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Chp6_331 - 6 Differential Form of the Basic Eq'ns The...

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6. Differential Form of the Basic Eq'ns The continuum assumption has allowed the treatment of fluid properties as fields, scalar or vector, which are functions of space and time. scalar fields vector fields dens. - ( x , y , z ; t ) vorticity - ( x , y , z ; t ) press. - P ( x , y , z ; t ) velocity - V ( x , y , z ; t ) temp. - T ( x , y , z ; t ) Total change in a scalar field, , due to change in position, r = position vector = x i + y j + z k d φ = ∂φ x d x + ∂φ y d y + ∂φ z d z d φ = φ .d r φ = ∂φ x i + ∂φ y j + ∂φ z k d r = d x i + d y j + d z k 56
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Consider P & Q to be 2 points on a surface ( x , y , z )=C. These points are chosen so that Q is a distance r d from P. Then moving from P to Q, The change in ( x , y , z )=C is given by, d φ = φ .d r = 0 since the 2 points are on the same surface C. This shows that is perpendicular to r d . Since r d may be in any direction from P, as long as it stays on the surface =C, point Q being restricted to the surface but having arbitrary direction, is seen as normal to the surface =C. Example : P 3 P 2 > P 3 P 1 > P 2 P isobars; curves of const. press. pressure gradient Recall that the total or substantial derivative is: D( ) D t = ( ) t + u ( ) x + v ( ) y + w ( ) z = ( ) t + V . ( ) 57
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6.1. Conservation of mass mass/unit area = ρ V V = u , v , w ; r = x , y , z Net flux of mass/time y x w z w z w z x v y v y v z y u x u x u z y x w z v y u x z y x V 58
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Conservation of mass states that the net flux of
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