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  • Bernoulli's Equation

"Conservation of energy applied to a flowing fluid."

Bernoulli's Equation

In fact, Bernoulli's equation is an application of the concepts of work and conservation of energy. For an incompressible fluid, laminar (non-turbulent) flow and no viscous acting forces (inverse fluid), the fluid energy will be constant at any point in the flow. Bernoulli's equation relates the kinetic and potential energy of the fluid to the absolute pressure at any two points, for the same tube of currents.

Bernoulli's equation applied to an incompressible fluid
Beornoulli equation can be written as $$ P + \ rho gy + \ rho v ^ 2 = constant, $$ or, based on the figure, $$ P_1+\rho g y_1 +\rho (v_1)^2 =P_2+\rho g y_2 +\rho (v_2)^2,$$ where \(P\) is the pressure and the work-related term of the pressure applied in an area, \(\rho gy\) is the term for energy and potential of the fluid \(\rho (v)^2\), called the dynamic pressure, is the term connected to kinetic energy.

Consequence of the Bernoulli equation

Changes in fluid velocities are due to pressure differences. When the speed of a fluid increases, the pressure decreases and vice versa.

Applications of the Bernoulli Equation

Torricelli's Theorem
In the figure above it is possible to apply the Torricelli's Theorem if and only if the fluid is incompressible and the flow is laminar. For this to occur, it is important that the width of the tank is much larger than that of the spout through which the fluid flows.
A density fluid \(\rho\) is contained in a container, a small hole is made in the side wall of the container, at a distance \(h\) from the surface of the liquid. The horizontal velocity with which the liquid will flow through the orifice has modulus $$ v = \sqrt{2gh},$$ where \(g\) is the acceleration of gravity.
Pitot Tube
The Pitot Tube is widely used to determine air velocity in airplanes, water in boats, and velocity of liquids and gases in some industries.
It is an instrument that, through the measurement of the pressure difference, at two different positions (point A (A) and B (B) of the figure), it is possible to obtain the flow velocity of the medium. The formula is given by $$ v_{meio} = \sqrt{2gh} \sqrt{{{\rho_{_f} }\over{\rho_{_{meio}}}}-1},$$ where \(v_{meio}\) is the velocity of the external fluid (e.g., air), \(\rho_{f}\) is the fluid density inside the tube, \(\rho_{meio}\) is the density of the \(g\) is the acceleration of gravity and \(h\) is the height difference between points \(A\) and \(B\). The point \(A\) is called the "stagnation point," the velocity is zero at that point and inside the tube.
Venturi Tubes
The Venturi meter measures the flow velocity from the pressure drop in a reduced section of the current tube.

The velocity of the fluid will be given by $$ v_1 = \frac{A_2} {\sqrt{{A_1}^2-{A_2}^2}} \sqrt{2g \Delta H}, $$ where \(v_1\) is the velocity of the fluid in the non-strangulated section, \(A_1\) the cross-sectional area 1, \(A_2\) the cross-sectional area 2, \(g\) is the acceleration of gravity e \(\Delta H\) the difference in height.

Another Venturi meter model uses a U-tube, and is shown in the figure below.

In this Venturi meter, the flow velocity is measured from the pressure drop in another liquid (yellow) on an all in "u".
In this case, the velocity of the fluid will be given by $$ v_1 = \frac{A_2} {\sqrt{{A_1}^2-{A_2}^2}} \sqrt{\frac{2 \color{goldenrod}{\rho_2} g \Delta h}{\color{blue}{\rho_1}}}, $$ \(v_1\) is the velocity of the density fluid \ (\ color {blue} {\ rho_1} \) in the section hho {\ color {blue} {\ rho_1} \(A_1\) the cross-sectional area 1, \(A_2\) the cross-sectional area 2, \(g\) is the acceleration of gravity and \(\Delta h\) the height difference of density yellow liquid (\ color {goldenrod} {\ rho_2} \).


The viscosity is related to the friction between the molecules of a fluid, the greater the friction between the molecules the greater its viscosity. Honey under normal conditions is much more viscous than water. A fluid that has no viscosity is called an inverse. Zero viscosity is only observed at very low temperatures, in superfluids. Some fluids have such a high viscosity that they are considered solids, glass, tar, magma, etc.

Reynolds number ( \(Re\) )
The transition from laminar to turbulent flow depends on geometry, surface roughness, flow velocity, surface temperature and fluid type, among other things. Reynolds found that the flow regime depends mainly on the relationship between the inertial forces and the viscous forces of the fluid. The formula for the Reynolds number for the internal flow in a circular tube is: $$ Re= {{\rho D v}\over{\eta}} $$ where: Fluid density \(=\rho\) , Diameter of the tube \(=D\) fluid velocity \(=v\) , viscosity \(=\eta\).

Significance ratio = Inertial force / viscous force.

In most practical conditions, the flow in a circular tube is laminar to \(Re\) <2300, turbulent to \(Re\) > 4000 and transition between these values. Curiosity. In carefully controlled experiments, the laminar flow has been maintained for Reynolds numbers up to 100,000.

The table shows the viscosity (approximate) for some fluids.
Fluids Viscosity (Pa.s)
Helium (2K) 0
Air (20 ° C) 0,0000183
water (20 oC) 0,00100
Olive oil (20 oC) 0.084
Shampoo (20 oC) 100
Honey (20 oC) 1000
Common glass (540 oC) 10¹³
Grades. The Pascal.second (Pa.s) is the unit of viscosity in the SI, which is equivalent to kg / ms. Note that only in conditions of extremely low temperatures, Helium is able to present one of the characteristics of the "super fluids", in this case, zero viscosity.


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