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 (Verallgemeinerungen)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: {Anzeigestil {frac {d}{dt}}int _{Omega (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.

Inhalt 1 Generelle Form 2 Form for a material element 3 A special case 3.1 Interpretation and reduction to one dimension 4 Siehe auch 5 Anmerkungen 6 Verweise 7 External links General form Reynolds transport theorem can be expressed as follows:[1][2][3] {Anzeigestil {frac {d}{dt}}int _{Omega (t)}mathbf {f} ,dV=int _{Omega (t)}{frac {teilweise mathbf {f} }{partial t}},dV+int _{partielles Omega (t)}links(mathbf {v} _{b}cdot mathbf {n} Rechts)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.