We observe that now mfuel is positive because we have already accounted for the

We observe that now mfuel is positive because we have

This preview shows page 75 - 77 out of 164 pages.

We observe that now ˙ m fuel is positive because we have already accounted for the sign in the solution process. Finally, we should point out that in this steady-state case we could have simply used ˙ m in = ˙ m out , but we would have to exercise some caution when evaluating ˙ m out to account for the fact that the airplane is moving with speed U p . By using the formal analysis procedure presented here, we have automatically accounted for such details. 3.4 Momentum Balance—the Navier–Stokes Equations In this section we will derive the equations of motion for incompressible fluid flows, i.e. , the Navier– Stokes (N.–S.) equations. We begin by stating a general force balance consistent with Newton’s second law of motion, and then formulate this specifically for a control volume consisting of a fluid element. Following this we will employ the Reynolds transport theorem which we have already discussed, and an argument analogous to that used in deriving the continuity equation to obtain the differential form of the momentum equations. We then develop a multi-dimensional form of Newton’s law of viscosity to evaluate surface forces appearing in this equation and finally arrive at the N.–S. equations. 3.4.1 A basic force balance; Newton’s second law of motion We begin by recalling that because we cannot readily view fluids as consisting of point masses, it is not appropriate to apply Newton’s second law of motion in the usual form F = ma . Instead, we will use a more general form expressed in words as braceleftbigg time rate of change of momentum of a material region bracerightbigg = braceleftbigg sum of forces acting on the material region bracerightbigg . The somewhat vague terminology “material region” is widely used, and herein it will usually be simply a fluid element. But later when we develop the control-volume momentum equation the material region will be any region of interest in a given flow problem. We also remark that we are employing the actual version of Newton’s second law instead of the one usually presented in elementary physics. Namely, if we recall that momentum is mass times velocity, e.g. , mu in 1D, then the general statement of Newton’s second law is F = d ( mu ) dt , which collapses to the usual F = ma in the case of point masses that are independent of time. At this point it is worthwhile to recall the equation for conservation of mass, Eq. (3.20), which we write here in the abbreviated form ρ t + ( ρu ) x + ( ρv ) y + ( ρw ) z = 0 ,
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70 CHAPTER 3. THE EQUATIONS OF FLUID MOTION containing the dependent variables ρ , u , v and w . It will be convenient to express the momentum equations in terms of these same variables, and to this end we first observe that the product, e.g. , ρu , is momentum per unit volume (since ρ is mass per unit volume). Thus, yet another alternative expression of Newton’s second law is F/V = d dt ( ρu ) , or force per unit volume is equal to time-rate of change of momentum per unit volume . We are now prepared to develop formulas for the left- and right-hand sides of the word formula given above.
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