Reynolds transport theorem

Reynolds transport theorem Part of a series of articles about Calculus Fundamental theorem Leibniz integral rule Limits of functionsContinuity Mean value theoremRolle's theorem hide Differential Definitions Derivative (generalizações)Differential infinitesimalof a functiontotal Concepts Differentiation notationSecond derivativeImplicit differentiationLogarithmic differentiationRelated ratesTaylor's theorem Rules and identities SumProductChainPowerQuotientL'Hôpital's ruleInverseGeneral LeibnizFaà di Bruno's formulaReynolds show Integral show Series show Vector show Multivariable show Advanced show Specialized show Miscellaneous vte In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds (1842–1912), is a three-dimensional generalization of the Leibniz integral rule. It is used to recast time derivatives of integrated quantities and is useful in formulating the basic equations of continuum mechanics.

Consider integrating f = f(x,t) over the time-dependent region Ω(t) that has boundary ∂Ω(t), then taking the derivative with respect to time: {estilo de exibição {fratura {d}{dt}}int_{Ómega (t)}mathbf {f} ,dV.} If we wish to move the derivative within the integral, there are two issues: the time dependence of f, and the introduction of and removal of space from Ω due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.

Conteúdo 1 Forma geral 2 Form for a material element 3 A special case 3.1 Interpretation and reduction to one dimension 4 Veja também 5 Notas 6 Referências 7 External links General form Reynolds transport theorem can be expressed as follows:[1][2][3] {estilo de exibição {fratura {d}{dt}}int_{Ómega (t)}mathbf {f} ,dV=int _{Ómega (t)}{fratura {matemática parcial {f} }{partial t}},dV+int _{Ômega parcial (t)}deixei(mathbf {v} _{b}cdot mathbf {n} certo)mathbf {f} ,dA} in which n(x,t) is the outward-pointing unit normal vector, x is a point in the region and is the variable of integration, dV and dA are volume and surface elements at x, and vb(x,t) is the velocity of the area element (not the flow velocity). The function f 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 In continuum mechanics, this theorem is often used for material elements. These are parcels of fluids or solids which no material enters or leaves. If Ω(t) is a material element then there is a velocity function v = v(x,t), and the boundary elements obey {estilo de exibição mathbf {v} ^{b}cdot mathbf {n} = mathbf {v} cdot mathbf {n} .} This condition may be substituted to obtain:[5] {estilo de exibição {fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} ,dVright)=int_{Ómega (t)}{fratura {matemática parcial {f} }{partial t}},dV+int _{Ômega parcial (t)}(mathbf {v} cdot mathbf {n} )mathbf {f} ,dA.} Proof for a material element Let Ω0 be reference configuration of the region Ω(t). Let the motion and the deformation gradient be given by {estilo de exibição mathbf {x} ={símbolo em negrito {varphi }}(mathbf {X} ,t),} {estilo de exibição {símbolo em negrito {F}}(mathbf {X} ,t)={símbolo em negrito {nabla }}{símbolo em negrito {varphi }}.} Let J(X,t) = det F(X,t). Definir {estilo de exibição {chapéu {mathbf {f} }}(mathbf {X} ,t)= mathbf {f} ({símbolo em negrito {varphi }}(mathbf {X} ,t),t).} Then the integrals in the current and the reference configurations are related by {estilo de exibição {começar{alinhado}int_{Ómega (t)}mathbf {f} (mathbf {x} ,t),dV&=int _{Ômega _{0}}mathbf {f} ({símbolo em negrito {varphi }}(mathbf {X} ,t),t),J(mathbf {X} ,t),dV_{0}\&=int _{Ômega _{0}}{chapéu {mathbf {f} }}(mathbf {X} ,t),J(mathbf {X} ,t),dV_{0}.fim{alinhado}}} 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 as {estilo de exibição {fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} (mathbf {x} ,t),dVright)=lim_{Delta trightarrow 0}{fratura {1}{Delta t}}deixei(int_{Ómega (t+Delta t)}mathbf {f} (mathbf {x} ,t+Delta t),dV-int _{Ómega (t)}mathbf {f} (mathbf {x} ,t),dVright).} Converting into integrals over the reference configuration, Nós temos {estilo de exibição {fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} (mathbf {x} ,t),dVright)=lim_{Delta trightarrow 0}{fratura {1}{Delta t}}deixei(int_{Ômega _{0}}{chapéu {mathbf {f} }}(mathbf {X} ,t+Delta t),J(mathbf {X} ,t+Delta t),dV_{0}-int_{Ômega _{0}}{chapéu {mathbf {f} }}(mathbf {X} ,t),J(mathbf {X} ,t),dV_{0}certo).} Since Ω0 is independent of time, temos {estilo de exibição {começar{alinhado}{fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} (mathbf {x} ,t),dVright)&=int _{Ômega _{0}}deixei(lim_{Delta trightarrow 0}{fratura {{chapéu {mathbf {f} }}(mathbf {X} ,t+Delta t),J(mathbf {X} ,t+Delta t)-{chapéu {mathbf {f} }}(mathbf {X} ,t),J(mathbf {X} ,t)}{Delta t}}certo),dV_{0}\&=int _{Ômega _{0}}{fratura {parcial }{partial t}}deixei({chapéu {mathbf {f} }}(mathbf {X} ,t),J(mathbf {X} ,t)certo),dV_{0}\&=int _{Ômega _{0}}deixei({fratura {parcial }{partial t}}{grande (}{chapéu {mathbf {f} }}(mathbf {X} ,t){grande )},J(mathbf {X} ,t)+{chapéu {mathbf {f} }}(mathbf {X} ,t),{fratura {parcial }{partial t}}{grande (}J(mathbf {X} ,t){grande )}certo),dV_{0}.fim{alinhado}}} The time derivative of J is given by:[6] {estilo de exibição {começar{alinhado}{fratura {partial J(mathbf {X} ,t)}{partial t}}={fratura {parcial }{partial t}}(a {símbolo em negrito {F}})&=(a {símbolo em negrito {F}})({símbolo em negrito {nabla }}cdot mathbf {v} )\&=J(mathbf {X} ,t),{símbolo em negrito {nabla }}cdot mathbf {v} {grande (}{símbolo em negrito {varphi }}(mathbf {X} ,t),t{grande )}\&=J(mathbf {X} ,t),{símbolo em negrito {nabla }}cdot mathbf {v} (mathbf {x} ,t).fim{alinhado}}} Portanto, {estilo de exibição {começar{alinhado}{fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} (mathbf {x} ,t),dVright)&=int _{Ômega _{0}}deixei({fratura {parcial }{partial t}}deixei({chapéu {mathbf {f} }}(mathbf {X} ,t)certo),J(mathbf {X} ,t)+{chapéu {mathbf {f} }}(mathbf {X} ,t),J(mathbf {X} ,t),{símbolo em negrito {nabla }}cdot mathbf {v} (mathbf {x} ,t)certo),dV_{0}\&=int _{Ômega _{0}}deixei({fratura {parcial }{partial t}}deixei({chapéu {mathbf {f} }}(mathbf {X} ,t)certo)+{chapéu {mathbf {f} }}(mathbf {X} ,t),{símbolo em negrito {nabla }}cdot mathbf {v} (mathbf {x} ,t)certo),J(mathbf {X} ,t),dV_{0}\&=int _{Ómega (t)}deixei({ponto {mathbf {f} }}(mathbf {x} ,t)+mathbf {f} (mathbf {x} ,t),{símbolo em negrito {nabla }}cdot mathbf {v} (mathbf {x} ,t)certo),dV.end{alinhado}}} Onde {estilo de exibição {ponto {mathbf {f} }}} is the material time derivative of f. The material derivative is given by {estilo de exibição {ponto {mathbf {f} }}(mathbf {x} ,t)={fratura {matemática parcial {f} (mathbf {x} ,t)}{partial t}}+{grande (}{símbolo em negrito {nabla }}mathbf {f} (mathbf {x} ,t){grande )}cdot mathbf {v} (mathbf {x} ,t).} Portanto, {estilo de exibição {fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} (mathbf {x} ,t),dVright)=int_{Ómega (t)}deixei({fratura {matemática parcial {f} (mathbf {x} ,t)}{partial t}}+{grande (}{símbolo em negrito {nabla }}mathbf {f} (mathbf {x} ,t){grande )}cdot mathbf {v} (mathbf {x} ,t)+mathbf {f} (mathbf {x} ,t),{símbolo em negrito {nabla }}cdot mathbf {v} (mathbf {x} ,t)certo),dV,} ou, {estilo de exibição {fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} ,dVright)=int_{Ómega (t)}deixei({fratura {matemática parcial {f} }{partial t}}+{símbolo em negrito {nabla }}mathbf {f} cdot mathbf {v} +mathbf {f} ,{símbolo em negrito {nabla }}cdot mathbf {v} certo),dV.} Using the identity {estilo de exibição {símbolo em negrito {nabla }}cdot (mathbf {v} otimes mathbf {W} )= mathbf {v} ({símbolo em negrito {nabla }}cdot mathbf {W} )+{símbolo em negrito {nabla }}mathbf {v} cdot mathbf {W} ,} we then have {estilo de exibição {fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} ,dVright)=int_{Ómega (t)}deixei({fratura {matemática parcial {f} }{partial t}}+{símbolo em negrito {nabla }}cdot (mathbf {f} otimes mathbf {v} )certo),dV.} Using the divergence theorem and the identity (a ⊗ b) · n = (b · n)uma, temos {estilo de exibição {começar{alinhado}{fratura {d}{dt}}deixei(int_{Ómega (t)}mathbf {f} ,dVright)&=int _{Ómega (t)}{fratura {matemática parcial {f} }{partial t}},dV+int _{Ômega parcial (t)}(mathbf {f} otimes mathbf {v} )cdot mathbf {n} ,dA\&=int _{Ómega (t)}{fratura {matemática parcial {f} }{partial t}},dV+int _{Ômega parcial (t)}(mathbf {v} cdot mathbf {n} )mathbf {f} ,dA.qquad square end{alinhado}}} A special case If we take Ω to be constant with respect to time, then vb = 0 and the identity reduces to {estilo de exibição {fratura {d}{dt}}int_{Ómega }f,dV=int _{Ómega }{fratura {f parcial}{partial t}},dV.} como esperado. (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 The theorem is the higher-dimensional extension of differentiation under the integral sign and reduces to that expression in some cases. Suppose f is independent of y and z, and that Ω(t) is a unit square in the yz-plane and has x limits a(t) and b(t). Then Reynolds transport theorem reduces to {estilo de exibição {fratura {d}{dt}}int_{uma(t)}^{b(t)}f(x,t),dx=int_{uma(t)}^{b(t)}{fratura {f parcial}{partial t}},dx+{fratura {partial b(t)}{partial t}}f{grande (}b(t),t{grande )}-{fratura {partial a(t)}{partial t}}f{grande (}uma(t),t{grande )},,} que, up to swapping x and t, is the standard expression for differentiation under the integral sign.

See also Mathematics portal Leibniz integral rule – Differentiation under the integral sign formula Notes ^ L. G. Leal, 2007, p. 23. ^ O. Reynolds, 1903, Volume. 3, p. 12–13 ^ J.E. Marsden and A. Tromba, 5ª edição. 2003 ^ Yamaguchi, H. (2008). Engineering Fluid Mechanics. Dordrecht: Springer. p. 23. ISBN 978-1-4020-6741-9. ^ Belytschko, T.; Liu, C. K.; Moran, B. (2000). Nonlinear Finite Elements for Continua and Structures. Nova york: John Wiley e Filhos. ISBN 0-471-98773-5. ^ Gurtin, M. E. (1981). An Introduction to Continuum Mechanics. Nova york: Imprensa Acadêmica. p. 77. ISBN 0-12-309750-9. References Leal, eu. G. (2007). Advanced transport phenomena: fluid mechanics and convective transport processes. Cambridge University Press. ISBN 978-0-521-84910-4. Marsden, J. E.; Tromba, UMA. (2003). Vector Calculus (5ª edição). Nova york: C. H. Freeman. ISBN 978-0-7167-4992-9. Reynolds, O. (1903). Papers on Mechanical and Physical Subjects. Volume. 3, The Sub-Mechanics of the Universe. Cambridge: Cambridge University Press. External links Osborne Reynolds, Collected Papers on Mechanical and Physical Subjects, in three volumes, published circa 1903, now fully and freely available in digital format: Volume 1, Volume 2, Volume 3, "Module 6 - Reynolds Transport Theorem". ME6601: Introduction to Fluid Mechanics. Georgia Tech. Archived from the original on March 27, 2008. http://planetmath.org/reynoldstransporttheorem Categories: AerodynamicsChemical engineeringContinuum mechanicsEquations of fluid dynamicsFluid dynamicsFluid mechanicsMechanical engineering