The Continuity Equation: A PDE for Mass Conservation, from Gauss's Divergence Theorem

Name: The Continuity Equation: A PDE for Mass Conservation, from Gauss's Divergence Theorem
Uploaded: 2022-05-13T10:00:08.000Z
Duration: 38 min 1 s

Gauss's Divergence Theorem and Mass Conservation

Introduction to Gauss's Divergence Theorem

Gauss's divergence theorem is a powerful equality in vector calculus that relates the flux of a vector field through a surface to the divergence of the field within a volume.

It allows for two methods of calculating the flux: integrating the vector field across the surface or computing its divergence throughout the volume.

Application in Fluid Dynamics

The theorem was applied to derive the mass conservation equation, which describes how mass is conserved in fluid flow characterized by density (ρ).

The total rate of change of mass in a volume equals the net amount carried in and out by fluid flow, emphasizing basic accounting principles.

Transition from Surface Integral to Volume Integral

A positive value for f cdot n indicates fluid leaving, leading to a negative rate of change of mass inside the volume.

This transition from surface integral to volume integral using Gauss's theorem is crucial for deriving partial differential equations related to conservation laws.

Importance of Control Volumes

Control volumes can be large enough to encompass significant physical events (e.g., explosions), maintaining validity for conservation laws within these volumes.

Integrals over control volumes are essential for understanding complex systems and ensuring accurate calculations despite external influences.

Source Terms and Mass Conservation Assumptions

In scenarios where mass can be created or destroyed (e.g., nuclear reactions), source terms must be included in conservation equations.

These source terms account for local changes due to energy conversion processes, highlighting limitations on simple assumptions about mass conservation.

Conclusion on Conservation Laws

While basic fluid flows typically conserve mass without creation or destruction, more complex processes may require adjustments with source terms.

Understanding Divergence and Shock Waves in Fluid Dynamics

The Role of Discontinuities in Gauss's Theorem

The discussion begins with the concept of divergence, illustrating how local divergences can cancel out when there is an abrupt change in the vector field, such as a shock function or wave.

It is emphasized that if the functions (f or ρ) are discontinuous, applying Gauss's theorem becomes problematic as it relies on smoothness for valid integration.

Discontinuous functions like shock waves lead to infinite values when computing derivatives, complicating analysis and necessitating a control volume approach instead of traditional PDE methods.

Incompressible vs. Compressible Flow

The speaker highlights the importance of considering incompressible flow scenarios where density remains constant, using liquid water as a primary example due to its resistance to compression.

Air movement at low speeds is also described as approximately incompressible; significant changes in air density only occur near sound speed thresholds (e.g., Mach 0.3).

For compressible flows, such as those involving supersonic jets where density varies significantly, special conditions can be derived from differential equations based on mass conservation principles.

Mathematical Derivation for Incompressibility

A mathematical exploration begins by computing terms related to divergence under the assumption that ρ (density) may vary across space and time.

The expression for divergence involves partial derivatives with respect to spatial coordinates (x, y, z), leading to a more complex formulation when density is not constant.

By applying the chain rule during differentiation, two key components emerge: one representing ρ times the divergence of f and another representing the gradient of ρ dotted with f.

Implications for Fluid Dynamics Analysis

Understanding Incompressible Fluid Dynamics

Key Concepts of Incompressibility

The speaker discusses the assumption that mercury, like many liquids, is incompressible. This means that the density (ρ) remains constant regardless of fluid movement.

When a fluid is incompressible, the partial derivative of density with respect to time (∂ρ/∂t) equals zero, indicating no change in density over time or space.

The discussion leads to the conclusion that for an incompressible flow, both the divergence of the fluid vector field and certain terms in related equations must equal zero.

Mass Continuity Equation

The mass continuity equation states that for incompressible flows, the divergence of the fluid vector field (f) must be zero. This indicates that mass is conserved within a flowing liquid like water.

The importance of this equation lies in its ability to simplify complex expressions when dealing with incompressible fluids by eliminating unnecessary terms.

Conservation Principles

The conservation of mass for incompressible flows implies that if density does not change, then mass conservation can be effectively tracked through simpler mathematical expressions.

The speaker notes how additional complexities such as shock waves or discontinuities can affect these principles and lead to more intricate equations beyond basic continuity.

Momentum and Energy Conservation

Instead of focusing solely on mass conservation, one can also derive equations for momentum conservation in fluids—specifically referencing Navier-Stokes equations which govern fluid motion under various conditions.