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Find the variation in pressure and density

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"Sample Work Shikhar Mittal May 22, 2017 Question Find the variation in pressure and density with the depth(h) in a cylinder \u0000lled with a \u0000uid. The \u0000uid is NOT incompressible and its compressibility is \u0000 . The uid’s \u0000 free su..

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"Sample Work Shikhar Mittal May 22, 2017 Question Find the variation in pressure and density with the depth(h) in a cylinder \u0000lled with a \u0000uid. The \u0000uid is NOT incompressible and its compressibility is \u0000 . The uid’s \u0000 free surface is open to vacuum and the density at the free surface is \u0000 . o 1Sample Work Shikhar Mittal May 22, 2017 Solution From the de\u0000nition of isothermal compressibility, \u0000 \u0000 1 d\u0000 \u0000 = T \u0000 dP T d\u0000 \u0000 dP = T \u0000 Z Z P \u0000 d\u0000 \u0000 dP = T \u0000 0 \u0000 o \u0000 \u0000 \u0000 \u0000 P = ln (1.1) T \u0000 o dP =\u0000g dx Put the density from (??) in the above equation, \u0000 P T dP =\u0000 e g dx o Z Z P y \u0000 \u0000 P T e dP = \u0000g dx 0 0 \u0000 \u0000 P \u0000 \u0000 P T e \u0000 =\u0000 gy o \u0000 T 0 \u0000 \u0000 P T 1\u0000 e =\u0000 gy o \u0000 \u0000 1 P = ln(1\u0000 \u0000 \u0000 gy) T o \u0000 T \u0000 1 Note: For incompressible \u0000uid( \u0000 approaching 0), lim ln(1\u0000 \u0000 \u0000 gy) =\u0000 gy as expected. T T o o \u0000 !0 T \u0000 T 1"

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