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# Equation of Motion

In one dimensional problems, if the applied force is , we solve In three dimensional situations we have, in cartesians coordinates, we have This is written in concise form by using vector notation so that (1.11)

In a continuuous medium, a variation of pressure, , from one location to another produces a pressure force. is the force per unit area exerted on a surface by a fluid (liquid or gas) or plasma. Thus, if the force on a face is . This is illustrated in Figure 1.5. Consider a fluid element, lying between the points and . On the left hand side the pressure force is and on the right hand side the force is The minus sign arises because the force on the right hand side acts towards the left. Therefore, the net force in the direction PER UNIT VOLUME is Thus, the equation of motion (1.11) per unit volume for a liquid, gas or plasma of density (mass per unit volume) becomes (1.12)

Pressure gradients produce a force in the opposite direction to the actual gradient. Thus, the pressure gradient force acts from areas of high pressure to areas of low pressure and is equal in magnitude to the gradient of the pressure. This is illustrated in Figure 1.6. In three dimensional problems, where , (1.12) generalises to give three components as or in vector form (1.13)   Next: The Gradient Operator, Up: Introduction Previous: Scalar and Vector Products
Prof. Alan Hood
2000-11-06