When viscous stresses are negligible, a fluid parcel is accelerated primarily by pressure forces and body forces. Newton's second law then gives the Euler momentum equation.

Let

• $\rho(\mathbf r,t)$ be the fluid density;
• $\mathbf v(\mathbf r,t)$ the velocity field;
• $p(\mathbf r,t)$ the pressure;
• $\mathbf f_b$ the body force per unit mass, such as gravity.

The acceleration of a moving fluid parcel is the material derivative

$$\frac{D\mathbf v}{Dt} =\frac{\partial\mathbf v}{\partial t} +(\mathbf v\cdot\nabla)\mathbf v.$$

Pressure-gradient force

Pressure acts normally on the surface of a small fluid element. If pressure were uniform, the forces on opposite faces would cancel. A pressure gradient produces a net force toward lower pressure.

For a small element of volume $dV$, the net pressure force is

$$d\mathbf F_p=-\nabla p,dV.$$

Since its mass is

$$dm=\rho,dV,$$

the pressure-force acceleration is

$$\boxed{\mathbf a_p=-\frac1\rho\nabla p}.$$

Euler's equation

Applying Newton's second law per unit mass gives

$$\boxed{ \frac{D\mathbf v}{Dt} =-\frac1\rho\nabla p+\mathbf f_b }.$$

Equivalently,

$$\boxed{ \frac{\partial\mathbf v}{\partial t} +(\mathbf v\cdot\nabla)\mathbf v =-\frac1\rho\nabla p+\mathbf f_b }.$$

This is the momentum equation for an inviscid fluid. Inviscid means that viscous stresses are neglected in the momentum balance; it does not mean that the fluid has no pressure or that its velocity is spatially uniform.

Conservative body forces

A body force is conservative if it can be written as the negative gradient of a potential per unit mass $\Phi$:

$$\boxed{\mathbf f_b=-\nabla\Phi}.$$

For uniform gravity with vertical coordinate $z$ increasing upward,

$$\Phi=gz,$$

so

$$-\nabla\Phi=-g\hat{\mathbf z}.$$

Euler's equation then becomes

$$\boxed{ \frac{D\mathbf v}{Dt} =-\frac1\rho\nabla p-\nabla\Phi }.$$

Writing the body force as a gradient is especially useful when deriving Bernoulli and circulation results.

Constant-density form

If density is constant,

$$-\frac1\rho\nabla p =-\nabla\left(\frac p\rho\right).$$

Therefore

$$\boxed{ \frac{D\mathbf v}{Dt} =-\nabla\left(\frac p\rho+\Phi\right) }.$$

In this case the entire acceleration field is generated by gradients of scalar quantities, although the nonlinear relation between acceleration and velocity remains through the material derivative.

Worked example: horizontal acceleration from a pressure gradient

Suppose water of density

$$\rho=1000,\mathrm{kg/m^3}$$

has an approximately uniform horizontal pressure gradient

$$\frac{\partial p}{\partial x}=-2000,\mathrm{Pa/m}.$$

Neglect viscosity and horizontal body forces. Euler's equation gives

$$\frac{Dv_x}{Dt} =-\frac1\rho\frac{\partial p}{\partial x}.$$

Thus

$$\frac{Dv_x}{Dt} =-\frac1{1000}(-2000) =\boxed{2.0,\mathrm{m/s^2}}.$$

The negative pressure gradient means pressure decreases toward positive $x$, so the fluid accelerates toward positive $x$.

Relation to Navier-Stokes

For an incompressible Newtonian fluid of constant density and viscosity, Navier-Stokes contains an additional viscous acceleration

$$\nu\nabla^2\mathbf v,$$

where

$$\nu=\frac{\mu}{\rho}$$

is the kinematic viscosity.

Euler's equation is recovered when viscous stresses are negligible at the scale of interest.

This can occur in the bulk of a high-Reynolds-number flow even though thin boundary layers or wakes remain strongly viscous. An inviscid model is therefore often a regional approximation, not a claim that the real fluid has zero viscosity everywhere.

Euler's equation is the local momentum foundation for ideal-flow theory: pressure and body forces determine how an inviscid fluid parcel accelerates.