Fluid mechanics_inroductionIt is about materials enginerring course similar to william callister subject code is MT30001.It is our sirs slides iam from iit kgp and applications_Classes 1 to 3.pptx

It is about fluid mechanics

1. Fluid Mechanics Dr. Vanteru Mahendra Reddy Associate Professor Mechanical Engineering Department Indian Institute of Technology, Kharagpur Kharagpur , India Mob: [phone] [email protected] ; [email protected]

2. Fluid : A substance in the liquid or gas phase that cannot resist shear stress and continuously deforms under its influence, no matter how small the applied force. Introduction A solid resists deformation and stops straining under constant shear, while a fluid deforms continuously under constant shear stress. In solids, stress is proportional to strain; in fluids, stress is proportional to strain rate. In a shear experiment: Solid (e.g., rubber): Deforms and returns to original shape when force is removed. Fluid : Flows continuously with the moving plate, velocity decreases with depth due to viscous effects. Liquids maintain constant volume due to strong molecular cohesion and form a free surface under gravity. Deformation of a rubber block between parallel plates under shear force, showing shear stress on the rubber and an equal, opposite stress on the upper plate.

3. Molecular structure : Solids : Densely packed, strong intermolecular forces, fixed structure. Liquids : Slightly less dense, molecules free to move but cohesive. Gases : Molecules far apart, weak intermolecular forces, high mobility. Introduction The arrangement of atoms in different phases: (a) molecules are at relatively fixed positions in a solid, (b) groups of molecules move about each other in the liquid phase, and Unlike a liquid, a gas does not form a free surface, and it expands to fill the entire available space.

4. Applications

5. Text Books Fluid Mechanics- Fox, McDonald, Pritchard, 8th Edition, Wiley Introduction to Fluid Mechanics and Fluid Machines- Som , Biswas and Chakraborty, 3rd edition, Mcgrawhill Fluid Mechanics- F M White, 8th Edition, McGrawhill Fluid Mechanics- Kundu, Cohen, Dowling, 6th Edition Academic Press Cengel , Y. and Cimbala , J., 2013.  Ebook : Fluid mechanics fundamentals and applications ( si units) . McGraw Hill.

6. System It is a quantity of matter (working substance) or region in space in which working substance is contained or flowing. In other words system is used to identify the subject of the analysis/ interest. Once the system is defined and the relevant interactions with other systems are identified, one or more physical laws or relations are applied. System: System Surroundings Process in solving Engineering Mechanics problems M echanics free body F orces exerted on it by other bodies Newton’s second law of motion

7. The mass or region outside the system is called surrounding. This surrounding should have interaction with system in terms of mass & energy (heat & work) responsible for energy transformation & variation in properties of working substance. The system is distinguished from its surroundings by a specified boundary, which may be at rest or flexible or in motion System Boundary Surroundings System Mass Energy Balloon and oven

8. Example of a control volume (open system). An automobile engine. ( i ) C losed system is defined when a particular quantity of matter is under study. A closed system always contains the same matter. There can be no transfer of mass across its boundary. (ii) Open System a region within a prescribed boundary is studied. The region is called a control volume. Mass may cross the boundary of a control volume. Types of systems Closed system Open system Isolated system Closed system: A gas in a piston–cylinder assembly (iii) Isolated system A special type of closed system that does not interact in any way with its surroundings is called an isolated system.

9. Property, State, and Process Properties of the system: These properties are Extensive/Intensive properties A property is a macroscopic characteristic of a system such as mass, volume, energy, pressure, and temperature to which a numerical value can be assigned at a given time without knowledge of the previous behavior ( history ) of the system. Properties are the characteristics of the working substance . Thermodynamics also deals with quantities that are not properties, such as mass flow rates and energy transfers by work and heat.

10. E xtensive properties A property is called extensive if its value for an overall system is the sum of its values for the parts into which the system is divided. Mass, volume, energy, etc. Extensive properties are dependent of mass of substance. The extensive properties of a system can change with time In the experimental analysis/thermodynamic analyses, a careful observation is required to record the changes in extensive properties such as mass and energy as a system interacts with its surroundings.

11. Intensive properties: Their values are independent of the size or extent of a system and may vary from place to place within the system at any moment. Intensive properties are independent of mass of substance. Extensive properties per unit volume mass are called specific properties which are intensive properties In tensive properties

12. Measuring Mass, Length, Time, and Force When engineering calculations are performed, it is necessary to be concerned with the units of the physical quantities involved. A unit is any specified amount of a quantity by comparison with which any other quantity of the same kind is measured. Centimeters , kilometers, feet, inches, and miles are all units of length. Seconds, minutes, and hours are alternative time units. P hysical quantities are related by definitions and laws, a relatively small number of physical quantities suffice to conceive of and measure all others. These may be called primary dimensions. The others may be measured in terms of the primary dimensions and are called secondary. Two commonly used sets of primary dimensions that suffice for applications in mechanics are (1) mass, length, and time and (2) force, mass, length, and time. Additional primary dimensions are required when additional physical phenomena come under consideration. Temperature is included for thermodynamics, and electric current is introduced for applications involving electricity. Once a set of primary dimensions is adopted, a base unit for each primary dimension is specified.

13. SI Units SI is the abbreviation for System International Unites (International System of Units), which is the legally accepted system in most countries. The mass standard for the United States is maintained by the National Institute of Standards and Technology. U nit of length is the meter ( metre ), m, defined as the length of the path traveled by light in a vacuum during a specified time interval. The base unit of time is the second, s. The second is defined as the duration of 9,192,631,770 cycles of the radiation associated with a specified transition of the cesium atom. The SI unit of force, called the newton, is a secondary unit, defined in terms of the base units for mass, length, and time.

14. F ma   The newton is defined so that the proportionality constant in the expression is equal to unity. That is, Newton’s second law is expressed as the equality F = ma The newton, N, is the force required to accelerate a mass of 1 kilogram at the rate of 1 meter per second per second. 1 N = ( 1 kg ) (1 m/ ) = 1 kg. m/   English Engineering Units: Newton’s second law of motion states that the net force acting on a body is proportional to the product of the mass and the acceleration, written

15. Continuum: Matter is made up of atoms that are widely spaced in the gas phase. The atomic nature of a substance and it posses as continuous, homogeneous matter with no holes, that is, a continuum . The continuum idealization allows us to treat properties as point functions and to assume the properties vary continually in space with no jump discontinuities. This idealization is valid as long as the size of the system is large relative to the space between the molecules. Ex: The density of water in a glass is the same at any point Continuum Despite the large gaps between molecules, a substance can be treated as a continuum because of the very large number of molecules even in an extremely small volume.

16. Concept of continuum The continuum assumption treats a fluid as a continuous, homogeneous medium, ignoring its molecular structure. This allows fluid properties such as density and pressure to be defined as point functions that vary smoothly in space. Fluids are composed of discrete molecules, but in most engineering applications, they are treated as continuous media. The continuum model is valid when the characteristic length of the system (L) is much larger than the mean free path (λ) the average distance between molecular collisions. Example (oxygen at 1 atm, 20°C): Molecular diameter ≈ 3×10 −10 m Mean free path (λ) ≈ 6.3×10 −8 m 1 mm³ contains ≈ 3×10 16 molecules At low pressures or high altitudes , λ increases and may become comparable to L, invalidating the continuum assumption. The Knudsen number ( Kn ) quantifies this relationship: Kn =λ/L If Kn < 0.01 → Continuum assumption is valid If Kn > 0.1 → Molecular (rarefied gas) effects become important A gas can be treated as a continuum due to the vast number of molecules, despite large gaps between them. The mean free path ( λ ) is the average distance a molecule travels between consecutive collisions with other molecules in a gas. 

17. Density and Specific Volume Density is defined as mass per unit volume Specific volume (v) is the reciprocal of density (ρ): For a small fluid element: Density depends on temperature and pressure: For gases : Density ∝ Pressure Density ∝ 1 / Temperature For liquids and solids : Nearly incompressible (little change with pressure) Density variation with temperature is more significant Example – Water at 20°C : At 1 atm: ρ = 998 kg/m³ At 100 atm: ρ = 1003 kg/m³ (only 0.5% increase) Example – Water at 1 atm : At 20°C: ρ = 998 kg/m³ At 75°C: ρ = 975 kg/m³ (2.3% decrease)

18. Specific Gravity and Specific Weight The specific gravity , or relative density, and is defined as the ratio of the density of a substance to the density of some standard substance at a specified temperature (usually water at 4°C, for which 𝜌 H2O = 1000 kg/m 3 ). The weight of a unit volume of a substance is called specific weight or weight density and is expressed as where g is the gravitational acceleration.

19. Pressure Pressure is defined as a normal force exerted by a fluid per unit area. The fluid on one side of the area exerts a compressive force on it that is normal to the area, F normal . An equal but oppositely directed force is exerted on the area by the fluid on the other side. The pressure p at the specified point is defined as the limit P   T he component normal to the area is called the normal stress, The two components in the plane of the area are termed shear stresses. The SI unit of pressure and stress is the pascal. 1 pascal = 1 N/   1 standard atmosphere ( 1 atm ) = 101325 Pa The word pressure deal with a gas or a liquid. The pressure in solids is normal stress.

20. When the local atmospheric pressure is greater than the pressure in the system, the term vacuum pressure is used. P ( vacuum ) = P atm ( absolute ) – P ( absolute ) A bsolute pressure A gage pressure or a vacuum pressure. The term gage pressure is applied when the pressure in the system is greater than the local atmospheric pressure, P atm . P ( gage ) = P ( absolute ) – P atm ( absolute )

21. p = gh  

22. Pressure Measurement Two commonly used devices for measuring pressure are the manometer and the Bourdon tube. Manometers measure pressure differences in terms of the length of a column of liquid such as water, mercury, or oil. p - p atm = gL   A Bourdon tube gage is one of the pressure measuring device using metal expansion. Pressure can be measured by other means as well. An important class of sensors utilize the piezoelectric effect : A charge is generated within certain solid materials when they are deformed. Pressure measurement by a manometer Pressure measurement by a Bourdon tube gage. Pressure sensor with automatic data acquisition.

23. Vapour pressure At a given pressure, the temperature at which a pure substance changes phase is called the saturation temperature T sat . Likewise, at a given temperature, the pressure at which a pure substance changes phase is called the saturation pressure P sat . The vapor pressure (saturation pressure) of a pure substance (e.g., water) is the pressure exerted by its vapor molecules when the system is in phase equilibrium with its liquid molecules at a given temperature. P v is a property of the pure substance, and turns out to be identical to the saturation pressure Psat of the liquid ( P v = P sat ). Vapor pressure increases with temperature. Thus, a substance at higher pressure boils at higher temperature. Ex: water boils at 134°C in a pressure cooker operating at 3 atm absolute pressure, but it boils at 93°C in an ordinary pan at a 2000-m elevation, where the atmospheric pressure is 0.8 atm. Vapor pressure of a pure substance at phase equilibrium between liquid and vapor at a given temperature.

24. Measuring Temperature The Temperature is termed as degree of hotness or coldness Thermal Equilibrium means equal temperature throughout the system   Hot body   cold body Q Heat interaction When a system undergoes a process while enclosed by an adiabatic wall, it experiences no thermal interaction with its surroundings. Such a process is called an adiabatic process. A process that occurs at constant temperature is an isothermal process. An adiabatic process is not necessarily an isothermal process, nor is an isothermal process necessarily adiabatic. Assume two bodies at different temperature, brought into the contact with each other (system). Heat transfer occurs between them and reached to uniform temperature after a while. When the two blocks temperatures are equal: Those (system) are in thermal equilibrium.    

25. Thermometers Sensors known as thermocouples are based on the principle that when two dissimilar metals are joined, an electromotive force (emf ) that is primarily a function of temperature will exist in a circuit. Devices using conductors are known as resistance temperature detectors . Semiconductor types are called thermistors . A variety of instruments measure temperature by sensing radiation, They are known by terms such as radiation thermometers and optical pyrometers . Thermometers.( a ) Liquid-in-glass. ( b ) Infrared sensing ear thermometer

26. Kelvin Scale Temperature scales It is the basis of measurement of temperature. All temperature scales are based on some early reproduceable state i.e ice point and steam point. The temperature of ice and water (liquid) in equilibrium is called ice point and the temperature of steam and water (liquid) in equilibrium is called steam point . The Kelvin scale is an absolute thermodynamic temperature scale that provides a continuous definition of temperature, valid over all ranges of temperature. Empirical measures of temperature, with different thermometers, can be related to the Kelvin scale. Kelvin scale has a zero of 0 K, and lower temperatures than this are not defined . The different temperature scales are ( i ) Celsius scale (ii) Farenheat scale (iii) Kelvin scale or thermodynamic temperature scale. Comparison of temperature scales

27. Fourier Law Joseph Fourier (1768-1830) - French mathematician and physicist Fourier law  E xpression for the local heat flux thermal conductivity coefficient T1 > T2 k is often treated as a constant, though this is not always true. k of a material usually varies with temperature However, variation can be small over a significant range of temperatures Heat flux, q =   - ve sign in Eq.  H eat transfer in the + ve x direction is a + ve quantity.

28. Heat Transfer Rates: Conduction Heat rate (W): Application to one-dimensional, steady conduction across a plane wall of constant thermal conductivity: Conduction: General (vector) form of Fourier ’ s Law: Heat flux Thermal conductivity Temperature gradient Heat Transfer Rates

29. Thermal conductivity Thermal conductivity coefficient Nonhomogeneous substances Anisotropic substances The thermal conductivity of a material is a measure of the ability of the material to conduct heat. A high value for thermal conductivity indicates that the material is a good heat conductor Low value indicates that the material is a poor heat conductor or insulator.

30. Material Thermal conductivity (W/m K)* Diamond 1000 Silver 406.0 Copper 385.0 Gold 314 Brass 109.0 Aluminum 205.0 Iron 79.5 Steel 50.2 Lead 34.7 Mercury 8.3 Ice 1.6 Glass,ordinary 0.8 Concrete 0.8 Water at 20° C 0.6 Asbestos 0.08 Conduction Fiberglass 0.04 Brick,insulating 0.15 Brick, red 0.6 Cork board 0.04 Wool felt 0.04 Rock wool 0.04 Polystyrene (styrofoam) 0.033 Polyurethane 0.02 Wood 0.12-0.04 Air at 0° C 0.024 Helium (20°C) 0.138 Hydrogen(20°C) 0.172 Nitrogen(20°C) 0.0234 Oxygen(20°C) 0.0238 Silica aerogel 0.003 Material Thermal conductivity (W/m K)*

31. Ideal gas Molecular motion defines heat transfer, diffusion, and viscosity The thermal conductivities of gases such as air vary by a factor of 10 4 from those of pure metals such as copper. In a liquid or gas, the kinetic energy (KE) of the molecules is due to their random translational motion as well as their vibrational and rotational motions. When two molecules collide, part of the KE of the more energetic (higher-temperature) molecule is transferred to the less energetic (lower temperature) molecule. The higher the temperature, the faster the molecules move and the higher the number of such collisions, and the better the heat transfer. Thermal conductivity Thermal conductivity of helium (M 4) is much higher than those of air (M 29) and argon (M 40).

32. Range of thermal conductivity of various materials At room temperature In liquids the molecules are more closely spaced, and they exert a stronger intermolecular force field. The thermal conductivities of liquids usually lie between those of solids and gases

33. Pure metals have high thermal conductivities k of an alloy of two metals is usually much lower than that of either metal The thermal conductivities of materials vary with temperature. Thermal conductivity

34. Compressibility The coefficient of compressibility , also known as the bulk modulus of elasticity (κ), is a property that quantifies how much a fluid's volume changes in response to a change in pressure. Fluids typically expand when depressurized or heated and contract when pressurized or cooled, but the degree of volume change varies between fluids. It can also be expressed approximately in terms of finite changes as Fluids, like solids, compress under increased pressure from P1 to P2​. The coefficient of compressibility is introduced to quantify a fluid’s resistance to compression , similar to Young’s modulus for solids. It captures how a fluid responds to pressure changes , behaving like an elastic solid under compression. Under increased pressure , fluids contract ; under reduced pressure , they expand . This property is crucial for analyzing fluid behavior in high-pressure applications and compressible flows . Helps in accurately modeling and predicting fluid dynamics under varying pressure conditions.

35. Note that volume and pressure are inversely proportional (volume decreases as pressure is increased and thus ∂P/∂v is a negative quantity), and the negative sign in the definition ensures that κ is a positive quantity. Isothermal Compressibility ( α): The isothermal compressibility of a fluid represents the fractional change in volume or density corresponding to a unit change in pressure. Compressibility

36. Coefficient of Volume Expansion Coefficient of Volume Expansion (β): The coefficient of volume expansion is a thermodynamic property that quantifies the change in fluid density with respect to temperature at constant pressure. It is defined as the fractional change in volume (or density) per unit change in temperature. This property explains temperature-induced density variations responsible for natural phenomena such as atmospheric winds, ocean currents, natural convection, and buoyant rise in hot-air balloons and chimneys. A large β indicates a significant density change with temperature. The term βΔT represents the fractional volume change for a temperature change ΔT at constant pressure. where T is the absolute temperature (K).

37. In buoyancy-driven flows, the density difference between the hot/cold fluid and the surrounding fluid is approximated by: where 𝜌∞ is the density and T∞ is the temperature of the quiescent fluid from the confined hot or cold fluid pocket. Natural Convection and Buoyancy: The strength of natural convection increases with the temperature difference, as buoyancy force is directly proportional to this density variation. Combined Pressure and Temperature Effects: Volume change due to both pressure and temperature is given by: leading to the fractional volume/density change: where α is the isothermal compressibility. Coefficient of Volume Expansion

38. Speed of sound speed of sound: The speed at which an infinitesimally small pressure wave travels through a medium. It is a key parameter in the analysis of compressible flow. Noting that the gas constant R has a fixed value for a specified ideal gas and the specific heat ratio k of an ideal gas is, at most, a function of temperature. The speed of sound changes with temperature and varies with the fluid. Represents the propagation velocity of small pressure disturbances in a fluid. Originates from tiny disturbances that cause slight changes In local pressure. Depends on the compressibility and elasticity of the fluid medium. Critical for understanding phenomena like shock waves, supersonic flow, and rapid pressure changes in high-speed flow systems. Propagation of a small pressure wave along a duct.

39. Mach number The Mach number (Ma): It is defined as the ratio of the actual speed of a fluid (or an object moving through a fluid) to the speed of sound in that fluid under the same thermodynamic conditions It is a fundamental parameter in compressible fluid flow, named after Austrian physicist Ernst Mach. Subsonic (Ma<1) Sonic (Ma=1) Supersonic (Ma>1) Transonic (Ma≈1) Hypersonic (Ma≫1 generally Ma>5) V = fluid velocity C = local speed of sound It also represents the ratio of inertial forces to elastic (compressibility) forces in a flow. Compressibility effects are usually negligible for Ma<1/3 allowing such flows to be treated as incompressible. Since speed of sound depends on fluid state (temperature, pressure, etc.), the Mach number changes with altitude or ambient conditions.

40. Viscosity Viscosity : A property that represents a fluid's internal resistance to flow or deformation. It quantifies how “thick” or “fluid” a substance is. Drag Force : The resistive force exerted by a fluid on a body moving through it, acting in the direction opposite to the relative motion. Its magnitude depends partly on the fluid's viscosity. The fluid in contact with the upper plate sticks to the plate surface and moves with it at the same speed and the shear stress acting on this fluid layer is where 𝜇 is called the coefficient of viscosity or the dynamic (or absolute) viscosity of the fluid (kg/ m·s ), A common viscosity unit is poise.

41. Viscosity The shear force acting on a Newtonian fluid layer (or, by Newton’s third law, the force acting on the plate) is where again A is the contact area between the plate and the fluid. Then the force F required to move the upper plate in F

42. Surface Tension (σₛ) : A force per unit length acting along the surface of a liquid due to molecular attraction. It causes the liquid surface to behave like a stretched elastic membrane. Units: N/m or J/m² Also referred to as surface energy per unit area. Surface Energy : The energy required to increase the surface area of a liquid by one unit. It is numerically equivalent to surface tension and expressed in J/m² or N·m /m². Surface tension Cause : Arises from the imbalance of molecular forces at the liquid’s surface. Molecules at the surface experience net inward attraction because they lack neighboring molecules above them. Effect : The fluid tends to minimize its surface area, resulting in spherical droplet shapes (e.g., raindrops, mist, soap bubbles), which have the lowest surface area for a given volume.

43. Surface tension P i​ and P o are the internal and external pressures of a droplet or bubble. In atmospheric conditions, P o ​ = atmospheric pressure. A soap bubble has two surfaces, so the pressure difference includes an extra factor of 2 in the force balance. A curved fluid interface results in a pressure difference (ΔP), with the pressure being higher on the concave side. This phenomenon is observed in cases like liquid droplets in air, gas bubbles in water, or soap bubbles in air. The excess pressure arises due to surface tension acting along the circumference of the interface and pressure acting over its area. By analyzing the force balance through a free-body diagram, this pressure jump across the interface can be quantified.

44. Surface tension The surface tension of a liquid droplet can be understood by analyzing a small increase in its radius caused by adding a differential mass. This expansion leads to an increase in surface area and, consequently, surface energy. Surface tension is thus interpreted as the increase in surface energy per unit area during this differential expansion. The expansion work done during this differential process is determined by multiplying the force by distance to obtain Note: The excess pressure in a droplet or bubble is inversely proportional to the radius

45. Capillary Effect Capillary effect is the rise or fall of a liquid in a small-diameter tube (capillary) due to surface tension and intermolecular forces. Common in porous materials (e.g., cotton wicks, soil, plant tissues). Capillaries : Narrow tubes or confined channels in which the capillary effect occurs. Meniscus : The curved surface of a liquid inside a capillary tube at the point of contact with the solid wall. Contact Angle ( ϕ) : The angle between the tangent to the liquid surface and the solid surface at the point of contact: ϕ<90∘: Wetting (e.g., water on glass) ϕ>90∘: Non-wetting (e.g., mercury on glass) Cohesive Forces : Attractive forces between like molecules (e.g., water-water). Adhesive Forces : Attractive forces between unlike molecules (e.g., water-glass). The contact angle for wetting and nonwetting fluids. The capillary rise of water and the capillary fall of mercury in a small-diameter glass tube.

46. The forces acting on a liquid column that has risen in a tube due to the capillary effect. For non-wetting liquids (ϕ > 90°), cos ϕ is negative, resulting in a negative capillary height (h) indicating a capillary drop instead of a rise. Capillary Effect The capillary rise in a circular tube is determined by performing a force balance on the liquid column of height h . Since the bottom of the liquid column is level with the free surface of the reservoir, the pressure at both ends is atmospheric and cancels out. Therefore, the rise is governed by the balance between the surface tension force along the tube wall and the weight of the liquid column.

47. Fluid flow over a stationary surface The development of a velocity profile due to the no-slip condition as a fluid flows over a blunt nose. A fluid flowing over a stationary surface comes to a complete stop at the surface because of the no-slip condition. No-Slip Condition: When a fluid flows over a solid, nonporous surface, its velocity becomes zero at the point of contact due to viscosity. This leads to the formation of a velocity gradient near the wall. Boundary Layer: The region adjacent to the surface where viscous effects are significant and velocity gradients develop is called the boundary layer . It originates from the no-slip condition. Skin Friction Drag: Due to the no-slip condition, the fluid exerts a drag force along the surface in the direction of flow, known as skin friction . Flow Separation: On curved surfaces (e.g., the rear of a cylinder), the boundary layer may detach from the surface this is called flow separation , though the no-slip condition still holds at the wall. No-Temperature-Jump Condition: In heat transfer, a similar principle applies: at the contact point between a fluid and solid at different temperatures, both assume the same temperature , ensuring continuous temperature distribution.

48. Viscous versus Inviscid Regions of Flow The flow of an originally uniform fluid stream over a flat plate, and the regions of viscous flow (next to the plate on both sides) and inviscid flow (away from the plate). Friction Between Layers: When fluid layers move at different velocities, a frictional force arises, with slower layers resisting faster ones this interaction is governed by viscosity. Viscous Flow: Flows where frictional (viscous) effects are significant, especially near solid boundaries. Inviscid Flow: Regions where viscous forces are negligible compared to inertial or pressure forces. These regions allow for simplified analysis and often occur away from solid surfaces. All real fluids exhibit viscous effects to some degree; ideal inviscid flow is an approximation. When a flat plate is introduced into a uniform fluid stream, the boundary layer forms along the surface due to the no-slip condition , marking the viscous region . Flow away from the surface, unaffected by viscosity, forms the inviscid region .

49. Internal versus External Flow Internal Flow: Occurs when the fluid is fully enclosed by solid boundaries (e.g., water flow inside a pipe). Viscous effects influence the entire flow field. External Flow: Fluid flows over exposed surfaces (e.g., air over a ball or pipe). Viscous effects are confined to boundary layers and wake regions. Open-Channel Flow: Occurs when a duct is partially filled with liquid, exposing a free surface (e.g., river or irrigation flow). External flow over a tennis ball, and the turbulent wake region behind. External flow over a tennis ball, and the turbulent wake region behind.

50. Laminar Flow: Smooth, orderly fluid motion in parallel layers (laminae); typical in high-viscosity fluids at low velocities . Turbulent Flow: Chaotic, disordered motion with velocity fluctuations; common in low-viscosity fluids at high velocities . Transitional Flow: A flow regime that alternates between laminar and turbulent . Reynolds Number, (Re) = A dimensionless number introduced by Osborne Reynolds in the 1880s to characterize flow regimes in pipes: Low Re → Laminar High Re → Turbulent Intermediate Re → Transitional   Laminar vs Turbulent Flow Laminar, transitional, and turbulent flows over a flat plate.

51. Steady versus Unsteady Flow Steady Flow: No change in fluid properties (velocity, temperature, pressure) at a fixed point over time. Unsteady Flow – properties vary with time at a fixed point. Uniform Flow: No change in fluid properties with location over a specified region. Transient Flow: A type of unsteady flow associated with changes over time before reaching a steady state (e.g., rocket startup). ➤ Not strictly synonymous with unsteady, but often used during developing flows. Periodic Flow: A form of unsteady flow where properties oscillate around a steady mean value (e.g., vortex shedding).

52. One-, Two-, and Three-Dimensional Flows Flow Dimensionality: 1D Flow: Velocity varies in one direction only. 2D Flow: Velocity varies in two directions. 3D Flow: Velocity varies in all three spatial directions (e.g., V( x,y,z ) or V( r,θ,z )) Fully Developed Flow: In internal flow (e.g., pipe), after a certain distance from the inlet, the velocity profile becomes constant in the flow direction. Fully developed flow is often treated as 1D in cylindrical coordinates (variation only in radial direction, not in angular or axial). Coordinate System Influence: Flow dimensionality can vary with coordinate system choice. Pipe flow is 1D in cylindrical but 2D in Cartesian coordinates. 2D Flow Approximation: When one dimension dominates (high aspect ratio) and velocity change along the longer axis is negligible, the flow may be modeled as 2D. Example: Airflow over a long antenna (length ≫ diameter).

53. Enthalpy and specific heats Enthalpy is a thermodynamic property that represents the total heat content of a system. It is defined as the sum of the internal energy of the system and the product of its pressure and volume. where P/𝜌 is the flow energy, also called the flow work A system without magnetic, electric, or surface tension effects called simple compressible system and its total energy consists of e = u + ke + pe The fluid entering or leaving a control volume possesses an additional form of energy the flow energy P/𝜌. Then the total energy of a flowing fluid on a unit-mass basis becomes where h = P/𝜌 + u is the enthalpy, V is the magnitude of velocity, and z is the elevation of the system relative to some external reference point.

54. Enthalpy and specific heats The differential and finite changes in the internal energy and enthalpy of an ideal gas can be expressed in terms of the specific heats Specific Heat at Constant Volume ( Cv ): It is the amount of heat required to raise the temperature of a unit mass of a substance by one degree (usually °C or K) while keeping the volume constant . Cv ​=(∂T/∂u​)v​ Specific Heat at Constant Pressure (Cp): It is the amount of heat required to raise the temperature of a unit mass of a substance by one degree while keeping the pressure constant . Cp​=(∂ T∂h ​)p​ Using specific heat values at the average temperature, the finite changes in internal energy and enthalpy can be expressed approximately as For incompressible substances, the constant-volume and constant-pressure specific heats are identical. Therefore, cp ≅ cv ≅ c for liquids, and the change in the internal energy of liquids can be expressed as Δu ≅ cavg ΔT. Noting that 𝜌 = constant for incompressible substances, the differentiation of enthalpy h = u + P/𝜌 gives dh = du + dP /𝜌. Integrating, the enthalpy change becomes