Physlib

Physlib.FluidDynamics.FluidFlow.Continuity

Continuity equation for fluid flows

i. Overview

This module defines the classical conservative mass-balance equation for a fluid flow and the corresponding continuity residual. These definitions are independent of a particular momentum equation, so they can be reused by Navier-Stokes, Euler, and other fluid models.

ii. Key results

- `FluidFlow.ClassicalContinuityEquation` : Classical conservation of mass in conservative form. - `FluidFlow.continuityResidual` : The scalar residual `partial_t rho + div (rho u)`. - `FluidFlow.SmoothContinuityEquation` : Continuity for globally differentiable fields. - `FluidFlow.classicalContinuityEquation_of_smoothContinuityEquation` : A smooth continuity equation implies the classical continuity equation.

iii. Table of contents

  • A. Continuity equations

iv. References

A. Continuity equations

4 declarations

definition

Classical continuity equation ∂ρ∂t+∇⋅(ρu)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0

For a fluid flow in dd-dimensional space with density ρ(t,x)\rho(t, x) and velocity field u(t,x)\mathbf{u}(t, x), the classical continuity equation asserts that for all times tt and positions xx, if ρ\rho is differentiable with respect to time at tt and the mass flux ρu\rho \mathbf{u} is spatially differentiable at xx, then the following conservation law holds: ∂ρ∂t(t,x)+∇⋅(ρ(t,x)u(t,x))=0\frac{\partial \rho}{\partial t}(t, x) + \nabla \cdot (\rho(t, x) \mathbf{u}(t, x)) = 0 where ∂∂t\frac{\partial}{\partial t} denotes the time derivative and ∇⋅\nabla \cdot denotes the spatial divergence operator.

definition

Continuity residual ∂ρ∂t+∇⋅(ρu)\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u})

For a fluid flow in dd dimensions with mass density ρ\rho and velocity field u\mathbf{u}, the continuity residual at time tt and position xx is defined as: ∂ρ∂t(t,x)+∇⋅(ρ(t,x)u(t,x))\frac{\partial \rho}{\partial t}(t, x) + \nabla \cdot (\rho(t, x) \mathbf{u}(t, x)) where ∂∂t\frac{\partial}{\partial t} is the partial derivative with respect to time, and ∇⋅\nabla \cdot is the spatial divergence operator. This residual represents the local rate of change of mass plus the divergence of mass flux; in a physical flow, this quantity vanishes to satisfy the conservation of mass.

definition

Smooth continuity equation ∂ρ∂t+∇⋅(ρu)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0

For a fluid flow in dd dimensions with mass density ρ\rho and velocity field u\mathbf{u}, the smooth continuity equation is the proposition that: 1. For every position xx, the density function t↦ρ(t,x)t \mapsto \rho(t, x) is differentiable. 2. For every time tt, the mass flux function x↦ρ(t,x)u(t,x)x \mapsto \rho(t, x) \mathbf{u}(t, x) is differentiable. 3. For all times tt and positions xx, the continuity residual vanishes: ∂ρ∂t(t,x)+∇⋅(ρ(t,x)u(t,x))=0\frac{\partial \rho}{\partial t}(t, x) + \nabla \cdot (\rho(t, x) \mathbf{u}(t, x)) = 0 where ∂∂t\frac{\partial}{\partial t} denotes the partial derivative with respect to time and ∇⋅\nabla \cdot denotes the spatial divergence operator.

theorem

Smooth Continuity Equation implies Classical Continuity Equation

For a fluid flow in dd-dimensional space with density ρ\rho and velocity field u\mathbf{u}, if the flow satisfies the smooth continuity equation—meaning that ρ\rho is differentiable with respect to time, the mass flux ρu\rho \mathbf{u} is spatially differentiable, and the equation ∂ρ∂t+∇⋅(ρu)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 holds for all times tt and positions xx—then it also satisfies the classical continuity equation.