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Chapter 9: Problem 37

A model is placed in an airflow at standard conditions with a given velocity and then placed in water flow at standard conditions with the same velocity. If the drag coefficients are the same between these two cases, how do the drag forces compare between the two fluids?

Short Answer

Expert verified
The drag force in water under the same conditions is approximately 816 times the drag force in air. This is because the drag force is directly proportional to the fluid's density, and water is much denser than air.

Step by step solution

01

Understanding the Drag Force Equation

The drag force equation, given as \(F_d = 0.5 * Cd * rho * A * V^2\), where \(F_d\) is the drag force, \(Cd\) is the drag coefficient, \(rho\) is the fluid's density, \(A\) is the area upon which force is applied, and \(V\) is the flow velocity.
02

Application of the Drag force equation for air

Substitute the known values into the equation for air. The drag coefficient \(Cd\) and the velocity \(V\) are the same as in the case of water. The density of air under standard conditions is approximately \(1.225 kg/m^3\). The area \(A\) will be same in both conditions, as it depends on the model not on the fluid. Let's denote the drag force in air as \(F_{d_{air}}\). So, \(F_{d_{air}} = 0.5 * Cd * 1.225 * A * V^2\).
03

Application of the Drag force equation for water

Substitute the known values into equation for water. As mentioned before, the drag coefficient \(Cd\) and the velocity \(V\) remain the same. The density of water under standard conditions is approximately \(1000 kg/m^3\). The area \(A\) remains the same. Let's denote this drag force in water as \(F_{d_{water}}\). So, \(F_{d_{water}} = 0.5 * Cd * 1000 * A * V^2\).
04

Comparing the drag forces in air and water

By looking at the two equations for \(F_{d_{air}}\) and \(F_{d_{water}}\), it can be clearly observed that, since all other factors are the same, the drag force is directly proportional to the fluid's density. Hence, \(F_{d_{water}} = F_{d_{air}} * (1000/1.225)\). This means that drag force in water is approximately 816 times the drag force in air under same conditions. This is due to the much higher density of water compared to air.

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Most popular questions from this chapter

A full-sized automobile has a frontal area of \(24 \mathrm{ft}^{2},\) and a compact car has a frontal area of \(13 \mathrm{ft}^{2}\). Both have a drag coefficient of 0.5 based on the frontal area. Find the horsepower required to move each automobile along a level road in still air at 55 mph. Assume that the power required to deform the tires continuously at this speed (called rolling resistance) is equal to the power to overcome the air resistance. Estimate the gas mileage of both automobiles if the energy supplied to the drive wheels is \(\frac{1}{4}\) that available in the fuel. A gallon of fuel has \(1.0 \times 10^{8} \mathrm{ft} \cdot 1 \mathrm{b}\) available energy.

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