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327lec-momeq1 - 530.327 Introduction to Fluid Mechanics Su...

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Unformatted text preview: 530.327 - Introduction to Fluid Mechanics - Su Applications of the momentum equation. Reading: Text, § 4.4 (The following examples are discussed in F.M. White, Fluid Mechanics , 5th ed.). Example 1. Vector nature of Newton’s second law. V 1 V 2 q 1 2 Figure 1: A streamtube embedded in a steady flow. This example emphasizes that Newton’s second law deals with vector quantities. Figure 1 depicts a streamtube in a steady flow, where a streamtube is defined as follows– • Streamtube : a closed surface whose sides are everywhere parallel to the local velocity vector. For convenience, we will also say that the surfaces labeled 1 and 2 on the ends of the streamtube are perpendicular to the local velocity vector, and that density and velocity are uniform on those surfaces. What we want to know is, what force must act on the streamtube to cause the flow to bend as shown? First, we will write the equation for conservation of mass for the system. (In general, even in problems where we’re concerned with forces and momenta, we need to start by ensuring that mass conservation is satisfied.) The general form of the conservation of mass is ∂ ∂t Z CV ρ dV + Z CS ρ ( −→ V · −→ dA ) = 0 . (1) We will define the control volume here as being exactly the streamtube depicted in Fig. 1. In this case, the flow is steady, so the first term in the above equation is zero. Also, the only mass inflow/outflow terms on the control surface are on surfaces 1 and 2, because the velocity is always...
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327lec-momeq1 - 530.327 Introduction to Fluid Mechanics Su...

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