27.5: Compressibility of a Fluid
- Page ID
- 28630
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When the pressure is uniform on all sides of an object in a fluid, the pressure will squeeze the object resulting in a smaller volume. When we increase the pressure by \(ΔP\) on a material of volume \(v_{o}\), then the volume of the material will change by \(ΔV < 0\) and consequently the density of the material will also change. Define the bulk stress by the increase in pressure change
\[\sigma_{B} \equiv \Delta P \nonumber \]
Define the bulk strain by the ratio
\[\varepsilon_{B} \equiv \frac{\Delta V}{V_{0}} \nonumber \]
For many materials, for small pressure changes, the bulk stress is linearly proportional to the bulk strain,
\[\Delta P=-B \frac{\Delta V}{V_{0}} \label{27.5.3} \]
where the constant of proportionality \(B\) is called the bulk modulus. The SI unit for bulk modulus is the pascal. If the bulk modulus of a material is very large, a large pressure change will result in only a small volume change. In that case the material is called incompressible. In Table 27.2, the bulk modulus is tabulated for various materials. \[\text {Table 27.2 Bulk Modulus for Various Materials} \nonumber \]
\[\begin{array}{|l|l|}
\hline \text { Material } & \text { Bulk Modulus, } Y \text { , (Pa) } \\
\hline \text { Diamond } & 4.4 \times 10^{11} \\
\hline \text { Iron } & 1.6 \times 10^{11} \\
\hline \text { Nickel } & 1.7 \times 10^{11} \\
\hline \text { Steel } & 1.6 \times 10^{11} \\
\hline \text { Copper } & 1.4 \times 10^{11} \\
\hline \text { Brass } & 6.0 \times 10^{10} \\
\hline \text { Aluminum } & 7.5 \times 10^{10} \\
\hline \text { Crown Glass } & 5.0 \times 10^{10} \\
\hline \text { Lead } & 4.1 \times 10^{10} \\
\hline \begin{array}{l}
\text { Water (value increases } \\
\text { at higher pressure) }
\end{array} & 2.2 \times 10^{9} \\
\hline \begin{array}{l}
\text { Air (adiabatic bulk } \\
\text { modulus) }
\end{array} & 1.42 \times 10^{5} \\
\hline \begin{array}{l}
\text { Air (isothermal bulk } \\
\text { modulus) }
\end{array} & 1.01 \times 10^{5} \\
\hline
\end{array} \nonumber \]
Example 27.3: Compressibility of Water
Determine the percentage decrease in a fixed volume of water at a depth of 4 km where the pressure difference is 40 Mpa, with respect to sea level.
Solution
The bulk modulus of water is \(2.2 \times 10^{9} \text Pa\). From Equation \ref{27.5.3},
\[\frac{\Delta V}{V_{0}}=-\frac{\Delta P}{B}=-\frac{40 \times 10^{6} \mathrm{Pa}}{2.2 \times 10^{9} \mathrm{Pa}}=-0.018. \nonumber \]
There is only a 1.8% decrease in volume. Water is essentially incompressible even at great depths in ocean, justifying our assumption that the density of water is uniform in the ocean in Example 27.1.