Principles of Fluid Mechanics, Dimensions, and Units

Concept of Fluid Mechanics

Fluid mechanics is the discipline that studies the laws governing the motion of fluids (liquids and gases) and their interactions with boundaries. Its core principles are founded on three fundamental physical laws: conservation of mass, conservation of momentum, and conservation of energy.

 

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Principles of Fluid Mechanics

Fluid mechanics is a branch of physics that deals with fluids at rest or in motion. In a broad sense, a fluid is a substance that undergoes continuous deformation when subjected to shear (tangential) stress. Unlike solids, whose atoms are tightly bonded and essentially rigid, fluid molecules can move freely relative to one another.

As a result, fluids lack the ability to resist deformation; once a shear stress is applied, they continue to flow.

Solids also deform under shear, but they quickly reach equilibrium when internal elastic forces balance the applied force. In contrast, the rate at which a fluid flows under shear depends on the magnitude of the applied force and on the fluid’s viscosity, which is a measure of the fluid’s resistance to shear deformation or flow.

Because fluids are capable of flowing, they do not have a fixed shape and instead take the shape of their container. Both liquids and gases are fluids, despite their substantial differences in properties. Together with solids and plasmas, these two types of fluids constitute the four states of matter.

In liquids, molecules are closely packed. As a result, liquids have relatively high density and viscosity, the latter arising from intermolecular attractive forces. Liquids are essentially incompressible, meaning that their volume remains nearly constant and is largely unaffected by changes in pressure and temperature.

In contrast, gas molecules are spaced much farther apart than those in liquids. Consequently, gases have relatively low density and viscosity and are highly compressible.

When liquids and gases come into contact, an interface (surface) forms between them. Owing to the imbalance of molecular forces at this interface, a surface tension arises, causing the surface to behave like a stretched membrane. Specifically, because liquid molecules are closely spaced, strong attractive forces exist within the liquid, whereas the forces exerted by gas molecules outside the liquid are relatively weak (and can often be neglected).

Surface tension plays a crucial role in the formation of liquid jets and droplets, which are the foundation of inkjet printing technology. Fluids also possess many other properties that collectively describe their kinematic, transport, and thermodynamic behavior. The development and design of inkjet systems require a deep understanding of these properties, as well as the ability to predict fluid behavior accurately.

 

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Dimensions and Units in Fluid Mechanics

In fluid mechanics, dimensions and units form the fundamental framework for describing physical quantities. A dimension represents the intrinsic nature of a physical quantity (such as length, time, or mass), whereas a unit is the specific measure used to quantify that dimension (such as meters, seconds, or kilograms).

A solid understanding of dimensions and units is essential for theoretical derivations, experimental design, and numerical simulations in fluid mechanics.

I. Basic Physical Quantities and Their Units

Examples of Common Derived Physical Quantities

II. Key Dimensionless Numbers

Dimensionless numbers are formed by combinations of physical quantities and have no units.
They are used to characterize flow similarity, flow regimes, and dominant physical effects.

 

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Note

Dimensionless numbers form the theoretical basis of similarity laws in experiments (such as wind tunnel testing).

III. Key Points of Unit Conversion

In fluid mechanics, unit consistency is critical to avoid calculation errors caused by mixed units.

 

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1. Pressure Units

• 1 atm = 101,325 Pa ≈ 1.013 bar
  
  

• 1 bar = 10⁵ Pa
  
  

• In engineering practice, kPa (10³ Pa) or MPa (10⁶ Pa) are commonly used.
  
  

 

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2. Viscosity Units

• Dynamic viscosity (μ)
  
  

• 1 Pa·s = 10 Poise (P)
  
  

• Kinematic viscosity (ν)
  
  

• 1 m²/s = 10⁴ Stokes (St)
  
  

 

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3. Energy Units

• 1 J = 1 N·m = 1 kg·m²/s²
  
  

• 1 cal = 4.184 J (commonly used in thermodynamic calculations)

IV. Applications of Dimensional Analysis

 

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1. Verifying the Correctness of Equations

Any physical equation must satisfy dimensional homogeneity, meaning that all terms in the equation must have the same dimensions.

Example: Bernoulli Equation

P+12ρv2+ρghP + \frac{1}{2}\rho v^2 + \rho g hP+21ρv2+ρgh

• Pressure term PPP:
  ML−1T−2M L^{-1} T^{-2}ML−1T−2
  
  

• Dynamic pressure term 12ρv2\frac{1}{2}\rho v^221ρv2:
  (ML−3)(L2T−2)=ML−1T−2(M L^{-3})(L^2 T^{-2}) = M L^{-1} T^{-2}(ML−3)(L2T−2)=ML−1T−2

• Potential energy term ρgh\rho g hρgh:
  (ML−3)(LT−2)(L)=ML−1T−2(M L^{-3})(L T^{-2})(L) = M L^{-1} T^{-2}(ML−3)(LT−2)(L)=ML−1T−2

➡ All three terms have identical dimensions, therefore the equation is dimensionally valid.

 

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2. Deriving Physical Laws (Buckingham π Theorem)

If a physical problem involves n physical variables and k fundamental dimensions, then there exist:

n−kn - kn−k

independent dimensionless numbers, which can be used to construct experimental or theoretical models.

Example: Pressure drop in pipe flow

The pressure drop ΔP\Delta PΔP depends on:

ρ, μ, v, L, D (pipe diameter)\rho,\ \mu,\ v,\ L,\ D \ (\text{pipe diameter})ρ, μ, v, L, D (pipe diameter)

• Number of variables:
  n=6n = 6n=6
  
  

• Fundamental dimensions:
  k=3k = 3k=3 (L, M, T)
  
  

• Number of dimensionless groups:
  n−k=3n - k = 3n−k=3

Thus, the relationship can be written as:

Eu=f(Re, L/D)\mathrm{Eu} = f(\mathrm{Re},\ L/D)Eu=f(Re, L/D)

Where:

• Eu = Euler number
  
  

• Re = Reynolds number
  
  

• L/D = geometric ratio
  
  

➡ The Euler number depends on the Reynolds number and the geometry of the system.