Density, S. Wight & Viscosity

3.1 Density:

The density  (rho) or more strictly, mass density, of a fluid is its mass per unit volume, while the specific weight  (gamma) is its weight per unit volume. In the British Gravitational (BG) system density  will be in slugs per cubic foot (kg/m³ in SI units), which can also be expressed as units of lb.sec² /ft⁴ (  in SI units).

Specific weight g represents the force exerted by gravity on a unit volume of fluid, and therefore must have the units of force per unit volume, such as pounds per cubic foot (N/m³ in SI units). Density and specific weight of a fluid are related as:

Since the physical equations are dimensionally homogeneous, the dimensions of density in SI units are:

So,

Note that density is absolute, since it depends on mass, which is independent of location. Specific weight , on the other hand, is not absolute, since it depends on the value of the gravitational acceleration g, which varies with location, primarily latitude and elevation above mean sea level. In general, density of a fluid decreases with increase in temperature. It increases with increase in pressure but density is highly variable in gases nearly proportional to the pressure and the density of liquids depends more strongly on temperature than it does on pressure.

Ideal gas equation of state: 

It expressed by the giving equation

The above equation is used to find the density of any fluid, if the pressure (P) and temperature (T) are known.

Any equation that relates the pressure, temperature, and specific volume of a substance is called an equation of state. It is experimentally observed that at a low pressure the volume o f a gas is proportional to its temperature.

R is the gas universal constant, and it is different for each gas R = 8.314 (kJ/kmol.K).

3.2 Specific Weight:
        Specific weight is the weight possessed by unit volume of a fluid. It is denoted by ( ). Its unit is (N/m³). Specific weight varies from place to place due to the change of acceleration due to gravity (g).

Specific weight g of pure water as a function of temperature and pressure for the condition where    g = 9.81 m/s²

3.3 Viscosity:

          The viscosity a fluid is a measure of its resistance to shear or angular deformation. Motor oil, for example, has high viscosity and resistance to shear, is cohesive, and feels “sticky,” whereas gasoline has low viscosity. The friction forces in flowing fluid result from the cohesion and momentum interchange between molecules

      

Figure 4 indicates how the viscosities of typical fluids depend on temperature. As the temperature increases, the viscosities of all liquids decrease, while the viscosities of all gases increase. This is because the force of cohesion, which diminishes with temperature, predominates with liquids, while with gases the predominating factor is the interchange of molecules between the layers of different velocities. Thus a rapidly-moving gas molecule shifting into a slower moving layer tends to speed up the latter. And a slow-moving molecule entering a faster-moving layer tends to slow down the faster-moving layer. This molecular interchange sets up a shear, or produces a friction force between adjacent layers. At higher temperatures molecular activity increases, so causing the   viscosity of gases to increase with temperature.                                                                                         

Fig 4 a graphically present numerical values of absolute and kinematic viscosities for a variety of liquids and gases, and show how they vary with temperature.

Consider the classic case of two parallel plates (Fig. 5), sufficiently large that we can neglect edge conditions, a small distance Y apart, with fluid filling the space between. The lower plate is stationary, while the upper one moves parallel to it with a velocity U due to a force F corresponding to some area A of the moving plate. At boundaries, particles of fluid adhere to the walls, and so their velocities are zero relative to the wall. This so-called no-slip condition occurs with all viscous fluids. Thus in Fig. 5 the fluid velocities must be U where in contact with the plate at the upper boundary and zero at the lower boundary. We call the form of the velocity variation with distance between these two extremes, as depicted in Fig. 5, a velocity profile. If the separation distance Y is not too great, if the velocity U is not too high, and if there is no net flow of fluid through the space, the velocity profile will be linear, as in Fig. 5a. If, in addition, there is a small amount of bulk fluid transport between the plates, as could result from pressure-fed lubrication for example, the velocity profile becomes the sum of the previous linear profile plus a parabolic profile (Fig. 5b); the parabolic additions to (or subtractions from) the linear profile are zero at the walls (plates) and maximum at the centerline. The behavior of the fluid is much as if it consisted of a series of thin layers, each of which slips a little relative to the next.

Fig-5

For a large class of fluids under the conditions of Fig. 5a, experiments have shown that

We see from similar triangles that we can replace U/Y by the velocity gradient. If we now introduce a constant of proportionality  (mu), we can express the shearing stress  (tau) between any two thin sheets of fluid by:

This equation called ‘’Newton’s equation of viscosity’’

   v Viscosity of liquids & gases: