Conv_hw1_sol - 1) D u v w Dt x y z t = + + + d dx dy dz dt...

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Unformatted text preview: 1) D u v w Dt x y z t = + + + d dx dy dz dt x y z t = + + + d dx dy dz dt x dt y dt z dt t = + + + d u v w dt x y z t = + + + d D dt Dt = 2) Multiply x-momentum by u yx xx Du P u u u u uX Dt x x y = - + + + Multiply y-momentum by v yy xy Dv P v v v v vY Dt y x y = - + + + Add together yx yy xy xx Du Dv P P u v u u u uX v v v vY Dt Dt x x y y x y + = - + + +- + + + Divide by ( 29 1 1 1 yy yx xy xx Du Dv P P u v u v u v u v uX vY Dt Dt x y x x y y + = - + + + + + + + Substitute Identity ( 29 2 2 1 2 Du Dv D u v u v Dt Dt Dt + = + Final Form ( 29 ( 29 2 2 1 1 1 1 2 yy yx xy xx D P P u v u v u v u v uX vY Dt x y x x y y + = - + + + + + + + 3) The figure below shows a differential control volume using spherical coordinates. The conservation of mass holds that with Unlike the Cartesian coordinate system, spherical coordinates do not allow all of the entering and exiting areas to be equivalent; each area must be calculated separately to ensure they are done correctly. The derivation will be divided into three steps (one for each coordinate). R Coordinate Leaving off the higher terms, a Taylor series expansion can be used to approximate () r+r Inserting the Taylor expansion into the expression for the rate of mass entering the C.V. less the Rate of Mass Rate of Mass Rate of Mass Accumulation Entering the C.V. Exiting the C.V....
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Conv_hw1_sol - 1) D u v w Dt x y z t = + + + d dx dy dz dt...

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