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4-8. Unsteady flow applications are considered in Chaps. 9 and 10. 1 Derive Eq. 8b). 2 With a equal to the acceleration DV/Dt, determine Da/Dt in terms of partial derivatives with respect to x and t. 3 Consider a body in a wind tunnel, as shown in the sketch: where Fx is the x component of the force on the body by the fluid. Assume steady, one-dimensional flow with no body forces and negligible shear on the wind-tunnel wall, and derive an equation for Fx. Assume that flow conditions are uniform at xl and ;c2, and start with the second equation in the subsection headed "Momentum," Sec.
3 Forces on the differential element of Fig. 2. x)2) terms are ignored. This component is zero only when ( d A / d x ) = 0. The surface pressure forces sum to dA . dA , dp , •-TJ— d^: 4- Fp^— dx = — -5—AA dx --A&dxdx dx dx OX —— JTI r\ VJ. 9) Observe that the acceleration DV/Dt is positive when the pressure gradient is negative. A decreasing pressure, called a favorable pressure gradient, thus accelerates a particle of fluid. Energy The elemental slice of material in Fig. 2 is a particle of fluid that constitutes a closed adiabatic system for which the first law of thermodynamics applies.
Note that (1 + z)n = 1 + nz + • • • for small z. This page intentionally left blank 4. STEADY STREAMTUBE FLOW The equations from Chap. 3 are next applied to a steady, adiabatic streamtube flow in which _i = 0 3t ' ^= A dx dx' 2_ Dt y d_ dx For this application, Eq. 1) and the entropy is a constant. Recall that with a constant entropy the flow is isentropic. The analysis is simplified by assuming a perfect gas and introducing the Mach number. , with the Mach number as the independent variable. These streamtube results are explored in what follows and are summarized in Appendix C.