User:Egm6322.s12.team2.steele.m2/Mtg19+20

=EGM6321 - Principles of Engineering Analysis 1, Fall 2011=

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Notes, Sec 19 Plan: Example: Exact N2-ODE p.18-5 cont’d Linear systems and Reynolds Transport Theorem 1-D case 3-D case Material time derivative Transform integral on $$\mathcal{B}_t$$ to integral on $$\mathcal{B} \equiv \mathcal{B}_0$$ Velocity of material point $$X \equiv x_0$$ Material description Spatial description Summation convention on repeated indices Conservation of mass equation Reynolds Transport Theorem (RTT) Application					(Note p.3-2)

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Example: Exact N2-ODE p.18-5 cont’d 2 Choices: 1)	From [[media:Pea1.f11.mtg18.djvu|Eqn(3) p.18-4]], assume that

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2)	From (3) p.18-4, assume that

R*3.16: Finish the story.

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=Linear Systems and Reynolds Transport Theorem= Recall [[media:Pea1.f11.mtg15.djvu|Eqn(1) p.15-2]] and R*3.7 p.15-1: Differentiate an integral with time-dependent integration limits.

See graphic on this page with the following parameters: $$\mathcal{B}_0, \mathcal{B}_{t1}, \mathcal{B}_{t2}, \mathcal{B}_{t3}, x_0, x_{t1} = \phi_{t1}(x_0), x_{t2}, x_{t3}$$



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1-D case: More general than (1) p.15-2

$$\color{blue}{\alpha} = \frac{d}{dt} \left[F(t,b(t))-F(t,a(t))\right]$$ $$=\frac{\partial F(t,b(t))}{\partial s} \frac{db(t)}{dt}-\frac{\partial F(t,a(t))}{\partial s}\frac{da(t)}{dt}$$

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$$=f(t,b(t)) \, \frac{db(t)}{dt}-f(t,a(t)) \, \frac{da(t)}{dt}$$ $$+\left[\frac{\partial}{\partial t} \int f(t,s) ds \right]_{s=a(t)}^{s=b(t)}$$ $$\left[ \int \frac{\partial f(t,s)}{\partial t} ds \right]_{s=a(t)}^{s=b(t)} = \int_{s=a(t)}^{s=b(t)} \frac{\partial f(t,s)}{\partial t} ds$$

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2-D and 3-D cases: Reynolds transport theorem

Mass density

Conservation of mass: $$\frac{dM}{dt}|_{X \{\rm fixed} =: \frac{DM}{Dt} = 0$$	(4) $$\frac{d(\cdot)}{dt}|_{X \{\rm fixed} =: \frac{D(\cdot)}{Dt}$$ = Material time derivative, i.e., total time derivative keeping the material point $$X \equiv x_0$$ fixed.

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Transform integral on $$\mathcal{B}_t$$ to integral on $$\mathcal{B} \equiv \mathcal{B}_0$$to take the total time derivative

See, e.g., Kolmogrov scales, Time rate of Jacobian http://en.wikiversity.org/wiki/User:Egm6936.f09/Kilmogorov_scales Velocity of material point$$X \equiv x_0$$:

U is material description

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(2)-(4) p.19-5 and [[media:Pea1.f11.mtg2.djvu|Eqn(3) p.2-4]]:

Summation convention on repeated indices

Or evuivalently in component form:

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$$\frac{DM}{Dt} = \int_{\mathcal B} \left[\frac{\partial \rho}{\partial t} + {\rm div} (\rho \mathbf u) \right] \, J \, d \mathcal B = 0$$
 * 1) (2) p. 19-7 and (1) p.19-8:

Hence the conservation of mass equation:

(3) Instead of the mass density $$\rho(x,t)$$ in (1) p.19-7 and (2), consider a general function f(x,t). Then the equivalent of (1) p.19-7 and (2) is the:

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Reynolds Transport Theorem (RTT):

See, e.g., Gross & Reusken 2011, Numerical methods for incompressible two-phase flows, p.3, (1.3). Application: Consider

(1)	And (3) p.19-9:

See, e.g., Malvern 1969, Intro to the mechanics of a continuous medium, p.211, (5.2.18). R*3.17: Show(3).

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How to relate the RTT to the 1-D case in (1) p.19-4? A: Apply the divergence theorem to the second term in (1) p.19-10: $$\frac{D}{Dt} \int_{\mathcal B_t}f(x,t) \, d \mathcal B_t = \int_{\mathcal B_t} \frac{\partial f}{\partial t} \, d \mathcal B_t + \int_{\partial \mathcal B_t} \mathbf n \cdot (f \mathbf u) \, d(\partial \mathcal B_t)$$	(1) See illustration of volume with the following parameters: a normal vector – an outward normal to boundary $$\partial \mathcal{B}_t, \partial{\mathcal{B}_t}, \mathcal{B}_t, x_t = \phi(X,t)$$

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(1)	P.19-11 is another version of the RTT stated in (1) p.19-10; see a different and direct derivation of (1) p.19-11 in Malvern 1969 p.211 (5.2.17).

R*3.18: Obtain (1) p.19-4 from (1) p.19-11.

R4.1: Provide a different and direct derivation of (1) p.19-11; see, e.g., Malvern 1969 p.211 (5.2.17).