Bernoulli’s equation for the flow of an ideal fluid, which is discussed in Chap. 5 , can be written

$\frac{p}{\gamma}+z+\frac{V^2}{2 g}=\text { constant }$ ( 5.7)

where $p=$ pressure, $\gamma=$ specific weight, $z=$ elevation, $V=$ mean flow velocity, and $g=$ acceleration of gravity. Demonstrate that this equation is dimensionally homogeneous, i.e., that all terms have the same dimensions

Question

Bernoulli's equation for the flow of an ideal fluid, which is discussed in Chap. 5 , can be written

[latex] \frac{p}{\gamma}+z+\frac{V^2}{2 g}= ext { constant }[/latex] ( 5.7)

where [latex]p=[/latex] pressure, [latex]\gamma=[/latex] specific weight, [latex]z=[/latex] elevation, [latex]V=[/latex] mean flow velocity, and [latex]g=[/latex] acceleration of gravity. Demonstrate that this equation is dimensionally homogeneous, i.e., that all terms have the same dimensions

Accepted Answer

Term 1: Dimensions of [latex]\frac{p}{\gamma}=\frac{F / L^2}{F / L^3}=L[/latex]

Term 2: Dimensions of [latex]z=L[/latex]

Term 3: Dimensions of [latex]\frac{V^2}{2 g}=\frac{(L / T)^2}{L / T^2}=L[/latex]

So all the terms have the same dimensions, [latex]L[/latex], which must also be the dimensions of the constant at the right-hand side of Eq. (5.7).

Chapter 1

Q. 1.SP.1

Q. 1.SP.1

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