Problem 5.17

# Problem 5.17 - 2 ) ( dxdydz x v m net y ) ( dxdydz x w m...

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Problem 5.17 [Difficulty: 4] Given: Conservation of mass in rectangular coordinates Find: Identical result to Equation 5.1a by expanding products of density and velocity in a Taylor Series. Solution: We will use the diagram in Figure 5.1 (shown here). We will apply the conservation of mass evaluating the derivatives at point O:    0 t w z v y u x Governing Equations: (Continuity equation - Eqn 5.1a) Assumptions: Expansion of density and velocity via Taylor series is valid around point O. udydz udA m x In the x-direction, the mass flux is:  dydz dx x u u m dx x 2 2 At the right face: (out of the volume)  dydz dx x u u m dx x 2 2 At the left face: (into the volume) The net mass flux out of the volume in the x-direction would then be:      dxdydz x u dydz dx x u u dydz dx x u u m m m dx x dx x net x 2 2 2
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Unformatted text preview: 2 ) ( dxdydz x v m net y ) ( dxdydz x w m net z ) ( Similarly, the net mass fluxes in the y-direction and z-direction are: dxdydz t dt dm vol The rate of mass accumulation in the volume is: Now the net outflux must balance the accumulation: ) ( vol net dt dm m dxdydz t dxdydz x w dxdydz x v dxdydz x u Therefore we may write: t x w x v x u We may divide the volume out of all terms: (Q.E.D.) t x w x v x u...
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