The Bernoulli equation can be used in flows where the fluid can be considered to be
incompressible and where the pressure losses due to viscosity in the fluid are small compared to the changes in static pressure, elevation pressure and/or dynamic pressure.
In devices where there are significant changes in velocity, such as the flow meters considered earlier or in the discharge of fluids from a large reservoir through a small hole, then the Bernoulli equation can be applied with good accuracy. However, in situations such as flows in
A pitot-static probe is attached to the wing of a light aircraft flying at an altitude of 2000 m, where the air temperature is 2 °C and the pressure is 0.8 bar. The pressure difference measured by the probe is 3.3 kPa. Take the gas constant for air to be 287.1 J kgⴚ1Kⴚ1. What is the speed of the plane in miles per hour?
For a pitot-static probe p ps12 v2
v2 2(3.3 10)
R p
T
28 0 7 .8
.1
1
2 0 7
5
1.013 kg m5 1 Velocity 80.7 m s1 181 miles h1
Figure 3.67
Figure 3.68 3.3 kPa
h
1
2
long horizontal pipes of constant diameter (e.g. pipelines), the pressure changes along the pipe are dominated by frictional effects due to the viscosity of the fluid. These situations will be considered later in this chapter.
Liquids can be considered incompressible under most practical situations and so the Bernoulli equation can be applied. It is generally found that in gas flows where the velocity of the gas is less than 30 per cent of the sonic velocity (the speed of sound), then compressibility effects are generally negligible and the Bernoulli equation can be applied with good accuracy. However, at higher velocities the Bernoulli equation can give large errors, for example in the discharge of compressed air from pipes. As guide, the sonic velocity a in a gas is given by the equation a 兹RT苶, where R is the specific gas constant, T is the absolute temperature and is the ratio of the specific heat capacity at constant pressure to the specific heat capacity at constant volume for a gas. For air, is approximately 1.4.
The study of compressible flow in gases is beyond the scope of this book and readers are advised to consult a more advanced fluid mechanics text.
3.4 Viscous (real) fluids
Viscosity
A fluid offers resistance to motion due to its viscosity or ‘internal friction’. The greater the resistance to flow, the greater the viscosity. For example, water flows (can be poured) more readily than some oils and we say that oil has a higher viscosity.
Viscosity arises from two effects: in liquids, intermolecular forces act as drag between layers of fluid moving at different velocities, and this acts like friction; in gases, it is the mixing of molecules as they move, in continuous random motion, between regions of faster and slower moving fluid that causes friction due to momentum transfer. Slower layers tend to retard faster layers, hence resistance.
Viscosity can be defined in terms of the forces generated at rate of shear or velocity gradient.
Consider the straight and parallel flow of a fluid over a fixed, horizontal surface as shown in Figure 3.69. Fluid in direct contact with the surface has zero velocity because surface irregularities trap molecules of the fluid. A short distance from the surface, the fluid has a relatively low velocity, but in the free-stream region, the velocity is vf.
The higher the viscosity of a fluid, the greater is the resistance to motion between fluid layers, and, for a given applied shear stress, the lower is the rate of shear deformations between layers.
Figure 3.69Velocity gradient near a fixed surface
vf Zero velocity gradient,
∴ zero τ Free-stream
velocity, vf
Fluid flow
Fixed surface
Velocity profile
Velocity, v Maximum velocity gradient
∴ maximum τ vf
v +δv v v +δv
v+δy y
v δy
δv Distance perpendicular to surface, y
0
The velocity of flow increases continuously from zero at the fixed surface to vf, in the main stream and is usually represented by a smooth curve, known as the velocity profile. At a distance y, let the velocity be v and at a distance y y, let the velocity be v v. The ratio
yv
is the average velocity gradient (rate of change of velocity with distance) over the distance y, but, as y tends to zero,
yv ➝ the value of the differential d d
yv at a point such as point A;
d d
yv at point A is the true velocity gradient, or rate of shear, at point A.
For most fluids used in engineering, it is found that the shear stress is directly proportional to the velocity gradient when straight and parallel flow is involved. Thus:
d d
yv or constant冢ddyv冣
The constant of proportionality is called the dynamic viscosity or often just the viscosity of the fluid, and is denoted by . Hence
This is Newton’s law of viscosity, and fluids that obey it are known as Newtonian fluids.
The equation is limited to straight and parallel (laminar) flow. Only if the flow is of this form does dv represent the time rate of sliding of one layer over another. If angular velocity is involved, the velocity gradient is not necessarily equal to the rate of shear.
The equation indicates that, in the main stream where d d
yv 0, 0, and, at the fixed surface where d
d
yv has its maximum value, is a maximum.
The value of depends on the type of fluid, on molecular motion between fluid layers, and on inter-molecular (cohesive) forces. However, for a fluid exhibiting Newtonian properties, is independent of velocity gradient and shear stress but depends considerably on fluid temperature and, to a very small extent, on fluid pressure. For liquids decreases as temperature increases due to reduced cohesive forces between molecules. For gases, increases as temperature increases due to increasing momentum of molecules leading to greater momentum exchange.
In some fluids the viscosity varies with the rate of shear and these are known as non-Newtonianfluids. Figure 3.70 shows the relationship between
d d
yv and for Newtonian and non-Newtonian fluids. For a Newtonian fluid there is a linear relationship and the gradient of the line, , is constant for each fluid at a given temperature. For fluids, subject to a given and a given temperature, varies inversely with the velocity gradient.
d d yv
Figure 3.70Relationship between velocity gradient and shear stress for Newtonian and non-Newtonian fluids
Fluid 1
Fluid 2 τ
δvδy 0 —
Typical non-Newtonian fluid μ1>μ2 at
given τ and given temperature
There are different kinds of non-Newtonian fluid in which the viscosity varies in different ways with the applied shear stress. In some fluids (a common example is tomato ketchup) the viscosity can be made to decrease when a high rate of shear is applied for a period of time; so if you shake the bottle the ketchup flows more easily. Conversely, some fluids increase in viscosity when the rate of shear increases.
The units of can be found by considering the dimensions of the quantities involved.
[ ] ⬅