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The model BODY is useful but limited. Thermodynamic systems don't just displace; they deform, grow, contract or flow. Extension of Newton's Second Law of Motion is our objective. When mass enters a system, it does so with a velocity, hence as mass enters, momentum enters also. It stands to reason a system equation admitting momentum crossing the system boundary would have momentum transfer terms. The mass equation and momentum equation are used together. For a system with flowing mass we have:
Passage of mass "into" and / or "out of" a
As mass flows, so also
While the above equation is correct, it is useful to identify the forces explicitly, especially to include the effect of pressure-forces. Surroundings, realistically, are not bodies but distributed matter. As a reminder, this equation includes the pressure-force part of the sum of forces is listed specifically.
Pressure-force is important in fluid flow. We separate
The surface integral can be complicated. It sums the differential vector forces over the surface of the system boundary. A negative sign precedes the integral because by convention the differential area (a vector) has as its unit vector directed normal to and outward from the surface. Since the pressure force acts toward the surface; the negative sign is appropriate. The integrand has two components, normal pressure-force, p and shear-force (friction-force) per area, τ (both with units - F/L2). Since friction is a topic for later; the τ is struck out: (X).
Serious usage of the momentum equation is a topic of fluid mechanics. The examples below, introductory, show the consistency of method of the mass, momentum and energy equations.