Newtons Laws of Mechanics

We have shown how a pointwise DE can be derived by application of the macroscopic principle of mass conservation to a material (control) volume of fluid. In this section, we consider the derivation of differential equations of motion by application of Newton s second law of motion, and its generalization from linear to angular momentum, to the same material control volume. It may be noted that introductory chemical engineering courses in transport phenomena often approach the derivation of these same equations of motion as an application of the conservation of linear and angular momentum applied to a fixed control volume. In my view, this obscures the fact that the equations of motion in fluid mechanics are nothing more than the familiar laws of Newtonian mechanics that are generally introduced in freshman physics. 

We begin with Newton s second law, which may be stated in the form 

This can be applied directly to the material (control) volume of fluid, Vm(t), which was introduced in the last section. As required for application of (2 21), this is a fixed body of material in the continuum sense. The resulting equation is 

The fact that the material control volume has a time-dependent shape does not lead to any complication of principle in applying Newton s second law. To proceed further, we must consider the types of forces that appear on the right-hand side of (2-22). 

With the necessity for body and surface forces thus identified, we can complete the mathematical statement of Newton s second law for our material control volume  

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Thus, his view of the elements allowed Mendeleev to maintain the vahd-ity of the periodic law even in instances where observational evidence seemed to point against it. Such boldness may have resulted from a deeply held beliefs that the periodic law applied to the abstract elements as basic substances and that this law was as fundamental and equal in status to Newtons laws of mechanics. Had he been more of a positivist, Mendeleev might easily have lost sight of the importance of the periodic law and might have harbored doubts about some of his predictions. 

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