Chapter Summary

10.1 Flow Rate and Its Relation to Velocity

Flow rate [latex]Q[/latex] is defined to be the volume [latex]V[/latex] flowing past a point in time[latex]t[/latex], or[latex]Q = \frac{V}{t}[/latex] where
[latex]V[/latex] is volume and[latex]t[/latex] is time.
The SI unit of volume is [latex]\text{m}^{3}[/latex].
Another common unit is the liter (L), which is [latex]\text{10}^{- 3} \text{m}^{3}[/latex].
Flow rate and velocity are related by [latex]Q = A \bar{v}[/latex] where [latex]A[/latex] is the cross-sectional area of the flow and[latex]\bar{v}[/latex] is its average velocity.
For incompressible fluids, flow rate at various points is constant. That is, [latex]\left\begin{matrix} Q_{1} = Q_{2} \\ A_{1} \left(\bar{v}\right)_{1} = A_{2} \left(\bar{v}\right)_{2} \\ n_{1} A_{1} \left(\bar{v}\right)_{1} = n_{2} A_{2} \left(\bar{v}\right)_{2} \end{matrix}\right} .[/latex]

Bernoulli’s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid:
[latex]P_{1} + \frac{1}{2} ρv_{1}^{2} + \rho gh_{1} = P_{2} + \frac{1}{2} ρv_{2}^{2} + \rho \text{gh}_{2} .[/latex]
Bernoulli’s principle is Bernoulli’s equation applied to situations in which depth is constant. The terms involving depth (or height h ) subtract out, yielding
[latex]P_{1} + \frac{1}{2} ρv_{1}^{2} = P_{2} + \frac{1}{2} ρv_{2}^{2} .[/latex]
Bernoulli’s principle has many applications, including entrainment, wings and sails, and velocity measurement.

Power in fluid flow is given by the equation [latex]\left(P_{1} + \frac{1}{2} ρv^{2} + \rho \text{gh}\right) Q = \text{power} ,[/latex] where the first term is power associated with pressure, the second is power associated with velocity, and the third is power associated with height.

Laminar flow is characterized by smooth flow of the fluid in layers that do not mix.
Turbulence is characterized by eddies and swirls that mix layers of fluid together.
Fluid viscosity [latex]\eta[/latex] is due to friction within a fluid. Representative values are given in Table 10.1. Viscosity has units of [latex]\left(\right. \text{N}/\text{m }^{2} \left.\right) \text{s }[/latex] or [latex]\text{Pa } \cdot \text{s }[/latex].
Flow is proportional to pressure difference and inversely proportional to resistance:
[latex]Q = \frac{P_{2} - P_{1}}{R} .[/latex]
For laminar flow in a tube, Poiseuille’s law for resistance states that
[latex]R = \frac{8 \eta l}{πr^{4}} .[/latex]
Poiseuille’s law for flow in a tube is
[latex]Q = \frac{\left(\right. P_{2} - P_{1} \left.\right) \pi r^{4}}{8 \eta l} .[/latex]
The pressure drop caused by flow and resistance is given by
[latex]P_{2} - P_{1} = R Q .[/latex]

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