Fluid Mechanics Principles, Dimensions, and Units
Fluid Mechanics Concept
Fluid mechanics is the study of the behavior of fluids (liquids and gases) in motion and their interactions with boundaries. Its core principles are based on three fundamental physical laws: conservation of mass, conservation of momentum, and conservation of energy.
Fluid Mechanics Principles
Fluid mechanics is a branch of physics that studies both stationary and moving fluids. Broadly speaking, a fluid is a substance that will continue to deform when subjected to shear (tangential) stress. Unlike solid materials, where atoms are closely bound together and are rigid, fluid molecules can move freely relative to each other.
Thus, fluids lack the ability to resist deformation. As soon as shear stress is applied, fluids will flow.
Solids also deform under shear forces, but they reach equilibrium quickly when internal elastic forces balance out the applied forces. The flow rate of a fluid under shear is determined by the size of the applied force and its viscosity, which is the property of the fluid that resists shear deformation or flow.
Because fluids can flow, they do not have a fixed shape but take the shape of their container. Both liquids and gases are fluids, though they differ significantly in properties. In liquids, molecules are tightly packed, so liquids have relatively high density and viscosity, the latter due to intermolecular attractive forces.
Liquids are essentially incompressible, meaning their volume remains relatively constant and is unaffected by pressure and temperature. In contrast, gas molecules are spaced farther apart compared to liquids. Therefore, gases have lower density and viscosity and can be highly compressed.
When liquids and gases meet, they form an interface (surface). Due to unbalanced intermolecular forces at the interface, surface tension causes the surface to behave like a membrane. Specifically, because the molecular distance in liquids is close, strong attractive forces exist internally, while the forces from the external gas molecules are relatively weak (essentially negligible).
Surface tension plays a crucial role in the formation of liquid jets and droplets, which are the basis of inkjet printing technology. Fluids also have many other properties that describe their kinematics, transport, and thermodynamic behavior. Understanding these properties and being able to predict fluid behavior are key in the development and design of inkjet systems.
Fluid Mechanics Dimensions (Dimension) and Units (Unit)
In fluid mechanics, dimensions and units provide the basic framework for describing physical quantities. Dimension refers to the inherent properties of physical quantities (such as length, time, and mass), while units are specific measurements for these quantities (such as meters, seconds, kilograms). Mastery of dimensions and units is crucial for theoretical derivations, experimental design, and numerical simulations.
Density
Density is defined as the mass per unit volume of a fluid. In the International System of Units (SI), the unit for density is kilograms per cubic meter (kg/m³). The density depends on the mass of the molecules composing the fluid and the number of molecules per unit volume.
The latter may depend on factors such as temperature and pressure that affect the intermolecular spacing. For example, as temperature increases, fluid molecules become more energetic and separate, leading to a decrease in the number of molecules per unit volume and, hence, a decrease in density. The density of liquids mainly depends on their molecular composition and to a lesser extent on temperature, which contrasts with gases.
For an ideal gas, the equation pV=nRTpV = nRTpV=nRT, where:
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ppp is the absolute pressure,
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VVV is the gas volume,
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nnn is the number of moles,
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RRR is the gas constant,
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TTT is the absolute temperature.
The density of a gas is given by ρ=MwpRT\rho = \frac{Mwp}{RT}ρ=RTMwp, where MwM_wMw is the molecular weight. Therefore, the density of an ideal gas is closely related to its molecular weight, pressure, and temperature.
The specific gravity sss of a fluid is the ratio of its density ρ\rhoρ to the reference fluid density ρsc\rho_{sc}ρsc under standard conditions. For liquids, ρsc\rho_{sc}ρsc is typically the density of water at 4°C, which is 1000 kg/m³ (1 g/mL). For gases, ρsc\rho_{sc}ρsc may be considered as the density of air at 60°F and 14.7 psi (1 bar) or 0°C at one absolute atmospheric pressure.
Viscosity
Viscosity is a property of fluids that determines their ability to resist shear deformation or flow. This resistance is caused by intermolecular cohesion, which generates friction between adjacent fluid layers during relative motion.
Viscosity depends on the intermolecular attraction and the momentum exchanged between molecules in adjacent fluid layers during relative motion.
Viscosity is affected by temperature but in different ways for liquids and gases.
In liquids, molecular cohesion dictates viscosity. As the temperature rises, the molecules in a liquid become more energetic and separate, leading to reduced cohesion and, thus, a decrease in viscosity.
In gases, viscosity is primarily determined by the exchange of momentum between adjacent layers during relative motion. As temperature increases, the frequency of molecular collisions and intermolecular forces increases, resulting in higher viscosity.
For example, in the diagram with two parallel plates separated by a small gap, where one plate moves under the influence of a force FFF while the other remains stationary, the fluid molecules adhere to the plates. The velocity uuu of the fluid is consistent with the speed of the plate at the boundary.
This is referred to as the no-slip boundary condition, where at the upper boundary, u=Uu = Uu=U, and at the lower boundary, u=0u = 0u=0. The velocity changes linearly with distance yyy between the plates if the gap is small.
Newtonian Fluids
Experiments show that for many fluids, the applied shear stress τ=FA\tau = \frac{F}{A}τ=AF is directly proportional to the velocity gradient.
The constant of proportionality μ\muμ is called the viscosity coefficient, also known as absolute viscosity, dynamic viscosity, or simply viscosity.
Kinematic viscosity is the ratio of dynamic viscosity μ\muμ to density ρ\rhoρ, expressed as ν=μρ\nu = \frac{\mu}{\rho}ν=ρμ.
The ratio U/hU/hU/h is known as the shear deformation rate or shear velocity, and the gradient du/dydu/dydu/dy is called the local shear velocity or strain rate, typically expressed as γ\gammaγ.
Fluids that follow Newton’s law of viscosity (i.e., μ\muμ is constant regardless of the shear velocity) are called Newtonian fluids.
Non-Newtonian Fluids
Fluids for which viscosity varies with the shear rate are called non-Newtonian fluids. Non-Newtonian fluids can be classified into three types: time-independent fluids, time-dependent fluids, and viscoelastic fluids.
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Time-independent fluids: The shear rate at a point is solely determined by the shear stress at that point at the given time. These are known as pure viscous fluids, non-elastic fluids, or generalized Newtonian fluids.
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Time-dependent fluids: The relationship between shear stress and shear rate depends on the duration of shear and the fluid’s history.
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Viscoelastic fluids: These exhibit both ideal fluid and elastic solid characteristics and show partial elastic recovery after deformation.
Types of Non-Newtonian Fluids:
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Plastic: Shear stress must reach a minimum threshold before flow begins.
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Bingham plastic: Similar to plastic but requires a minimum shear stress.
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Pseudoplastic: No minimum shear stress required, and viscosity decreases with increasing shear rate.
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Dilatant: Viscosity increases with shear rate, such as in quicksand.
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Thixotropic: Viscosity decreases as shear force is applied over time.
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Rheopectic: Viscosity increases as shear force is applied over time.
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Viscoelastic materials: These behave like Newtonian fluids under normal conditions but exhibit plastic behavior when exposed to sudden shear changes.
Knowledge Expansion
Viscosity / Viscosity Coefficient μ\muμ
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Definition: A measure of a fluid’s resistance to shear deformation or flow, reflecting the ratio between shear stress and shear rate in the fluid (τ=μ∗dudy\tau = \mu * \frac{du}{dy}τ=μ∗dydu).
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Units: Pascal-seconds (Pa·s) or Newton-seconds per square meter (N·s/m²).
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Commonly used in centipoise (cP, where 1 cP = 0.001 Pa·s).
Importance: Viscosity is one of the core transmission properties in fluid dynamics. It determines:
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Flow resistance (friction loss).
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Boundary layer formation and development.
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The generation and development of turbulence.
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Wetting ability.