Differential_Balances_web

Differential_Balances_web - ENU 4133 Differential Balances...

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Unformatted text preview: ENU 4133 Differential Balances January 31, 2011 Chapter 4 Differential Relations for Fluid Flow In scope of Exam #1 and #5 I 4.1 acceleration field of a fluid I 4.2 differential mass conservation (continuity) I 4.3 differential momentum conservation I 4.3, 4.6, 4.10 Navier-Stokes equation and solutions I 4.5 differential energy equation Sections 4.7-4.9 are intermediate topics, covered later in the course. Acceleration Field of a Fluid General velocity field (Cartesian coordinates Eulerian/fixed frame): ~ V ( ~ r , t ) = ~ iu ( ~ r , t ) + ~ jv ( ~ r , t ) + ~ kw ( ~ r , t ) (1) Acceleration field differentiate w/r/t t . ~ a = d ~ V dt = ~ i du dt + ~ j dv dt + ~ k dw dt (2) Note: total time derivatives, not partial derivative. Considering u / x component: du dt = u t + u x dx dt + u y dy dt + u z dz dt (3) du dt = u t + u u x + v u y + w u z (4) du dt = u t + ~ V u (5) Acceleration Field of a Fluid (2) du dt = u t + ~ V u (6) dv dt = v t + ~ V v (7) dw dt = w t + ~ V w (8) ~ a = d ~ V dt = ~ V t + ~ V ~ V (9) Acceleration composed of local and convective parts. In general, this is termed the substantial or material derivative of B use of the DB / Dt notation is helpful, but not universal (particularly in White) DB Dt = B t + ~ V B (10) Differential Mass Conservation Equation I Take control volume as differential volume dx dy dz = dV I Identify generation and destruction terms (hint: very easy) I Identify inflow and outflow terms I Compile terms & do a little math I Transform resulting equation into vector form Process for differential momentum & energy equations not dissimilar. Inlet and Outlet Mass Flows (x-direction) Figure 4.1, White Inlet and Outlet Mass Flows In x-direction, from figure: ( AV ) in , x = udydz (11) ( AV ) out , x = u + x ( u ) dx dydz (12) ( AV ) out , x- ( AV ) in , x = x ( u ) dxdydz (13) y and z directions, similar: ( AV ) out , y- ( AV ) in , y = y ( v ) dydxdz (14) ( AV ) out , z- ( AV ) in , z = z ( w ) dzdxdy (15) Deriving Continuity Z CV t d V t dxdydz (16) t dxdydz + x ( u ) dxdydz + y ( v ) dydxdz + z ( w ) dzdxdy = 0 (17) t + x ( u ) +...
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Differential_Balances_web - ENU 4133 Differential Balances...

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