APPENDIX ASign Convention for the Forceon a Fluid Particle

The equations of motion are a statement of Newton’s law of motion applied to asystem that consists of a fluid particle. The latter can be thought of as a constituentpart of a fluid continuum that moves with the local fluid velocity and has a constantinfinitesimal mass; however, it need not have constant volume and density sincethese two quantities can vary in time while maintaining constant fluid particlemass. The fluid particle is a useful system since its mass is so small that in thelimit of vanishingly small volume, its properties become those of a point in thefluid continuum that moves with the local fluid velocity.

To apply Newton’s law of motion to a fluid particle, we need to consider theforce �Fp exerted on a fluid particle by the fluid surrounding it. The total stresstensor σ by definition operates on the local unit normal vector �n to a surface to

cause a force. The convention employed here defines �Fp to be the force exertedon the surface of the particle by the fluid on the side into which the normal vectoris pointing; this is shown in Figure A.1-1 and is given by

�Fp = −∫

S

�nσ dS (A.1-1)

It follows that the force �Ff that the fluid particle exerts on the surrounding fluidis given by

�Ff = +∫

S

�nσ dS (A.1-2)

An entirely equivalent definition is the following: σij is the total stress exerted inthe +j -direction on a fluid surface defined by a plane of constant i by the fluidin the region of lesser i. For example, σyx is the total stress exerted in the +x-direction on a fluid surface defined by a plane of constant y by the fluid locatedbeneath this surface.

Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approachto Model Building and the Art of Approximation, By William B. KrantzCopyright 2007 John Wiley & Sons, Inc.

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SIGN CONVENTION FOR THE FORCE ON A FLUID PARTICLE 481

Total surfacearea S

n

Fluid particle

→

Figure A.1-1 Fluid particle of surface area S showing the unit normal vector to surface �n.

It is important to note that there is no agreement in the technical literatureregarding the sign convention for the force acting on a fluid particle. Popular engi-neering textbooks such as Transport Phenomena by Bird et al.1 employ the signconvention implied by equations (A.1-1) and (A.1-2) whereas other books employa convention in which the signs are reversed in these two equations. One can useeither convention, of course. However, in doing so, one must be certain to use com-patible signs in the constitutive equations that relate the stress to the rate of strain;that is, to ensure that positive forces are exerted in the positive coordinate direc-tion, the sign convention given by equations (A.1-1) and (A.1-2) implies a certainset of signs in the constitutive equations that relate the stress to the rate of strain.Hence, the signs given for Newton’s constitutive equation in Appendixes B and Dare compatible with the sign convention for the force on a fluid particle given byequations (A.1-1) and (A.1-2) When using constitutive equations taken from thetechnical literature, one has to be careful to employ a form of the equations ofmotion that has a compatible sign convention.

1R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, Hoboken, NJ,2002, p. 13.