By Frank M. White

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**Example text**

What is the excess pressure within the bubble? 0712 N/m. For a droplet or bubble with one spherical surface, from Eq. 0712 N/m) = ≈ 28500 Pa R (5E−6 m) Ans. 63 Derive Eq. 37) by making a force balance on the fluid interface in Fig. 9c. Solution: The surface tension forces YdL1 and YdL2 have a slight vertical component. Thus summation of forces in the vertical gives the result å Fz = 0 = 2YdL 2 sin(dθ1 /2) Fig. 9c + 2YdL1 sin(dθ 2 /2) − ∆p dA But dA = dL1dL2 and sin(dθ/2) ≈ dθ/2, so we may solve for the pressure difference: ∆p = Y æ dθ æ 1 dL 2 dθ1 + dL1dθ 2 dθ ö 1 ö = Yç 1 + 2 ÷ = Yç + ÷ dL1dL 2 è dL1 dL 2 ø è R1 R 2 ø Ans.

65 The system in Fig. 65 is used to estimate the pressure p1 in the tank by measuring the 15-cm height of liquid in the 1-mm-diameter tube. The fluid is at 60°C. Calculate the true fluid height in the tube and the percent error due to capillarity if the fluid is (a) water; and (b) mercury. Ans. Fig. 65 Solutions Manual • Fluid Mechanics, Fifth Edition 36 Solution: This is a somewhat more realistic variation of Ex. 9. 25 cm (+22% error) Ans. 91cm ( −6%error) Ans. 66 A thin wire ring, 3 cm in diameter, is lifted from a water surface at 20°C.

He lost his sight in 1766 but continued to work, aided by a prodigious memory, and produced a vast output of scientific papers, dealing with mathematics, optics, mechanics, hydrodynamics, and celestial mechanics (for which he is most famous today). His famous paper of 1755 on fluid flow derived the full inviscid equations of fluid motion (Eqs. 36 of the text) now called Euler’s equations. He used a fixed coordinate system, now called the Eulerian frame of reference. The paper also presented, for the first time, the correct form of Bernoulli’s equation (Eq.