Unit vectors along and normal to the streamline are denoted by  and  respectively. We assume that the fluid is inviscid and so viscous forces, , is zero. For steady flow, we apply Newton’s second law along the streamline direction, s, giving us

where  represents the sum of the s components of al the forces acting on the particle, which has mass , and  is the acceleration in the s direction. Here,  is the particle volume. This equation is valid for both compressible and incompressible fluids implying that the density need not be constant throughout the flow flied.

The gravity force of the particle can be written as , where  is the specific weight of the fluid. Hence, the component of the weight force in the direction of the streamline is

If the streamline is horizontal at the point of interest, then . There is no component of the particle weight contributing to the acceleration.

Discussed previously in the earlier chapters, the pressure is not constant throughout a stationary fluid because of the fluid weight. Likewise, in a flowing fluid the pressure is usually not constant. In general, for steady flow, . if the pressure at the center of the particle is denoted by p, then it average value on the two end faces that are perpendicular to the streamline are  and . Since we are considering a small particle, we can use  one-term Taylor series expansion for the pressure field, as done previously, to obtain

Thus, if  is the net pressure force on the particle in the streamline direction, it follows that

and rearranging to give

Notice that the actual level of the pressure, p, is not important. The net pressure force is produced from the fact that pressure is not constant throughout the fluid. The nonzero pressure gradient, , is what provides the net pressure force in the particle.

Hence, the net force acting in the streamline direction on the particle is given by

By combining this with our previous equation, we obtain the following equation of motion along the streamline direction:

This is a representation of the fact that it is the fluid density, not the mass, of the fluid particle that is important. The physical interpretation of this equation is that a change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle weight along the streamline. For a flowing fluid the pressure and weight forces do not necessarily balance. This unbalance force is what causes the appropriate acceleration resulting in particle motion.

This is the equation we have after applying Newton’s second law. We have left the density to be a variable. Depending on incompressible or compressible fluids, this density function must be manipulated accordingly. We need to be mindful of this when integrating the function, which will be our next job.