriptions of the body and

Thus it will be necessary to introduce physical

This preview shows page 77 - 79 out of 164 pages.

Thus, it will be necessary to introduce physical descriptions of the body and surface forces in terms of convenient variables and to manipulate the equation to obtain a form in which only a volume integral appears—as was done for conservation of mass in the previous section. We first note that the body forces are generally easy to treat without much further consideration. In particular, the most common such force arising in practice is a buoyancy force due to gravitational acceleration; i.e. , typically F B = ρ g where g is gravitational acceleration, often taken as constant. We note in passing that numerous other body forces are possible, including electromagnetic and rotational effects. We shall not be concerned with these in the present lectures. The surface forces require considerable effort for their treatment, and we will delay this until after we have further simplified Eq. (3.29) by moving differentiation inside the integral as we did earlier in deriving the equation representing conservation of mass. As should be expected we will employ a transport theorem to accomplish this. But in contrast to our analysis of the continuity equation we will use the Reynolds transport theorem , Eq. (3.10), which we repeat here for easy reference: D Dt integraldisplay R ( t ) Φ dV = integraldisplay R ( t ) Φ ∂t dV + integraldisplay S ( t ) Φ U · n dA . (3.30) Recall that Φ is in general a vector field, but here we will work with only a single component at a time, so we can replace this with the scalar Φ, and for the present discussions set Φ = ρu , the x component of momentum per unit volume. Substitution of this into Eq. (3.30) results in D Dt integraldisplay R ( t ) ρu dV = integraldisplay R ( t ) ∂ρu ∂t dV + integraldisplay S ( t ) ρu U · n dA , and applying Gauss’s theorem to the surface integral yields integraldisplay R ( t ) ∂ρu ∂t + ∇ · ( ρu U ) dV (3.31) on the right-hand side of the above expression. We next simplify the second term in the integrand of (3.31). First apply product-rule differen- tiation to obtain ∇ · ( ρu U ) = U · ∇ ( ρu ) + ρu ∇ · U . Now we make use of the divergence-free condition of incompressible flow ( i.e. , ∇ · U = 0) and constant density ρ to write ∇ · ( ρu U ) = ρ U · ∇ u .
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72 CHAPTER 3. THE EQUATIONS OF FLUID MOTION Substitution of this result into Eq. (3.31) shows that D Dt integraldisplay R ( t ) ρu dV = integraldisplay R ( t ) ρ ∂u ∂t + ρ U · ∇ u dV = integraldisplay R ( t ) ρ Du Dt dV , (3.32) where we have used constant density and definition of the substantial derivative on the right-hand side. Thus, we have succeeded in interchanging differentiation (this time, total) with integration over the fluid element R ( t ). We remark that the assumption of incompressible flow used above is not actually needed to obtain this result; but it simplifies the derivation, and we will be making use of it in the sequel in any case. We leave as an exercise to the reader the task of obtaining Eq. (3.32) without the incompressibility assumption as well as deriving analogous formulas for the other two components of momentum.
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