- Source: Reynolds transport theorem
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:
d
d
t
∫
Ω
(
t
)
f
d
V
.
{\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 of f, and the introduction of and removal of space from Ω due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.
General form
Reynolds transport theorem can be expressed as follows:
d
d
t
∫
Ω
(
t
)
f
d
V
=
∫
Ω
(
t
)
∂
f
∂
t
d
V
+
∫
∂
Ω
(
t
)
(
v
b
⋅
n
)
f
d
A
{\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 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. 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
v
b
⋅
n
=
v
⋅
n
.
{\displaystyle \mathbf {v} _{b}\cdot \mathbf {n} =\mathbf {v} \cdot \mathbf {n} .}
This condition may be substituted to obtain:
d
d
t
(
∫
Ω
(
t
)
f
d
V
)
=
∫
Ω
(
t
)
∂
f
∂
t
d
V
+
∫
∂
Ω
(
t
)
(
v
⋅
n
)
f
d
A
.
{\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.}
A special case
If we take Ω to be constant with respect to time, then vb = 0 and the identity reduces to
d
d
t
∫
Ω
f
d
V
=
∫
Ω
∂
f
∂
t
d
V
.
{\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
=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
d
d
t
∫
a
(
t
)
b
(
t
)
f
(
x
,
t
)
d
x
=
∫
a
(
t
)
b
(
t
)
∂
f
∂
t
d
x
+
∂
b
(
t
)
∂
t
f
(
b
(
t
)
,
t
)
−
∂
a
(
t
)
∂
t
f
(
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 swapping x and t, is the standard expression for differentiation under the integral sign.
See also
Leibniz integral rule – Differentiation under the integral sign formula
References
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.
"Reynolds transport theorem". planetmath.org. Retrieved 2024-04-22.
Kata Kunci Pencarian:
- Reynolds transport theorem
- Transport theorem
- Reynolds number
- List of things named after Gottfried Leibniz
- Osborne Reynolds
- Reynolds
- Leibniz integral rule
- List of theorems
- Fluid dynamics
- Reynolds analogy
