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Reynolds transport theorem

From Wikipedia, the free encyclopedia
3D generalization of the Leibniz integral rule
Part of a series of articles about
Calculus
∫abf′(t)dt=f(b)−f(a){\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}

Indifferential calculus, theReynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply theReynolds theorem, named afterOsborne Reynolds (1842–1912), is a three-dimensional generalization of theLeibniz integral rule. It is used to recast time derivatives of integrated quantities and is useful in formulating the basic equations ofcontinuum mechanics.

Consider integratingf =f(x,t) over the time-dependent regionΩ(t) that has boundary∂Ω(t), then taking the derivative with respect to time:ddt∫Ω(t)fdV.{\displaystyle {\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} \,dV.}If we wish to move the derivative into the integral, there are two issues: the time dependence off, and the introduction of and removal of space fromΩ due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.

General form

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Reynolds transport theorem can be expressed as follows:[1][2][3]ddt∫Ω(t)fdV=∫Ω(t)∂f∂tdV+∫∂Ω(t)(vb⋅n)fdA{\displaystyle {\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} \,dV=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}\,dV+\int _{\partial \Omega (t)}\left(\mathbf {v} _{b}\cdot \mathbf {n} \right)\mathbf {f} \,dA}in whichn(x,t) is the outward-pointing unit normal vector,x is a point in the region and is the variable of integration,dV anddA are volume and surface elements atx, andvb(x,t) is the velocity of the area element (not the flow velocity). The functionf may be tensor-, vector- or scalar-valued.[4] Note that the integral on the left hand side is a function solely of time, and so the total derivative has been used.

Form for a material element

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In continuum mechanics, this theorem is often used formaterial elements. These are parcels of fluids or solids which no material enters or leaves. IfΩ(t) is a material element then there is a velocity functionv =v(x,t), and the boundary elements obeyvb⋅n=v⋅n.{\displaystyle \mathbf {v} _{b}\cdot \mathbf {n} =\mathbf {v} \cdot \mathbf {n} .}This condition may be substituted to obtain:[5]ddt(∫Ω(t)fdV)=∫Ω(t)∂f∂tdV+∫∂Ω(t)(v⋅n)fdA.{\displaystyle {\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} \,dV\right)=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}\,dV+\int _{\partial \Omega (t)}(\mathbf {v} \cdot \mathbf {n} )\mathbf {f} \,dA.}

Proof for a material element

LetΩ0 be reference configuration of the regionΩ(t). Letthe motion and thedeformation gradient be given byx=φ(X,t),F(X,t)=∇φ.{\displaystyle {\begin{aligned}\mathbf {x} &={\boldsymbol {\varphi }}(\mathbf {X} ,t),\\{\boldsymbol {F}}(\mathbf {X} ,t)&={\boldsymbol {\nabla }}{\boldsymbol {\varphi }}.\end{aligned}}}

LetJ(X,t) = detF(X,t). Definef^(X,t)=f(φ(X,t),t).{\displaystyle {\hat {\mathbf {f} }}(\mathbf {X} ,t)=\mathbf {f} ({\boldsymbol {\varphi }}(\mathbf {X} ,t),t).}Then the integrals in the current and the reference configurations are related by∫Ω(t)f(x,t)dV=∫Ω0f(φ(X,t),t)J(X,t)dV0=∫Ω0f^(X,t)J(X,t)dV0.{\displaystyle {\begin{aligned}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV&=\int _{\Omega _{0}}\mathbf {f} ({\boldsymbol {\varphi }}(\mathbf {X} ,t),t)\,J(\mathbf {X} ,t)\,dV_{0}\\&=\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)\,J(\mathbf {X} ,t)\,dV_{0}.\end{aligned}}}

That this derivation is for a material element is implicit in the time constancy of the reference configuration: it is constant in material coordinates. The time derivative of an integral over a volume is defined asddt∫Ω(t)f(x,t)dV=limΔt→01Δt(∫Ω(t+Δt)f(x,t+Δt)dV−∫Ω(t)f(x,t)dV).{\displaystyle {\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV=\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\left(\int _{\Omega (t+\Delta t)}\mathbf {f} (\mathbf {x} ,\,t{+}\Delta t)\,dV-\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV\right).}

Converting into integrals over the reference configuration, we getddt∫Ω(t)f(x,t)dV=limΔt→01Δt(∫Ω0f^(X,t+Δt)J(X,t+Δt)dV0−∫Ω0f^(X,t)J(X,t)dV0).{\displaystyle {\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV=\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\left(\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,\,t{+}\Delta t)\,J(\mathbf {X} ,\,t{+}\Delta t)\,dV_{0}-\int _{\Omega _{0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t)\,J(\mathbf {X} ,t)\,dV_{0}\right).}

SinceΩ0 is independent of time, we haveddt∫Ω(t)f(x,t)dV=∫Ω0(limΔt→0f^(X,t+Δt)J(X,t+Δt)−f^(X,t)J(X,t)Δt)dV0=∫Ω0∂∂t(f^(X,t)J(X,t))dV0=∫Ω0(∂∂t(f^(X,t))J(X,t)+f^(X,t)∂∂t(J(X,t)))dV0.{\displaystyle {\begin{aligned}{\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV&=\int _{\Omega _{0}}\left(\lim _{\Delta t\to 0}{\frac {{\hat {\mathbf {f} }}(\mathbf {X} ,\,t{+}\Delta t)\,J(\mathbf {X} ,\,t{+}\Delta t)-{\hat {\mathbf {f} }}(\mathbf {X} ,t)\,J(\mathbf {X} ,t)}{\Delta t}}\right)\,dV_{0}\\&=\int _{\Omega _{0}}{\frac {\partial }{\partial t}}\left({\hat {\mathbf {f} }}(\mathbf {X} ,t)\,J(\mathbf {X} ,t)\right)\,dV_{0}\\&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}{\big (}{\hat {\mathbf {f} }}(\mathbf {X} ,t){\big )}\,J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)\,{\frac {\partial }{\partial t}}{\big (}J(\mathbf {X} ,t){\big )}\right)\,dV_{0}.\end{aligned}}}

The time derivative ofJ is given by:[6]∂∂tJ(X,t)=∂∂t(detF)=(detF)tr⁡(F−1∂F∂t)=(detF)tr⁡(∂X∂φ∂∂t(∂φ∂X))=(detF)tr⁡(∂X∂φ∂∂X(∂φ∂t))=(detF)tr⁡(∂∂x(∂φ∂t))=(detF)(∇⋅v)=J(X,t)∇⋅v(φ(X,t),t)=J(X,t)∇⋅v(x,t).{\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}J(\mathbf {X} ,t)&={\frac {\partial }{\partial t}}(\det {\boldsymbol {F}})\\&=(\det {\boldsymbol {F}})\operatorname {tr} \left({\boldsymbol {F}}^{-1}{\frac {\partial {\boldsymbol {F}}}{\partial t}}\right)\\&=(\det {\boldsymbol {F}})\operatorname {tr} \left({\frac {\partial {\boldsymbol {X}}}{\partial {\boldsymbol {\varphi }}}}{\frac {\partial }{\partial t}}\left({\frac {\partial {\boldsymbol {\varphi }}}{\partial {\boldsymbol {X}}}}\right)\right)\\&=(\det {\boldsymbol {F}})\operatorname {tr} \left({\frac {\partial {\boldsymbol {X}}}{\partial {\boldsymbol {\varphi }}}}{\frac {\partial }{\partial {\boldsymbol {X}}}}\left({\frac {\partial {\boldsymbol {\varphi }}}{\partial t}}\right)\right)\\&=(\det {\boldsymbol {F}})\operatorname {tr} \left({\frac {\partial }{\partial {\boldsymbol {x}}}}\left({\frac {\partial {\boldsymbol {\varphi }}}{\partial t}}\right)\right)\\&=(\det {\boldsymbol {F}})({\boldsymbol {\nabla }}\cdot \mathbf {v} )\\&=J(\mathbf {X} ,t)\,{\boldsymbol {\nabla }}\cdot \mathbf {v} {\big (}{\boldsymbol {\varphi }}(\mathbf {X} ,t),t{\big )}\\&=J(\mathbf {X} ,t)\,{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t).\end{aligned}}}

Therefore,ddt∫Ω(t)f(x,t)dV=∫Ω0(∂∂t(f^(X,t))J(X,t)+f^(X,t)J(X,t)∇⋅v(x,t))dV0=∫Ω0(∂∂t(f^(X,t))+f^(X,t)∇⋅v(x,t))J(X,t)dV0=∫Ω(t)(f˙(x,t)+f(x,t)∇⋅v(x,t))dV.{\displaystyle {\begin{aligned}{\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}\left({\hat {\mathbf {f} }}(\mathbf {X} ,t)\right)\,J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)\,J(\mathbf {X} ,t)\,{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)\,dV_{0}\\&=\int _{\Omega _{0}}\left({\frac {\partial }{\partial t}}\left({\hat {\mathbf {f} }}(\mathbf {X} ,t)\right)+{\hat {\mathbf {f} }}(\mathbf {X} ,t)\,{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)\,J(\mathbf {X} ,t)\,dV_{0}\\&=\int _{\Omega (t)}\left({\dot {\mathbf {f} }}(\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t)\,{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)\,dV.\end{aligned}}}wheref˙{\displaystyle {\dot {\mathbf {f} }}} is thematerial time derivative off. The material derivative is given byf˙(x,t)=∂f(x,t)∂t+(∇f(x,t))⋅v(x,t).{\displaystyle {\dot {\mathbf {f} }}(\mathbf {x} ,t)={\frac {\partial \mathbf {f} (\mathbf {x} ,t)}{\partial t}}+{\big (}{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} ,t){\big )}\cdot \mathbf {v} (\mathbf {x} ,t).}

Therefore,ddt∫Ω(t)f(x,t)dV=∫Ω(t)(∂f(x,t)∂t+(∇f(x,t))⋅v(x,t)+f(x,t)∇⋅v(x,t))dV,{\displaystyle {\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t)\,dV=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} (\mathbf {x} ,t)}{\partial t}}+{\big (}{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} ,t){\big )}\cdot \mathbf {v} (\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t)\,{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right)\,dV,}or,ddt∫Ω(t)fdV=∫Ω(t)(∂f∂t+∇f⋅v+f∇⋅v)dV.{\displaystyle {\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} \,dV=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {f} \cdot \mathbf {v} +\mathbf {f} \,{\boldsymbol {\nabla }}\cdot \mathbf {v} \right)\,dV.}

Using the identity∇⋅(v⊗w)=v(∇⋅w)+∇v⋅w,{\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {v} \otimes \mathbf {w} )=\mathbf {v} ({\boldsymbol {\nabla }}\cdot \mathbf {w} )+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {w} ,}we then haveddt(∫Ω(t)fdV)=∫Ω(t)(∂f∂t+∇⋅(f⊗v))dV.{\displaystyle {\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} \,dV\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\cdot (\mathbf {f} \otimes \mathbf {v} )\right)\,dV.}

Using thedivergence theorem and the identity(a ⊗b) ·n = (b ·n)a, we haveddt(∫Ω(t)fdV)=∫Ω(t)∂f∂tdV+∫∂Ω(t)(f⊗v)⋅ndA=∫Ω(t)∂f∂tdV+∫∂Ω(t)(v⋅n)fdA.{\displaystyle {\begin{aligned}{\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} \,dV\right)&=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}\,dV+\int _{\partial \Omega (t)}(\mathbf {f} \otimes \mathbf {v} )\cdot \mathbf {n} \,dA\\&=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}}\,dV+\int _{\partial \Omega (t)}(\mathbf {v} \cdot \mathbf {n} )\mathbf {f} \,dA.\end{aligned}}}Q.E.D.

A special case

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If we takeΩ to be constant with respect to time, thenvb = 0 and the identity reduces toddt∫ΩfdV=∫Ω∂f∂tdV.{\displaystyle {\frac {d}{dt}}\int _{\Omega }f\,dV=\int _{\Omega }{\frac {\partial f}{\partial t}}\,dV.}as expected. (This simplification is not possible if the flow velocity is incorrectly used in place of the velocity of an area element.)

Interpretation and reduction to one dimension

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The theorem is the higher-dimensional extension ofdifferentiation under the integral sign and reduces to that expression in some cases. Supposef is independent ofy andz, and thatΩ(t) is a unit square in theyz-plane and hasx limitsa(t) andb(t). Then Reynolds transport theorem reduces toddt∫a(t)b(t)f(x,t)dx=∫a(t)b(t)∂f∂tdx+∂b(t)∂tf(b(t),t)−∂a(t)∂tf(a(t),t),{\displaystyle {\frac {d}{dt}}\int _{a(t)}^{b(t)}f(x,t)\,dx=\int _{a(t)}^{b(t)}{\frac {\partial f}{\partial t}}\,dx+{\frac {\partial b(t)}{\partial t}}f{\big (}b(t),t{\big )}-{\frac {\partial a(t)}{\partial t}}f{\big (}a(t),t{\big )}\,,}which, up to swappingx andt, is the standard expression for differentiation under the integral sign.

See also

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References

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  1. ^Leal, L. G. (2007).Advanced transport phenomena: fluid mechanics and convective transport processes. Cambridge University Press. p. 23.ISBN 978-0-521-84910-4.
  2. ^Reynolds, O. (1903).Papers on Mechanical and Physical Subjects. Vol. 3, The Sub-Mechanics of the Universe. Cambridge: Cambridge University Press. pp. 12–13.
  3. ^Marsden, J. E.;Tromba, A. (2003).Vector Calculus (5th ed.). New York:W. H. Freeman.ISBN 978-0-7167-4992-9.
  4. ^Yamaguchi, H. (2008).Engineering Fluid Mechanics. Dordrecht: Springer. p. 23.ISBN 978-1-4020-6741-9.
  5. ^Belytschko, T.; Liu, W. K.; Moran, B. (2000).Nonlinear Finite Elements for Continua and Structures. New York: John Wiley and Sons.ISBN 0-471-98773-5.
  6. ^Gurtin, M. E. (1981).An Introduction to Continuum Mechanics. New York: Academic Press. p. 77.ISBN 0-12-309750-9.

External links

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