Ideal Fluid | Fluid Mechanics for Mechanical Engineering PDF Download

Consider a hypothetical fluid having a zero viscosity ( μ = 0). Such a fluid is called an ideal fluid and the resulting motion is called as ideal or inviscid flow. In an ideal flow, there is no existence of shear force because of vanishing viscosity.

• All the fluids in reality have viscosity (μ > 0) and hence they are termed as real fluid and their motion is known as viscous flow.
• Under certain situations of very high velocity flow of viscous fluids, an accurate analysis of flow field away from a solid surface can be made from the ideal flow theory.

• There are certain fluids where the linear relationship between the shear stress and the deformation rate (velocity gradient in parallel flow) as expressed by the    is not valid. For these fluids the viscosity varies with rate of deformation. 

• Due to the deviation from Newton's law of viscosity they are commonly termed as non-Newtonian fluids. Figure 2.1 shows the class of fluid for which this relationship is nonlinear.
  

  Fig 2.1: Shear stress & deformation rate relationship of different fluids
  
  

• The abscissa in Fig. 2.1 represents the behaviour of ideal fluids since for the ideal fluids the resistance to shearing deformation rate is always zero, and hence they exhibit zero shear stress under any condition of flow.
• The ordinate represents the ideal solid for there is no deformation rate under any loading condition.
• The Newtonian fluids behave according to the law that shear stress is linearly proportional to velocity gradient or rate of shear strain  t = μ du/dy. Thus for these fluids, the plot of shear stress against velocity gradient is a straight line through the origin. The slope of the line determines the viscosity. 
• The non-Newtonian fluids are further classified as pseudo-plastic, dilatant and Bingham plastic.

• Compressibility of any substance is the measure of its change in volume under the action of external forces.
• The normal compressive stress on any fluid element at rest is known as hydrostatic pressure p and arises as a result of innumerable molecular collisions in the entire fluid. 
• The degree of compressibility of a substance is characterized by the bulk modulus of elasticity E defined as:
            (2.3)
• Where Δ and Δp are the changes in the volume and pressure respectively, and  is the initial volume. The negative sign (-sign) is included to make E positive, since increase in pressure would decrease the volume i.e for Δp>0 , Δ<0) in volume.
• For a given mass of a substance, the change in its volume and density satisfies the relation
       (2.4)
  
    (2.5)
• Values of E for liquids are very high as compared with those of gases (except at very high pressures). Therefore, liquids are usually termed as incompressible fluids though, in fact, no substance is theoretically incompressible with a value of E as ∞  .
• For example, the bulk modulus of elasticity for water and air at atmospheric pressure are approximately 2 x 10⁶ kN/m ² and 101 kN/m ² respectively. It indicates that air is about 20,000 times more compressible than water. Hence water can be treated as incompressible. 
• For gases another characteristic parameter, known as compressibility K, is usually defined , it is the reciprocal of E
     (2.6)
• K is often expressed in terms of specific volume .
• For any gaseous substance, a change in pressure is generally associated with a change in volume and a change in temperature simultaneously. A functional relationship between the pressure, volume and temperature at any equilibrium state is known as thermodynamic equation of state for the gas. 
  
  For an ideal gas, the thermodynamic equation of state is given by:
    p = ρRT    (2.7)
   
• where T is the temperature in absolute thermodynamic or gas temperature scale (which are, in fact, identical), and R is known as the characteristic gas constant, the value of which depends upon a particular gas. However, this equation is also valid for the real gases which are thermodynamically far from their liquid phase. For air, the value of R is 287 J/kg K. 
• K and E generally depend on the nature of process

• In order to know, if it is necessary to take into account the compressibility of gases in fluid flow problems, we need to consider whether the change in pressure brought about by the fluid motion causes large change in volume or density.
  Using Bernoulli's equation 
  p + (1/2)ρV²= constant (V being the velocity of flow), change in pressure, Δp, in a flow field, is of the order of (1/2)ρV² (dynamic head).
  Invoking this relationship into:
                  (2.12)
  So if Δρ/ρ is very small, the flow of gases can be treated as incompressible with a good degree of approximation
• According to Laplace's equation, the velocity of sound is given by:
  
  where, Ma is the ratio of the velocity of flow to the acoustic velocity in the flowing medium at the condition and is known as Mach number. So we can conclude that the compressibility of gas in a flow can be neglected if Δρ/ρ is considerably smaller than unity, i.e. (1/2)Ma²<<1.
• In other words, if the flow velocity is small as compared to the local acoustic velocity, compressibility of gases can be neglected. Considering a maximum relative change in density of 5 per cent as the criterion of an incompressible flow, the upper limit of Mach number becomes approximately 0.33. In the case of air at standard pressure and temperature, the acoustic velocity is about 335.28 m/s. Hence a Mach number of 0.33 corresponds to a velocity of about 110 m/s. Therefore flow of air up to a velocity of 110 m/s under standard condition can be considered as incompressible flow.

Surface Tension of Liquids

• The phenomenon of surface tension arises due to the two kinds of intermolecular forces
  (i) Cohesion : The force of attraction between the molecules of a liquid by virtue of which they are bound to each other to remain as one assemblage of particles is known as the force of cohesion. This property enables the liquid to resist tensile stress.

  (ii) Adhesion : The force of attraction between unlike molecules, i.e. between the molecules of different liquids or between the molecules of a liquid and those of a solid body when they are in contact with each other, is known as the force of adhesion. This force enables two different liquids to adhere to each other or a liquid to adhere to a solid body or surface.
  
  
  Figure 2.3 The intermolecular cohesive force field in a bulk of liquid with a free surface
   

• A and B experience equal force of cohesion in all directions, C experiences a net force interior of the liquid The net force is maximum for D since it is at surface

• Work is done on each molecule arriving at surface against the action of an inward force. Thus mechanical work is performed in creating a free surface or in increasing the area of the surface. Therefore, a surface requires mechanical energy for its formation and the existence of a free surface implies the presence of stored mechanical energy known as free surface energy. Any system tries to attain the condition of stable equilibrium with its potential energy as minimum. Thus a quantity of liquid will adjust its shape until its surface area and consequently its free surface energy is a minimum.

• The magnitude of surface tension is defined as the tensile force acting across imaginary short and straight elemental line divided by  the length of the line. 

• The dimensional formula is F/L or MT^-2 . It is usually expressed in N/m in SI units.

• Surface tension is a binary property of the liquid and gas or two liquids which are in contact with each other and defines the  interface. It decreases slightly with increasing temperature. The surface tension of water in contact with air at 20°C is about 0.073 N/m. 

• It is due to surface tension that a curved liquid interface in equilibrium results in a greater pressure at the concave side of the surface than that at its convex side. 

Capillarity

• The interplay of the forces of cohesion and adhesion explains the phenomenon of capillarity. When a liquid is in contact with a  solid, if the forces of adhesion between the molecules of the liquid and the solid are greater than the forces of cohesion among the liquid molecules themselves, the liquid molecules crowd towards the solid surface. The area of contact between the liquid and solid increases and the liquid thus wets the solid surface.

• The reverse phenomenon takes place when the force of cohesion is greater than the force of adhesion. These adhesion and cohesion properties result in the phenomenon of capillarity by which a liquid either rises or falls in a tube dipped into the liquid depending upon whether the force of adhesion is more than that of cohesion or not (Fig.2.4).

• The angle θ as shown in Fig. 2.4, is the area wetting contact angle made by the interface with the solid surface.
  
  Fig 2.4: Phenomenon of Capillarity
• For pure water in contact with air in a clean glass tube, the capillary rise takes place with θ = 0 . Mercury causes capillary depression with an angle of contact of about 130⁰ in a clean glass in contact with air. Since h varies inversely with D as found from Eq.     an appreciable capillary rise or depression is observed in tubes of small diameter only.

• All liquids have a tendency to evaporate when exposed to a gaseous atmosphere. The rate of evaporation depends upon the molecular energy of the liquid which in turn depends upon the type of liquid and its temperature. 
• The vapour molecules exert a partial pressure in the space above the liquid, known as vapour pressure. If the space above the liquid is confined (Fig. 2.5) and the liquid is maintained at constant temperature, after sufficient time, the confined space above the liquid will contain vapour molecules to the extent that some of  them will be forced to enter the liquid. 
• Eventually an equilibrium condition will evolve when the rate at which the number of vapour molecules striking back the liquid surface and condensing is just equal to the rate at which they leave from the surface. The space above  the liquid then becomes saturated with vapour. 
• The vapour pressure of a given liquid is a function of temperature only and is equal to the saturation pressure for boiling corresponding to that temperature. Hence, the vapour pressure increases with the increase in temperature. Therefore the phenomenon of boiling of a liquid is closely related to the vapour pressure. 
• In fact, when the vapour pressure of a liquid becomes equal to the total pressure impressed on its surface, the liquid starts boiling. This concludes that boiling can be achieved either by raising the temperature of the liquid, so that its vapour pressure is elevated to the ambient pressure, or by lowering the pressure of the ambience (surrounding gas) to the liquid's vapour pressure at the existing temperature.
                                                    Fig 2.5: To & Fro movement of liquid molecules from an interface in a confined space as a closed surrounding