Unformatted text preview: or leaves? Mass flow through a surface element is ∙ Momentum flow would then be ∙ If S is the surface containing V we integrate all over the surface to get the total momentum entering (or leaving) V through S ∙ So the total rate of change of momentum in V is ∙ Now consider the forces Pressure acts inward The force due to viscosity which for now we merely assign a symbol And consider the body force (gravity, electric, magnetic, etc) If we use f as a force per unit mass We can now write Newton’s second law as an integral equation as ∙ local convective pressure body viscous derivative derivative force force This is the form we will use with control volume problems If we bring the time derivative inside the integral and apply the gradient theorem to the first integral on the right side we will have ∙ ∙ is a scalar and ̂ ̂ so we can rewrite this as three scalar equations ∙ ∙ ∙ We will work with the x equation, but the same steps can be applied to all three ∙
∙ And apply the divergence theorem to the seco...
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- Fall '10
- Noncototient, January 20, total rate