Physlib

Physlib.FluidDynamics.CauchyFlow.BodyForce

Body-force predicates for Cauchy flows

i. Overview

This module defines predicates for conservative specific body forces on `CauchyFlow`.

ii. Key results

- `CauchyFlow.HasBodyForcePotential` : Predicate encoding the convention `specificBodyForce = -grad Phi`. - `CauchyFlow.HasConservativeBodyForce` : Predicate saying the specific body force has some potential.

iii. Table of contents

  • A. Conservative-force convention

iv. References

A. Conservative-force convention

2 declarations

definition

`CauchyFlow` has body-force potential Φ\Phi when b=−∇Φ\mathbf{b} = -\nabla \Phi

A `CauchyFlow` in dd dimensions is said to have a body-force potential Φ:Space d→R\Phi: \text{Space } d \to \mathbb{R} if its specific body force b(t,x)\mathbf{b}(t, x) satisfies the condition b(t,x)=−∇Φ(x)\mathbf{b}(t, x) = -\nabla \Phi(x) for all times tt and spatial positions xx. Here, ∇Φ\nabla \Phi denotes the gradient of the potential function Φ\Phi.

definition

`CauchyFlow` has conservative body force when b=−∇Φ\mathbf{b} = -\nabla \Phi for some potential Φ\Phi

A `CauchyFlow` in dd dimensions is said to have a conservative body force if there exists a scalar potential function Φ:Space d→R\Phi: \text{Space } d \to \mathbb{R} such that the specific body force b\mathbf{b} of the flow satisfies b(t,x)=−∇Φ(x)\mathbf{b}(t, x) = -\nabla \Phi(x) for all times tt and spatial positions xx, where ∇Φ\nabla \Phi denotes the gradient of the potential.