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Chapter 9: Problem 48

Determine the drag on a small circular disk of \(0.01-\mathrm{ft}\) diameter moving \(0.01 \mathrm{ft} / \mathrm{s}\) through oil with a specific gravity of 0.87 and a viscosity 10,000 times that of water. The disk is oriented normal to the upstream velocity. By what percent is the drag reduced if the disk is oriented parallel to the flow?

Short Answer

Expert verified
will need to calculate the Reynolds number and assess whether the flow is laminar, turbulent, or transition. Using this information, the drag forces can be calculated under different orientations of the disk in relation to the flow. Then the percentage reduction in drag can be obtained.

Step by step solution

01

Identify Knowns and Unknowns

Known is: diameter d=0.01 ft, speed V=0.01 ft/s, the density of the oil (as specific gravity is given as 0.87 and density of water is known), and the viscosity of the oil is 10,000 times of water (whose viscosity is roughly \(1 \times 10^{-3} \, \mathrm{Pa.s}\) at room temperature). The unknown is the drag force in both scenarios and the reduction percentage between those scenarios.
02

Calculate Reynolds number

Reynolds number (\(Re\)) is a dimensionless quantity that helps predict flow patterns. The formula for the Reynolds number is \(Re=\frac{VDρ}{µ}\), where V is the velocity, D is the diameter of the disk, ρ is the density of the fluid and µ is the dynamic viscosity. First will need to convert the viscosity and the specific gravity to a consistent unit system. Once the conversion is done, substitute those values into the formula to calculate Reynolds number.
03

Calculate Drag Forces

In these conditions, the drag force acting on the disc when it is oriented normal to the upstream velocity can be calculated using Stokes Law (\(F_D=3πDµV\)) and when the disc is oriented parallel to the flow, the drag force can be calculated using the formula (\(F_D=1.6πDµV\)). Input the calculated viscosity along with other values into both formulas to get the respective drag forces.
04

Calculate percentage reduction

To find how much the drag is reduced, subtract the drag when the disk is parallel from the drag when the disk is normal, divide by the drag when the disk is normal, and then multiply by 100% to convert to percentage.

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