MIT Department of Mechanical Engineering 2.25 Advanced Fluid Mechanics Problem 1.10 This problem is from “Advanced Fluid Mechanics Problems” by A.H. Shapiro and A.A. Sonin The Swiss scientist Auguste Picard developed a navigable diving vessel, the “bathyscape”, to investigate the ocean at great depths (http://en.wikipedia.org/wiki/Bathyscaphe). In 1960, his son Jacques, accompanied by Lt. Don Walsh of the U.S. Navy, reached a depth of 10, 916 m in the Paciﬁc’s Mariana Trench. Suppose that the ocean is at constant temperature, has a density of 1030 kg/m3 at sea level, and is charac terized by a constant isothermal bulk compressibility κT ≡ 1 ρ ∂ρ ∂p = 4.6 × 10−10 m2 /N. (1.10a) T Compute the pressure at a depth of 11 km, (a) assuming the density is constant at the sea level value, and (b) taking the water’s compressibility into account. For part (b), derive an expression for the pressure as a function of depth below the surface, considering the sea level density ρ0 and pressure p0 , as well as κT , as given quantities. Courtesy of the U.S. Naval History Center. Photograph in the public domain. 1 Solution: (a) Assuming the density is constant at the sea level value and the pressure at sea level is p0 = 1.01×105 Pa, we ﬁnd that p = p0 − ρgh (1.10b) 5 3 2 = 1.01 × 10 Pa − (1030 kg/m )(9.8 m/s )(−11, 000 m) = 1.11 × 108 Pa = 111 MPa (b) Taking the water’s compressibility into account, the density of water ρ will vary with pressure p. First we will solve the compressibility equation [Eq. (1.10a)] by separating variables to get ρ in terms of p. p ρ 1 dρ ρ κT dp = p0 ρ0 κT (p − p0 ) = ln ρ − ln ρ0 = ln ρ ρ0 Solving for ρ, we ﬁnd: ρ = ρ0 eκT (p−p0 ) (1.10c) At this point, you may be inclined to substitute this expression for ρ into the pressure equation [Eq. (1.10b)] used in part (a). However, we note that this pressure distribution assumes a constant density ρ (see Kundu & Cohen [K&C] pp.11). Instead, we use the more general form of the pressure gradient [Eq. (1.8) in K&C] and substitute Eq. (1.10c) to give dp = −ρg dz = −ρ0 eκT (p−p0 ) g. Again, we separate variables and integrate: p h e −κT (p−p0 ) p0 1 − κT e dp = − ρ0 gdz 0 −κT (p−p0 ) − 1 = −ρ0 gh After some algebra, we ﬁnally have p = p0 − 1 ln(1 + κT ρ0 gh) κT = 1.01 × 105 Pa − (1.10d) 1 ln 1 + (4.6 × 10−10 )(1030)(9.8)(−11, 000) 4.6 × 10−10 m2 /N = 114 MPa (1.10e) As a check on our pressure equation [Eq. (1.10d)], take the limit as x = κT ρ0 gh is small. Note that ln(1 + x) ≈ x for small values of x. Thus, 1 ln(1 + κT ρ0 gh) κT 1 ≈ p0 − κT ρ0 gh κT = p0 − ρ0 gh p = p0 − 2 Note, the equation above is the the same as the incompressible pressure equation [Eq. (1.10b)] in part (a). D 3