Bernoulli's and Pascal’s Law

Principle

The static (no flow) hydrostatic pressure in a tube is visualized in Fig. 1 and is given by Pascal’s Law:

P₁+ ρgh₁ = P₂+ ρgh₂ = constant, (1)

where

ρ = fluid density (kg/m³);

g = acceleration due to gravity on Earth (m/s²);

h = height from an arbitrary point in the direction of gravity (m).

P = pressure somewhere imposed in the tube (N/m², Pa).

The second term is due to gravity. Fig. 1 illustrates the hydrostatic condition in the tube.

Fig. 1 Visualizing of Pascal’s law and Bernoulli’s equation in a tube.

By adding an inertia term, ½ρv², the kinetic energy per unit mass of the moving liquid with velocity v, Pascal’s Law evolves to Bernoulli's equation:

½ρv² + ρgh + P = constant , (2)

Resuming, in fluid dynamics, Bernoulli's equation, describes the behavior of a fluid (gases included) moving in the tube. More precisely, the flow is along a streamline (a line which is everywhere perpendicular to the local velocity flow). There are typically two different formulations of the equations; one applies to incompressible fluids and (2) holds. The other applies to compressible fluids (see More Info).

These assumptions must be met for the equation to apply:

· inviscid flow: viscosity (internal friction) = 0;

· steady flow;

· incompressible flow: ρ is constant along a streamline. However, ρ may vary from streamline to streamline;

· for constant-density steady flow, it applies throughout the entire flow field. Otherwise, the equation applies along a streamline.

A decrease in pressure occurs simultaneously with an increase in velocity, as predicted by the equation, is often called Bernoulli's principle.

Bernoulli’s equation (2) under the above conditions and 1D is a special, simple case of the Navier-Stokes equations.

Applications

In devices and instruments to calculate pressure in gas and liquid flows. In science, the Bunsen burner, the water-jet pump (venturi-principle), the Pitot tube, In the (path)physiology of the vascular and airways system to calculate and measure pressure, for instance before and after a stenosis (see Flow through a stenosis). For flow in the vascular and airways system many refinements are needed as those of More Info, and even more than that. Two approaches are feasible, a semi-analytic approach with applying practical ‘rules’ of technical liquid dynamics or a numerical approach with mathematical tools like the finite element approach.

More info

A second, more general form of Bernoulli's equation may be written for compressible fluids, in which case, following a streamline, we have:

v²/2 + φ + w = constant (3)

Here, φ is the gravitational potential energy per unit mass, which is just φ = gh in the case of a uniform gravitational field, and w is the fluid enthalpy per unit mass, which is also often written as h. Further it holds that:

w = ε + P/ρ. (4)

ε is the fluid thermodynamic energy per unit mass, also known as the specific internal energy or "sie". With constant P, the term on the right hand side is often called the Bernoulli constant and denoted b. For steady inviscid adiabatic flow (see Gas Laws) with no additional sources or sinks of energy, b is constant along any given streamline. Even more generally when b may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid.

Classically, the pressure drop over a tube (volume flow Q, length L and diameter d) with incompressible and with compressible flow is:

ΔP = k₁Q + k₂Q² (5)

The first term applies for the laminar, the viscous component of the resistance and the 2^nd for the turbulent part. Both are not implied in (2) and (3). The laminar part comprises the friction within the tube, and extra friction introduced by irregularities like non-smoothness of the tube wall, constrictions (see Flow through a stenosis), bends (see Flow in a bended tube) and bifurcations (see Flow in bifurcations). The same irregularities can also contribute to the second term when the flow through the irregularities is turbulent on basis of the Reynolds number. For transitional flow both terms contribute.

The influence of non-smoothness of the tube wall is expressed in a resistance factor λ. For Re<2300 it is irrelevant (always laminar). For 4000<Re<10⁵ the pressure drop ΔP is given by the Darcy-Weisbach equation:

ΔP = 0.5λ∙ρ∙L∙v²/d = 8∙π^–2∙λ∙ρ∙L∙d^–5∙Q², (6)

where

λ = 0.316/Re^0.25 (according to the Blasius boundary layer).

For gas flow in a smooth tube there is an expression covering the various types of flow:

ΔP_tube = 8∙π ^–7/4∙ η^1/4∙ρ^3/4∙L∙d^–19/4∙Q^7/4, (7)

where η is the dynamic gas viscosity (Pa∙s).

Literature

Wikipedia

Van Oosterom, A and Oostendorp, T.F. Medische Fysica, 2^nd edition, Elsevier gezondheidszorg, Maarssen, 2001.