327lec-momeq2 - 530.327 Introduction to Fluid Mechanics Su...

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Unformatted text preview: 530.327 - Introduction to Fluid Mechanics - Su Momentum equation – The effect of moving reference frames. Reading: Text, § 4.4, § 4.5. In using the momentum equation, we want to keep in mind two things: • The derivation of the Reynolds transport theorem, which was our starting point for arriving at both the equation for mass conservation and the momentum equation, is totally valid for moving control volumes as long as all velocities are measured relative to the moving control volume . As long as you use relative velocities that way, the equations basically can’t tell that the control volume is moving. • Newton’s second law, that the rate of change of the linear momentum of a system equals the net forces on that system, is only valid when the linear momentum is measured in an inertial (non-accelerating) reference frame. Thus, even though the Reynolds transport theorem is fine for any moving control volumes, the momentum equation as we’ve seen it only works for non-accelerating control volumes. To consider accelerating control volumes, we will need to make further modifications. Example 1. Moving, inertial reference frames. (This example is drawn from J.A. Roberson and C.T. Crowe, Engineering Fluid Mechanics , 3rd ed.) d V J H O 2 r =1000 kg/m 3 V S x y Figure 1: Schematic for example 1. Figure 1 shows a tank of water resting on a wheeled sled. A jet of water, with diameter d , issues from the orifice on the side of the tank. Assume that top of the tank is filled with pressurized air, and that the water jet velocity relative to the tank , V J , is fixed. The sled moves to the left at a constant speed, V S . The ground imposes a horizontal force, F gr , on the sled through the wheels (the exact nature of this force is not important). What is the magnitude and direction of this force? We’ll specify the control volume to be a rectangle containing the entire tank and sled, which cuts perpendicularly through the water jet, and with the bottom of the control surface being adjacent to the ground. The control volume moves at the same speed as the tank and sled; the speed is constant, so the control volume represents an inertial reference frame. The general form of the momentum equation is “I” z }| { −→ F CV = “II” z }| { ∂ ∂t Z CV ρ −→ V dV + “III” z }| { Z CS ρ −→ V ( −→ V · −→ dA ) . (1) 1 All of the velocities in (1) are measured relative to the moving control volume, as discussed above. Equation 1 is exactly the same momentum equation we’ve previously used for stationary control volumes; we are able to apply it here also because physics tells us that an observer in an moving reference frame can’t tell that the reference frame is moving as long as the motion is inertial. This is also why we need to use velocities measured in the frame of reference of the moving control volume. We further observe that only the x-component of this vector equation will be important, since the sled only moves in the horizontal plane.since the sled only moves in the horizontal plane....
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327lec-momeq2 - 530.327 Introduction to Fluid Mechanics Su...

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