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# Differential_Balances_web - ENU 4133 Dierential Balances...

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ENU 4133 – Differential Balances January 31, 2011

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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)

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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.

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Inlet and Outlet Mass Flows (x-direction) – Figure 4.1, White
Inlet and Outlet Mass Flows In x-direction, from figure: ( ρ AV ) in , x = ρ udydz (11) ( ρ AV ) out , x = ρ u + x ( ρ u ) dx dydz (12) ( ρ AV ) out , x - ( ρ AV ) in , x = x ( ρ u ) dxdydz (13) y and z directions, similar: ( ρ AV ) out , y - ( ρ AV ) in , y = y ( ρ v ) dydxdz (14) ( ρ AV ) out , z - ( ρ AV ) in , z = z ( ρ w ) dzdxdy (15)

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Deriving Continuity Z CV ∂ρ t d V ∂ρ t dxdydz (16) ∂ρ t dxdydz + x ( ρ u ) dxdydz + y ( ρ v ) dydxdz + z ( ρ w ) dzdxdy = 0 (17) ∂ρ t + x ( ρ u ) + y ( ρ v ) + z ( ρ w ) = 0 (18) ∂ρ t + ∇ · ρ ~ V = 0 (19) Equation 19 is the continuity equation and can be used in several coordinate systems (Cartesian, cylindrical, spherical); Equation 18 is the Cartesian form.
Special Cases Steady, compressible flow t = 0 (20) ∇ ·

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Differential_Balances_web - ENU 4133 Dierential Balances...

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