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 ideal or inviscid flow. In an ideal flow there is no existence of shear force because of vanishing viscosity.

• All fluids in reality possess viscosity (μ > 0) and are therefore termed real fluids; their motion is known as viscous flow.
• Under certain circumstances (for example, at high Reynolds numbers, and away from solid boundaries) an accurate analysis of the flow field can be obtained from the ideal flow theory. Ideal-flow solutions often form the first approximation for external flows and are used together with boundary-layer theory to obtain realistic results near solid surfaces.

Non-Newtonian Fluids

Certain fluids do not obey the linear relationship between shear stress and rate of deformation that is stated by Newton's law of viscosity. For such fluids the apparent viscosity varies with the rate of deformation; these fluids are called non-Newtonian fluids.

Newton's law of viscosity for parallel shear flow is

τ = μ (du/dy)

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

• The ideal fluid is represented by the abscissa in Fig 2.1 because an ideal fluid offers zero resistance to shearing and thus exhibits zero shear stress under any deformation rate.
• The ordinate represents an ideal solid for which there is no deformation rate under loading.
• For Newtonian fluids, shear stress is linearly proportional to the velocity gradient; the plot of τ against du/dy is a straight line through the origin and the slope is the viscosity μ.
• Non-Newtonian fluids display nonlinear τ versus du/dy behaviour. Common classes are pseudo-plastic (shear-thinning), dilatant (shear-thickening) and Bingham plastics (yield stress followed by linear viscous behaviour).

Compressibility

Compressibility is the measure of the change in volume of a substance under applied pressure. The normal compressive stress on a fluid element at rest is the hydrostatic pressure p, which arises from molecular collisions within the fluid.

Bulk modulus and compressibility coefficient

The degree of compressibility is characterised by the bulk modulus of elasticity E, defined by the relation

where ΔV and Δp are the changes in volume and pressure respectively, and V is the initial volume. The negative sign is used because an increase in pressure produces a decrease in volume (Δp > 0 ⇒ ΔV < 0), making E positive.

For a given mass, changes in volume and density satisfy the relations

• Liquids have very large values of E compared with gases (except at very high pressures). Consequently liquids are commonly treated as incompressible in many engineering analyses, although strictly speaking no material is perfectly incompressible (E → ∞ only in the ideal limit).
• For example, the bulk modulus of water and air at atmospheric pressure are approximately 2 × 10⁶ kN/m² and 101 kN/m² respectively. Thus air is about 20 000 times more compressible than water; water may therefore be treated as incompressible in most practical flows.

Compressibility coefficient

The reciprocal of E is the compressibility K:

K is often expressed using the specific volume .

Equation of state for gases

For gaseous substances, pressure, volume and temperature are related by a thermodynamic equation of state. For an ideal gas the relation is

p = ρ R T (2.7)

where T is the absolute temperature and R is the specific gas constant (for air R ≈ 287 J/kg·K).

Both K and E generally depend on the thermodynamic process followed (isothermal, adiabatic, etc.).

Distinction between Incompressible and Compressible Flow

• To decide if compressibility must be accounted for in a flow problem we examine whether pressure changes caused by the motion lead to significant changes in density.
• From Bernoulli's equation, p + (1/2) ρ V² = constant, the characteristic pressure change Δp in a flow is of the order of the dynamic head (1/2)ρV². Substituting this into the compressibility relation gives

• If the relative change in density Δρ/ρ is very small, the flow may be treated as incompressible to a good approximation.
• According to Laplace's relation, the velocity of sound a in the medium is

• The Mach number Ma is the ratio of flow velocity to local acoustic velocity. Compressibility effects are small when Ma is small. In terms of density change, the approximation (1/2)Ma² ≪ 1 applies.
• Using a practical criterion that a maximum relative change in density of 5% (Δρ/ρ = 0.05) is acceptable for treating a flow as incompressible yields an upper Mach number limit of approximately Ma ≈ 0.33.
• For air at standard conditions the acoustic velocity is about 335.28 m/s, so Ma = 0.33 corresponds to a velocity near 110 m/s. Thus airflows at speeds below ~110 m/s are usually treated as incompressible.

Surface Tension of Liquids

• The phenomenon of surface tension arises from intermolecular forces in a liquid: cohesion and adhesion.
• Cohesion: the attraction between like molecules of the liquid; it keeps the liquid as one assemblage and enables it to resist tensile stress.
• Adhesion: the attraction between unlike molecules, for instance between liquid molecules and a solid surface; this causes wetting of solids by liquids when adhesion forces dominate.

Figure 2.3 The intermolecular cohesive force field in a bulk of liquid with a free surface

• Molecules in the interior (points A and B) experience cohesive forces equally in all directions. Molecules near the surface experience a net inward force; work must be done to bring molecules to the surface.
• A free surface therefore stores mechanical energy called surface (or free-surface) energy. A body of liquid seeks the shape that minimises its surface area and thus its surface energy.
• Surface tension is defined as the tensile force acting across an imaginary short straight elemental line of unit length on the surface. Its dimensional formula is F/L or M T^-2; SI unit is N/m.
• Surface tension is a property of the interface between two phases (for example liquid-gas) and decreases slightly with temperature. The surface tension of water in contact with air at 20 °C is about 0.073 N/m.
• Because of surface tension a curved liquid interface in equilibrium produces a pressure difference between its concave and convex sides (Young-Laplace effect).

Capillarity

• Capillarity is the result of the balance between cohesion and adhesion. If adhesion between a liquid and a solid is stronger than cohesion among liquid molecules, the liquid wets the solid and moves up a narrow tube; if cohesion is stronger the liquid is depressed.
• The contact angle θ is the angle between the liquid interface and the solid surface (measured through the liquid). For pure water on clean glass θ ≈ 0° so a capillary rise occurs. For mercury on clean glass θ ≈ 130° causing capillary depression.

Fig 2.4: Phenomenon of Capillarity

• Capillary rise h varies inversely with the tube diameter D, as given by the capillarity relation (see equation):

• An appreciable capillary rise or depression is observed only in tubes of small diameter.

Vapour Pressure

• Liquids evaporate when exposed to a gas; vapour molecules above the liquid exert a partial pressure known as vapour pressure.
• If the above-liquid space is confined and temperature is held constant, an equilibrium (saturation) state develops in which the rate of evaporation equals the rate of condensation; the vapour in the space is then saturated.
• The vapour pressure of a liquid is a function of temperature only and equals the saturation pressure at that temperature. Thus vapour pressure increases with temperature.
• Boiling occurs when the vapour pressure of the liquid reaches the ambient (external) pressure. Boiling can therefore be produced either by increasing temperature (raising vapour pressure) or by reducing the ambient pressure to the vapour-pressure level at the existing temperature.

Fig 2.5: To & Fro movement of liquid molecules from an interface in a confined space as a closed surrounding

Applications, Limitations and Important Remarks

• Use of ideal flow theory: Ideal (inviscid) flow solutions - especially potential flow solutions (inviscid and irrotational) - are widely used to approximate external high-Reynolds-number flows away from walls and to obtain pressure distributions around bodies. They form the basis for many aerodynamic and hydrodynamic analyses.
• Limitations: Ideal flow neglects viscosity and therefore cannot predict boundary-layer formation, flow separation, wall shear stress or real drag. To treat flows near solid surfaces and to obtain forces due to friction one must include viscous effects (boundary-layer theory and Navier-Stokes equations).
• D'Alembert's paradox: For steady, inviscid, incompressible, irrotational flow about a body the theoretical drag is zero. This paradox emphasises the essential role of viscosity (however small) in producing realistic drag via boundary layers and wake formation.
• Compressibility criterion: For gases, compressibility effects may be neglected when Ma ≲ 0.3 (approx.). At higher Mach numbers, density variations and compressibility must be accounted for in the analysis.
• Non-Newtonian fluids: Many engineering materials (paints, slurries, blood, polymers) are non-Newtonian; their analysis requires appropriate constitutive models (power-law, Bingham, etc.).
• Surface phenomena and capillarity: Surface tension and capillarity are important in small-scale flows (microfluidics), wetting, droplet behaviour, and in many civil and environmental engineering problems (soil suction, rise of water in porous media).

Summary (optional): An ideal fluid is an inviscid model useful for approximating many flow fields away from boundaries. Real fluids are viscous and may be compressible or incompressible depending on flow speed and thermodynamic conditions. Surface tension, capillarity and vapour pressure are essential interfacial phenomena that govern small-scale liquid behaviour and phase change processes.