MECHANICAL PROPERTIES OF FLUIDS
1. INTRODUCTION
In this topic, we shall study some common physical properties of liquids and gases. Liquids and gases can flow and are therefore, called fluids. It is this property that distinguishes liquids and gases from solids in a basic way.
Fluids are everywhere around us. Earth has an envelop of
air and two-thirds of its surface is covered with water. Water is not only necessary for our existence; every mammalian
body constitute mostly of water. All the processes occurring
in living beings including plants are mediated by fluids. Thus understanding the behaviour and properties of fluids is important.
How are fluids different from solids? What is common in
liquids and gases? Unlike a solid, a fluid has no definite shape of its own. Solids and liquids have a fixed volume, whereas a gas fills the entire volume of its container. We have learnt in the previous chapter that the volume of solids can be changed by stress. The volume of solid, liquid or gas depends on the stress or pressure acting on it. When we talk about fixed volume of solid or liquid, we mean its volume
under atmospheric pressure. The difference between gases
and solids or liquids is that for solids or liquids the change in volume due to change of external pressure is rather small.
In other words solids and liquids have much lower
compressibility as compared to gases.
Shear stress can change the shape of a solid keeping its
volume fixed. The key property of fluids is that they offer very little resistance to shear stress; their shape changes by application of very small shear stress. The shearing stress of fluids is about million times smaller than that of solids.
A sharp needle when pressed against our skin pierces it. Our
skin, however, remains intact when a blunt object with a wider contact area (say the back of a spoon) is pressed against it with the same force. If an elephant were to step on a man’s chest, his ribs would crack. A circus performer across whose chest a large, light but strong wooden plank is
placed first, is saved from this accident. Such
everyday experiences convince us that both the
force and its coverage area are important. Smaller the area on which the force acts, greater is the impact. This impact is known as pressure. When an object is submerged in a fluid at rest, the fluid exerts a force on its surface. This force is always normal to the object’s surface.
This is so because if there were a component of
force parallel to the surface, the object will also
exert a force on the fluid parallel to it; as a
consequence of Newton’s third law. This force
will cause the fluid to flow parallel to the surface.
Since the fluid is at rest, this cannot happen. Hence, the force exerted by the fluid at rest has to be perpendicular to the surface in contact with it.
2. Atmospheric Pressure and
Gauge Pressure
The pressure of the atmosphere at any point is
equal to the weight of a column of air of unit
cross-sectional area extending from that point
to the top of the atmosphere. At sea level, it is
1.013 × 10^5 Pa (1 atm). Italian scientist Evangelista Torricelli (1608–1647) devised for
the first time a method for measuring
atmospheric pressure. A long glass tube closed
at one end and filled with mercury is inverted
into a trough of mercury.
This device is known as ‘mercury barometer’.
The space above the mercury column in the tube
contains only mercury vapour whose pressure P is so small that it may be neglected. Thus, the pressure at Point A=0. The pressure inside the coloumn at Point B must be the same as the pressure at Point C, which is atmospheric pressure, Pa.
Pa
= ρgh
where ρ is the density of mercury and h is the
height of the mercury column in the tube.
In the experiment it is found that the mercury
column in the barometer has a height of about
76 cm at sea level equivalent to one atmosphere
(1 atm). This can also be obtained using the value of ρ. A common way of stating
pressure is in terms of cm or mm of mercury
(Hg). A pressure equivalent of 1 mm is called a
torr (after Torricelli).
1 torr = 133 Pa.
The mm of Hg and torr are used in medicine
and physiology. In meteorology, a common unit
is the bar and millibar.
1 bar = 10^5
Pa
An open tube manometer is a useful
instrument for measuring pressure differences.
It consists of a U-tube containing a suitable liquid i.e., a low density liquid (such as oil) for measuring small pressure differences and a high density liquid (such as mercury) for large pressure differences. One end of the tube is open to the atmosphere and the other end is
connected to the system whose pressure we want
to measure. The pressure P at A is equal to pressure at point B. What we normally measure is the gauge pressure, which is P −Pa and is proportional to manometer height h.
3. Hydraulic Machines
Let us now consider what happens when we
change the pressure on a fluid contained in a
vessel. Consider a horizontal cylinder with a
piston and three vertical tubes at different
points. The pressure in the
horizontal cylinder is indicated by the height of
liquid column in the vertical tubes. It is necessarily the same in all.If we push the piston, the fluid level rises in all the tubes, again reaching the same level in each one of them.
This indicates that when the pressure on the
cylinder was increased, it was distributed
uniformly throughout. We can say whenever
external pressure is applied on any part of a
fluid contained in a vessel, it is transmitted
undiminished and equally in all directions.
This is another form of the Pascal’s law and it
has many applications in daily life.
A number of devices, such as hydraulic lift and hydraulic brakes, are based on the Pascal’s law. In these devices, fluids are used for transmitting pressure. In a hydraulic lift, two pistons are separated by the space filled with a liquid. A piston of small cross-section A1 is used to exert a force F1 directly on the liquid. The pressure P is
transmitted throughout the liquid to the larger
cylinder attached with a larger piston of area A2
, which results in an upward force of P × A2 .Therefore, the piston is capable of supporting a large force (large weight of, say a car, or a truck.
4. STREAMLINE FLOW
So far we have studied fluids at rest. The study
of the fluids in motion is known as fluid
dynamics. When a water tap is turned on slowly,
the water flow is smooth initially, but loses its smoothness when the speed of the outflow is increased. In studying the motion of fluids, we focus our attention on what is happening to various fluid particles at a particular point in space at a particular time. The flow of the fluid is said to be steady if at any given point, the velocity of each passing fluid particle remains constant in time. This does not mean that the velocity at different points in space is same. The velocity of a particular particle may change as it moves from one point to another. That is, at some
other point the particle may have a different
velocity, but every other particle which passes
the second point behaves exactly as the previous
particle that has just passed that point. Each particle follows a smooth path, and the paths of the particles do not cross each other. The meaning of streamlines. (a) A typical trajectory of a fluid particle. (b) A region of streamline flow.
The path taken by a fluid particle under a
steady flow is a streamline. It is defined as a
curve whose tangent at any point is in the
direction of the fluid velocity at that point.
Consider the path of a particle. The curve describes how a fluid particle moves with time. The curve PQ is like a permanent map of fluid flow, indicating how the fluid streams. No two streamlines can cross, for if they do, an oncoming fluid particle can go either one way or the other and the flow would not be steady. Hence, in steady flow, the map of flow is stationary in time. How do we draw closely
spaced streamlines ? If we intend to show
streamline of every flowing particle, we would
end up with a continuum of lines. Consider planes
perpendicular to the direction of fluid flow e.g., at three points P, R and Q . The
plane pieces are so chosen that their boundaries
be determined by the same set of streamlines. This means that number of fluid particles crossing the surfaces as indicated at P, R and Q is the same. If area of cross-sections at these points are AP ,AR and AQ and speeds of fluid particles are vP , vR and vQ , then mass of fluid ∆mP crossing at AP in a small interval of time ∆t is ρPAP vP ∆t. Similarly mass of fluid ∆mR flowing or crossing at AR in a small interval of time ∆t is ρRAR vR ∆t and mass of fluid ∆mQ
is ρQAQ vQ ∆t crossing at AQ. The mass of liquid flowing out equals the mass flowing in, holds in all cases.
5. VISCOSITY
Most of the fluids are not ideal ones and offer some
resistance to motion. This resistance to fluid motion is like an internal friction analogous to friction when a solid moves on a surface. It is called viscosity. This force exists when there is relative motion between layers of the liquid. Suppose we consider a fluid like oil enclosed between two glass plates. The bottom plate is fixed while the top plate is moved with a constant velocity v relative to the fixed plate. If oil is replaced by honey, a greater force is required to move the plate with the same velocity. Hence we say that honey is more viscous than oil. The fluid in contact with a surface has the same velocity as that of the surfaces. Hence, the layer of the liquid in contact with top surface moves with a velocity v and the layer of the liquid in contact with the fixed surface is stationary. The velocities of layers increase uniformly from bottom (zero velocity) to the top layer (velocity v). For any layer of liquid, its upper layer pulls it forward while lower layer pulls it backward. This results in force between the layers. This type of flow is known as laminar. The layers of liquid slide over one another as the pages of a book do when it is placed flat on a table and a horizontal force is applied to the top cover. When a fluid is flowing in a pipe or a tube, then velocity of the liquid layer along the axis of the tube is maximum and decreases gradually as we move towards the walls where it becomes zero. The velocity on a cylindrical surface in a tube is constant.
On account of this motion, a portion of liquid,
which at some instant has the shape ABCD, take the shape of AEFD after short interval of time(∆t). During this time interval the liquid has
undergone a shear strain of ∆x/l. Since, the strain in a flowing fluid
increases with time continuously. Unlike a solid, here the stress is found experimentally to depend on ‘rate of change of strain’ or ‘strain rate’ i.e. ∆x/(l ∆t) or v/l instead of strain itself. The coefficient of viscosity (pronounced ‘eta’) for a fluid is defined as the ratio of shearing stress to the strain rate.
Comments
Post a Comment