Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 10, Problem 19P
To determine

The velocities with actual form of stocks theorem and compare with actual determined value.

Expert Solution & Answer
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Answer to Problem 19P

The theoretical velocities for all different cases are 3.121mm/s, 12.12mm/s and 55.56mm/s. The difference between experimental and theoretical velocities for all three cases are 0.079mm/s, 0.68mm/s and 4.84mm/s.

Explanation of Solution

Given information:

Actual stokes law is FD=3πμDV+(9π/16)ρV2D2. Three aluminum balls of diameters 2mm, 4mm and 10mm.

Concept used:

For constant velocity of ball under glycerin, the relation for summation of forces on the ball is expressed as follows:

  FD+FB=W...... (1)

Here, drag force is FD, Buoyant force is FB, Weight of the ball is W.

Calculation:

Substitute 3πμDV+(9π/16)ρV2D2 for FD, ρalg×πD36 for W and ρglg×πD36 for FB in equation (1).

  3πμDV+(9π/16)ρV2D2+ρglg×πD36=ρalg×πD36( 9π 16ρ glD)V2+3μV+D2g6(ρ glρ al)=0

For first case:

Substitute 2mm for D, 1260Kg/m3 for ρgl, 2700Kg/m3 for ρal and 1Kg-m/s for μ in above equation.

  ( 9π 16×1260×( 2mm)( 1m 1000mm ))V2+3×1×V+ ( ( 2mm )( 1m 1000mm ) )2×9.816(12602700)=04.4532V2+3V0.0094176=0

On solving quadratic equation, the value of V are 0.003121m/s and 0.6768m/s. Velocity cannot negative for this case. So, the theoretical velocity for 2mm diameter ball is 0.003121m/s or 3.121mm/s.

Difference between actual and theoretical velocity is calculated as follows:

  ΔV=VexperimentalVtheoreticalΔV=3.2mm/s3.121mm/sΔV=0.079mm/s

For second case:

Substitute 4mm for D, 1260Kg/m3 for ρgl, 2700Kg/m3 for ρal and 1Kg-m/s for μ in above equation.

  ( 9π 16×1260×( 4mm)( 1m 1000mm ))V2+3×1×V+ ( ( 4mm )( 1m 1000mm ) )2×9.816(12602700)=08.9064V2+3V0.03767=0

On solving quadratic equation, the value of V are 0.01212m/s and 0.3489m/s. Velocity cannot negative for this case. So, the theoretical velocity for 4mm diameter ball is 0.01212m/s or 12.12mm/s.

Difference between actual and theoretical velocity is calculated as follows:

  ΔV=VexperimentalVtheoreticalΔV=12.8mm/s12.12mm/sΔV=0.68mm/s

For Third case:

Substitute 10mm for D, 1260Kg/m3 for ρgl, 2700Kg/m3 for ρal and 1Kg-m/s for μ in above equation.

  ( 9π 16×1260×( 10mm)( 1m 1000mm ))V2+3×1×V+ ( ( 10mm )( 1m 1000mm ) )2×9.816(12602700)=022.26V2+3V0.23544=0

On solving quadratic equation, the value of V are 0.05556m/s and 0.190339m/s. Velocity cannot negative for this case. So, the theoretical velocity for 10mm diameter ball is 0.05556m/s or 55.56mm/s.

Difference between actual and theoretical velocity is calculated as follows:

  ΔV=VexperimentalVtheoreticalΔV=60.4mm/s55.56mm/sΔV=4.84mm/s

Thus, the theoretical velocities for all different cases are 3.121mm/s, 12.12mm/s and 55.56mm/s. The difference between experimental and theoretical velocities for all three cases are 0.079mm/s, 0.68mm/s and 4.84mm/s.

Conclusion:

The theoretical velocities for all different cases are 3.121mm/s, 12.12mm/s and 55.56mm/s. The difference between experimental and theoretical velocities for all three cases are 0.079mm/s, 0.68mm/s and 4.84mm/s.

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