Solution_HW3 - Homework Set #3: SOLUTIONS Problem #1:...

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Homework Set #3: SOLUTIONS Problem #1: Discussed in class Problem #2: Solve problem 2A.2 at the end of Chapter 2 of BSL. Answer: Equation 2.3-21 is expressed as such: ( ) 4 PR w 8L π Δρ = μ The capillary radius can be solved for: 4 8Lw R P μ = πρΔ Inserting the data renders: () ( ) 53 13 4 2 4 4 35 8 4.03x10 0.5002 2.997x10 R 3.186x10 7.51x10 m 7.51x10 cm 0.9552x10 4.829x10 −− == = = π You can check the result by finding the Reynolds number: ( ) ( ) 3 z 45 3 2 2.997x10 Du 4w 2w Re 66.0 DR 7.51x10 4.03x10 0.9552x10 ρ = = = μπ μ π ν ρ π Therefore, the flow is laminar. Some difficulties with this type of measurement are: 1. Inability to account for departures from a straight, cylindrical wall geometry. 2. Inability to account for spatial and temporal variations of temperature, and therefore fluid density and viscosity. A simpler method is to measure the length L and mass m of a small slug of liquid mercury (or another liquid of known density) injected into the tube, and calculate the mean radius R of the slug as [] 1/2 m R L ⎛⎞ = ⎜⎟ ρπ ⎝⎠ , on the assumption that the slug is a right circular cylinder. This method allows comparisons of mean R values for various intervals of the tube length. Problem #3: Discussed in class For x-momentum rate transferred across z-plane at z = 0 (influxes)
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() ( ) z zx z x x x du Wx uu dx τρ μ ⎛⎞ Δ+ = Δ ⎜⎟ ⎝⎠ This rate is constant in z direction, thus the net rate is zero x-momentum rate transferred across y-plane is zero For x-momentum rate transferred across x-plane at x = x (influxes) and at x+ x (outfluxes) ( ) ( ) ( ) ; x xx x x x x x WL p u u WL p WL p ++ = x-momentum balance ( ) ( )( ) ( ) sin 0 x g x x WL p p F WL p p WL x g ρβ −+ = Δ = This equation just shows that pressure is a unique function of x which is not needed for determining velocity profile. Therefore, based on the assumptions discussed in the lecture, z- momentum balance is sufficient to determine velocity profile. Problem #4: Problem 2B.3 at the end of chapter 2 of BSL, parts (a), (b) and (c) only. Follow the terminology in the diagram in Figure 2B.3 on p. 63. Note that x = 0 in the center of the slit in this figure.
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This note was uploaded on 04/17/2008 for the course PGE 322K taught by Professor Dicarlo during the Fall '08 term at University of Texas at Austin.

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Solution_HW3 - Homework Set #3: SOLUTIONS Problem #1:...

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