Key Theories in Fluid Mechanics
Fluid mechanics is a fundamental branch of physics and engineering that studies the behavior
of fluids (liquids and gases) at rest and in motion. Below are the key theories and principles
that form the foundation of fluid mechanics:
1. Continuum Hypothesis
Fluid mechanics treats fluids as continuous media, assuming their properties (density,
pressure, velocity) vary smoothly across space and time. This approach ignores the molecular
structure of fluids and considers them as homogeneous substances.
2. Governing Equations
The motion and behavior of fluids are governed by three fundamental laws of physics:
Conservation of Mass (Continuity Equation):
This law ensures that mass is neither created nor destroyed in a control volume. For
incompressible fluids (e.g., water), the inflow equals the outflow. Mathematically:
∂ρ∂t+∇⋅(ρv⃗)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0
For incompressible fluids (ρ=constant\rho = \text{constant}):
∇⋅v⃗=0\nabla \cdot \vec{v} = 0
Conservation of Momentum (Navier-Stokes Equations):
These equations describe how the velocity of a fluid changes due to forces such as pressure
gradients, gravity, and viscosity. The general form is:
ρ(∂v⃗∂t+(v⃗⋅∇)v⃗)=−∇p+μ∇2v⃗+F⃗\rho \left( \frac{\partial \vec{v}}{\partial t} + (\vec{v}
\cdot \nabla) \vec{v} \right) = -\nabla p + \mu \nabla^2 \vec{v} + \vec{F}
Where:
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ρ\rho: density
v⃗\vec{v}: velocity
pp: pressure
μ\mu: dynamic viscosity
F⃗\vec{F}: external forces (e.g., gravity)
Conservation of Energy:
This equation governs how energy is transferred within a fluid, considering thermal, kinetic,
and potential energy. It is expressed as:
∂E∂t+∇⋅(v⃗(E+p))=Q+Φ\frac{\partial E}{\partial t} + \nabla \cdot (\vec{v} (E + p)) = Q +
\Phi
Where QQ is the heat added to the fluid, and Φ\Phi represents viscous dissipation.
3. Bernoulli's Equation
A simplified form of the Navier-Stokes equations, Bernoulli's principle applies to ideal (nonviscous), incompressible, and steady flows. It states that the total energy along a streamline is
constant:
p+12ρv2+ρgh=constantp + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}
Where:
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pp: static pressure
12ρv2\frac{1}{2} \rho v^2: dynamic pressure (velocity-related)
ρgh\rho g h: hydrostatic pressure (height-related)
This principle is widely used in applications like flow in pipes, aerodynamics (around wings),
and fluid flow meters.
4. Reynolds Number (ReRe)
The Reynolds number is a dimensionless parameter that characterizes the flow regime of a
fluid, comparing inertial forces to viscous forces:
Re=ρvLμRe = \frac{\rho v L}{\mu}
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Re≪1Re \ll 1: Dominated by viscous forces (laminar flow).
Re≫1Re \gg 1: Dominated by inertial forces (turbulent flow).
5. Laminar vs. Turbulent Flow
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Laminar Flow: Fluid particles move in parallel layers with minimal mixing (low
ReRe).
Turbulent Flow: Chaotic and irregular flow with vortices and mixing (high ReRe).
6. Pascal's Principle
This principle states that pressure applied to a confined fluid is transmitted uniformly in all
directions. It forms the basis of hydraulic systems:
p=ρghp = \rho g h
Applications
Fluid mechanics is applied in many fields, including:
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Piping systems and flow networks
Aerodynamics (airflow over aircraft)
Hydrodynamics (flow around ships and submarines)
HVAC systems (heating, ventilation, air conditioning)
Biomedical flows (e.g., blood circulation)