Modeling Fluid Drag The drag force on an object moving through a liquid or a gas is a function of the velocity. A commonly used model of the drag force D on an object is D = 1 2 𝜌ACDv² (1) where 𝜌 is the mass density of the fluid, A is the object’s cross-sectional area normal to the relative flow

Question

Modeling Fluid Drag

The drag force on an object moving through a liquid or a gas is a function of the velocity. A commonly used model of the drag force D on an object is

[latex]D=\frac{1}{2}\rho AC_{D}v^2 \quad (1)[/latex]

where 𝜌 is the mass density of the fluid, A is the object’s cross-sectional area normal to the relative flow, v is the object’s velocity relative to the fluid, and [latex]C_D[/latex] is the drag coefficient, which is usually determined from wind-tunnel or water-channel tests on models. Curve A in Figure 1.3.6 is a plot of this equation for an Aerobee rocket 1.25 ft in diameter, with [latex]C_D[/latex] = 0.4, moving through the lower atmosphere where 𝜌 = 0.0023 slug/ft³, for which equation (1) becomes

[latex]D=0.00056v^2 \quad (2)[/latex]

a. Obtain a linear approximation to this drag function valid near v = 600 ft/sec.
b. Obtain a linear approximation that gives a conservative (high) estimate of the drag force as a function of the velocity over the range 0 ≤ v ≤ 1000 ft/sec.

Accepted Answer

a. The Taylor series approximation of equation (2) near v = 600 is

[latex]D=D|_{v=600} + \frac{dD}{dv}|_{v=600} \ \ (v-600)=201.6+0.672(v-600)[/latex]

This straight line is labeled B in Figure 1.3.6. Note that it predicts that the drag force will be negative when the velocity is less than 300 ft/sec, a result that is obviously incorrect. This illustrates how we must be careful when using linear approximations.

b. The linear model that gives a conservative estimate of the drag force (that is, an estimate that is never less than the actual drag force) is the straight-line model that passes through the origin and the point at v = 1000. This is the equation D = 0.56v, shown by the straight line C in Figure 1.3.6.

Question 1.3.4: Modeling Fluid Drag The drag force on an object moving throu...

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