Equation of motion

The equation of motion is the conservation of momentum applied to fluid flow. It is used to determine the velocity field in a flowing continuum.

General development

We start with the general formulation of the microscopic balance using the Eulerian specification for the transport quantity of momentum, or $\mathrm{\Psi }=m\mathbf{v}$

$\frac{\partial \left(\rho \hat{t}\right)}{\partial t}=-\mathrm{\nabla }\cdot {\mathrm{\Phi }}_{\mathrm{\Psi }}+{\dot{r}}_{{\mathrm{\Psi }}_{}}$

(1)

where

• $\rho$ is the mass density
• $\hat{\psi }$ is the the transport quantity on a unit mass basis, which in this case is $\hat{\psi }=\mathbf{v}$
• ${\mathrm{\Phi }}_{\mathrm{\Psi }}=\left(\rho \mathbf{v}\right)\mathbf{v}+\mathit{\tau }$ is the convective momentum flux due to both bulk flow and molecular transfer mechanisms.
• ${\dot{r}}_{{\mathrm{\Psi }}_{}}=-\mathrm{\nabla }P+\rho \mathbf{g}$, which are the external forces per unit volume. Recall that the momentum of a system is conserved unless acted upon by an external force.

Eq. (1) therefore is expressed as

$\frac{\partial \left(\rho \mathbf{v}\right)}{\partial t}=-\mathrm{\nabla }\cdot \left(\left(\rho \mathbf{v}\right)\mathbf{v}\mathbf{+}\mathit{\tau }\right)-\mathrm{\nabla }P+\rho \mathbf{g}$

(2)

Applying the chain rule to the $\mathrm{\nabla }\cdot \left(\rho \mathbf{v}\right)\mathbf{v}$ term and rearranging, Eq. (2) can be expressed as

$\rho \left(\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot \mathrm{\nabla }\mathbf{v}\right)+\mathbf{v}\left(\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot \rho \mathbf{v}\right)=-\mathrm{\nabla }P-\mathrm{\nabla }\cdot \mathit{\tau }+\rho \mathbf{g}$

(3)

Noting that $\left(\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot \rho \mathbf{v}\right)=0$ from the continuity equation, Eq. (3) becomes

$\rho \left(\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot \mathrm{\nabla }\mathbf{v}\right)=-\mathrm{\nabla }P-(\mathrm{\nabla }\cdot \mathit{\tau })+\rho \mathbf{g}$

(4)

where

• $\rho \frac{\partial \mathbf{v}}{\partial t}$ is the local accelerative force per unit volume.
• $\rho (\mathbf{v}\cdot \mathrm{\nabla }\mathbf{v})$ is the advective or the inertial force per unit volume.
• $-\mathrm{\nabla }P$ is pressure force per unit volume
• $-(\mathrm{\nabla }\cdot \mathit{\tau })$ is the viscous force per unit volume
• $\rho \mathbf{g}$ is the body force per unit volume.

Compact form

The equation of motion in the material derivative form can be written compactly as

$\mathrm{\rho }\frac{D\mathbf{v}}{Dt}=-\mathrm{\nabla }P-(\mathrm{\nabla }\cdot \mathit{\tau })+\rho \mathbf{g}$

(5)

where the symbol $D/Dt$ represents the material derivative. Each term in the above equation has the units of a "body force" (force per unit volume). This equation can be interpreted in the context of Newton's second law of motion, for example, $m\frac{d\mathbf{v}}{dt}=\sum _i\mathbf{F}$, where the accelerative force acting on a fluid packet is equal to the sum of forces acting on the fluid packet.

Special forms

Incompresible, Newtonian fluid

Known as the Navier-Stokes equations, for an incompressible Newtonian fluid, the equation of motion is expressed as:

$\mathrm{\rho }\frac{D\mathbf{v}}{Dt}=-\mathrm{\nabla }P+\mathrm{\mu }{\mathrm{\nabla }}^2\mathbf{v}+\rho \mathbf{g}$

(6)

Remember, in an incompressible Newtonian fluids the following holds true: $\mathit{\tau }=-\mu \left(\mathrm{\nabla }\mathbf{v}+\mathrm{\nabla }{\mathbf{v}}^{⊺}\right)$

Inviscid flow

Known as the Euler equation, this simplified form of the equation of motion describes flows characterized by large Reynolds numbers, where inertial (advective) forces dominate over viscous forces.

$\rho \frac{D\mathbf{v}}{Dt}=-\mathrm{\nabla }p+\rho \mathbf{g}$

(7)

The Bernoulli's law is a direct consequence of assuming inviscid flow.

Creeping flow

Known as Stokes flow, this type of flow occurs when the viscous forces are much larger than the inertial (or advective) forces, This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. The simplified equation of motion is

$\rho \frac{\partial \mathbf{v}}{\partial t}=-\mathrm{\nabla }p-\left(\mathrm{\nabla }\cdot \mathit{\tau }\right)+\rho \mathbf{g}$

(8)