Definition of Stress
Consider a small area δA on the surface of a body (Fig. 1.1). The force acting on this area is δF.
This force can be resolved into two perpendicular components
Fig 1.1 Normal and Tangential Forces on a surface
When they are expressed as force per unit area they are called as normal stress and tangential stress respectively. The tangential stress is also called shear stress.
The normal stress
(1.1)
And shear stress
(1.2)
Definition of Fluid
A fluid is a substance that deforms continuously in the face of tangential or shear stress, irrespective of the magnitude of shear stress .This continuous deformation under the application of shear stress constitutes a flow.
Example : Consider Fig 1.2
Fig 1.2 Shear stress on a fluid body
If a shear stress τ is applied at any location in a fluid, the element 011' which is initially at rest, will move to 022', then to 033'. Further, it moves to 044' and continues to move in a similar fashion.
In other words, the tangential stress in a fluid body depends on velocity of deformation and vanishes as this velocity approaches zero. A good example is Newton's parallel plate experiment where dependence of shear force on the velocity of deformation was established.
Distinction Between Solid and Fluid
Fig 1.3 Deformation of a Solid Body
Concept of Continuum
Concept of Continuum - contd from previous slide
However, for the flow regimes considered in this course , K _{n} is always less than 0.01 and it is usual to say that the fluid is a continuum.
Other factor which checks the validity of continuum is the elapsed time between collisions. The time should be small enough so that the random statistical description of molecular activity holds good.
In continuum approach, fluid properties such as density, viscosity, thermal conductivity, temperature, etc. can be expressed as continuous functions of space and time.
Fluid Properties
Characteristics of a continuous fluid which are independent of the motion of the fluid are called basic properties of the fluid. Some of the basic properties are as discussed below.
Property | Symbol | Definition | ||||
Density (kg/m^{3)} | ρ | The density p of a fluid is its mass per unit volume . If a fluid element enclosing a point P has a volume Δ and mass Δm (Fig. 1.4), then density (ρ)at point P is written as However, in a medium where continuum model is valid one can write -
| ||||
Specific Weight (N/m^{3)} | γ | The specific weight is the weight of fluid per unit volume. The specific weight is given
Where g is the gravitational acceleration. Just as weight must be clearly distinguished from mass, so must the specific weight be distinguished from density. | ||||
Specific Volume (m^{3}/Kg) | v | The specific volume of a fluid is the volume occupied by unit mass of fluid. Thus
| ||||
Specific Gravity | s | For liquids, it is the ratio of density of a liquid at actual conditions to the density of pure water at 101 kN/m^{2} , and at 4 degreeC. The specific gravity of a gas is the ratio of its density to that of either hydrogen or air at some specified temperature or pressure. However, there is no general standard; so the conditions must be stated while referring to the specific gravity of a gas. |
Viscosity ( μ )
Viscosity is a fluid property whose effect is understood when the fluid is in motion.
In a flow of fluid, when the fluid elements move with different velocities, each element will feel some resistance due to fluid friction within the elements.
Therefore, shear stresses can be identified between the fluid elements with different velocities.
The relationship between the shear stress and the velocity field was given by Sir Isaac Newton.
Consider a flow (Fig. 1.5) in which all fluid particles are moving in the same direction in such a way that the fluid layers move parallel with different velocities.
Fig 1.5 Parallel flow of a fluid Fig 1.6 Two adjacent layers of a moving fluid.
The upper layer, which is moving faster, tries to draw the lower slowly moving layer along with it by means of a force F along the direction of flow on this layer. Similarly, the lower layer tries to retard the upper one, according to Newton's third law, with an equal and opposite force F on it (Figure 1.6).
Such a fluid flow where x-direction velocities, for example, change with y-coordinate is called shear flow of the fluid.
t = F/A
Viscosity ( μ )
Newton postualed that τ is proportional to the quantity Δu/ Δy where Δy is the distance of separation of the two layers and Δu is the difference in their velocities.
In the limiting case of , Δu / Δy equals du/dy, the velocity gradient at a point in a direction perpendicular to the direction of the motion of the layer.
where, the constant of proportionality μ is known as the coefficient of viscosity or simply viscosity which is a property of the fluid and depends on its state. Sign of τ depends upon the sign of du/dy. For the profile shown in Fig. 1.5, du/dy is positive everywhere and hence, τ is positive. Both the velocity and stress are considered positive in the positive direction of the coordinate parallel to them.
Equation
defining the viscosity of a fluid, is known as Newton's law of viscosity. Common fluids, viz. water, air, mercury obey Newton's law of viscosity and are known as Newtonian fluids.
Other classes of fluids, viz. paints, different polymer solution, blood do not obey the typical linear relationship, of τ and du/dy and are known as non-Newtonian fluids. In non-newtonian fluids viscosity itself may be a function of deformation rate as you will study in the next lecture.
Causes of Viscosity
Causes of Viscosity -
Fig 1.8: Change of Viscosity of Water and Air under 1 atm
No-slip Condition of Viscous Fluids