session1_lecture2_MTMFAO-combined

Session 1: Definition of Fluids

Definition of Fluids

Fluids are defined as substances that undergo continuous deformation when subjected to external stress, meaning they do not return to their original shape after the stress is removed. This category encompasses both liquids, which are incompressible and take the shape of their containers, and gases, which are compressible and expand to fill the entire volume available to them. The behavior of fluids is influenced significantly by their molecular interactions and thermodynamic properties.

Properties of Solids, Liquids, and Gases

  • Solids: Solids maintain a definite shape and volume, displaying rigidity. They do not deform significantly under small external stresses due to strong intermolecular forces, which keep particles tightly packed in a structured arrangement.

  • Fluids (Liquids and Gases):

    • Deformation: Fluids can deform continuously under the application of external forces, showing a significant difference in behavior from solids.

    • Irreversible Deformation: Once a fluid has been deformed, it does not retain the original shape. For example, pouring water will alter its container's shape, but the water itself does not have a fixed shape.

    • Microscopic Properties: The properties of fluids at a microscopic level are influenced by the arrangement and interaction of molecules, affecting the fluid's viscosity, density, and diffusion characteristics.

    • Macroscopic Properties: These are obtained from observing the average behavior of a large number of molecules, allowing us to describe fluids in terms of bulk properties like pressure, temperature, and flow rate.

Continuum Hypothesis

The Continuum Hypothesis posits that fluids can be treated as continuous media, ignoring the molecular composition for practical applications. This approach is valid under specific conditions:

  • The characteristic length scale of a fluid parcel (L) is significantly larger than the mean free path between molecular collisions (L >> mean free path).

  • The number of molecules contained within a given volume, represented as L^3, is substantial.

  • Local thermodynamic equilibrium exists within the fluid parcel, allowing for uniform temperature and pressure distributions throughout.

Variables:

  • Extensive Variables: These are properties that scale with the number of molecules present, such as mass (M) and volume (V).

  • Intensive Variables: These do not depend on the quantity of substance and remain constant regardless of scale, including pressure (P) and temperature (T).


  • Specific Property: This involves any extensive variable divided by the mass of the parcel, providing a unique value per unit mass (e.g., specific volume):[ \text{Specific Volume} = \frac{V}{M} ]

Scalar and Vector Fields

Fluids are characterized by fields that assign values to every point in space, with variations over time:

  • Scalar Fields: These fields have a single value at each point, such as temperature (T), which can influence the thermal behavior of a fluid.

  • Vector Fields: These are described by vectors at each location, such as velocity (( \mathbf{u} )), indicating both magnitude and direction.

Examples of Fields:

  • Pressure: Scalar

  • Flow speed: Scalar

  • Energy: Scalar

  • Position: Vector (( \mathbf{r} ))

  • Density: Scalar (( \rho ))

  • Acceleration: Vector (( \mathbf{a} ))

  • Spin (Vorticity): Vector (( \mathbf{ω} ))

  • Momentum: Vector (( \mathbf{p} = \rho \mathbf{u} ))

Relative Motion in Fluids



The complexity of fluid flow arises from the variations in velocity within the flow field. If every point within a fluid moved in unison, the flow would be simple and uninteresting. The Relative Velocity between two points can be described mathematically:[ \delta v = \mathbf{u} (\delta x, \delta y, t) - \mathbf{u} (0, 0, t) ]This can be further analyzed using Taylor expansion to understand the variations in each component of velocity and how they interact.

Key Terms:


  • Divergence: Represents the rate of expansion of a fluid and is defined as[


abla \cdot \mathbf{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} ]



  • Vorticity: Defined as[ \mathbf{ω} =
    abla \times \mathbf{u} ]which measures the rotation of fluid elements within the flow.

Flow Kinematics



Fluid flow can be characterized by its velocity field expressed as:[ \mathbf{u} = \mathbf{u}(\mathbf{r}, t) = (u, v, w) ]Kinematics deals with how fluid properties are transported through space, using a known velocity field, while dynamics focuses on deriving the velocity from the forces acting on the fluid.

Flow Visualization

To interpret complex fluid motion, various visualization techniques are utilized:

  • Trajectories: Paths traced by individual particles as they move through the fluid.

  • Streaklines: Lines that represent the position of dye injected into the flow, showing how points on the line evolve over time.

  • Streamlines: Curves that are tangent to the velocity field at a given instant; in steady flow, streamlines can not cross but show the direction of fluid motion.

Steady vs. Unsteady Flow

  • Steady Flow: In this type of flow, the characteristics of fluid, including trajectories and streamlines, remain constant over time, indicating that the fluid’s properties do not change with time at any point in space.

  • Unsteady Flow: This flow is characterized by changing properties over time, leading to trajectories that diverge from streamlines and can lead to complex patterns due to the exponential separation of fluid elements.

  • Chaotic Advection: Refers to the phenomenon where continuous separation of trajectories occurs, leading to unpredictable and intricate fluid behavior.

Eulerian and Lagrangian Perspectives

Two main descriptions of fluid motion are:


  • Eulerian Frame: Observes fluid processes at fixed points in space while examining how quantities change over time:[ Q = Q(\mathbf{r}, t) ]


  • Lagrangian Frame: Follows individual fluid parcels over time, describing how properties evolve along particle paths:[ Q = Q(a, t) ]



  • Material Derivative:[ \frac{DQ}{Dt} = \frac{\partial Q}{\partial t} + (\mathbf{u} \cdot
    abla) Q ]represents the change in a quantity as observed from the perspective of a fluid parcel, noting that tracers of properties are conserved following the movement of fluid elements.

Conservation Laws and Equations of Motion

The conservation of mass and momentum dictates that the solution to fluid motion equations must account for the various forces acting on fluid parcels, including gravitational forces (long-range) and intermolecular interactions (short-range), which play a critical role in determining fluid behavior.

Vorticity and Circulation


  • Vorticity: It is a measure of the local rotation of fluid elements, expressed as[ \mathbf{ω} =
    abla \times \mathbf{u} ]



  • Circulation: Refers to the total vorticity around a closed curve and is connected to the concept of vorticity through Stokes' theorem:[ \Gamma = \oint_C \mathbf{u} \cdot d\mathbf{l} = \iint_S (
    abla \times \mathbf{u}) \cdot d\mathbf{A} ]which relates circulation to the vorticity integrated over the surface enclosed by the curve.

  • Kelvin's Theorem: Suggests that in the absence of viscosity, the total circulation in a fluid remains constant over time, highlighting the conservation of vorticity during ideal fluid motion.

Potential Vorticity



  • Ertel Potential Vorticity: This quantity combines the concepts of vorticity and stratification; it remains conserved within a fluid parcel during frictionless, adiabatic flow:[ PV = \frac{\mathbf{ω}}{\rho} \frac{d\rho}{dz} ]thereby illustrating the relationship between fluid dynamics and atmospheric behavior.

Summary of Concepts

The dynamics of fluids are governed by a range of mechanisms that include divergence, deformation, and circulation. A grasp of these concepts is crucial for understanding more complex applications in meteorology and oceanography, where fluid flow plays a pivotal role in global systems and weather patterns.