Reynolds Transport Theorem

Contents

1. Overview
2. Etymology
3. Cultural Impact

The Reynolds transport theorem, a name that carries the weight of Osborne Reynolds himself, is essentially a more elaborate, three-dimensional rendition of the Leibniz integral rule . It’s not just some abstract mathematical nicety; it’s the tool you reach for when you need to untangle the time derivatives of quantities that are being integrated over regions that are themselves shifting and changing. Think of it as the foundational scaffolding for the core equations in continuum mechanics . Without it, trying to describe the motion and evolution of fluids or deformable solids would be a chaotic, if not impossible, endeavor.

Imagine you have a function, let’s call it f, that depends on both position x and time t: f( x, t). Now, picture this f being integrated over a region, Ω(t), which isn’t static. Its boundaries, ∂Ω(t), are in motion, expanding, contracting, or generally being reshaped by time. When you want to know how the integral of f over this dynamic region changes over time, you’re faced with a two-headed beast. First, f itself is evolving with time, regardless of the region. Second, the very space over which you’re integrating is changing, with new bits entering and old bits exiting the region. The Reynolds transport theorem is the sophisticated mechanism that allows us to reconcile these complexities, to move that pesky time derivative from outside the integral to inside, accounting for both the changes in the integrand and the changes in the domain of integration.

General Form

The theorem, in its general guise, lays out the transformation quite precisely. If we are considering the time derivative of an integral of f over a time-dependent region Ω(t), denoted as:

$${\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} ,dV$$

The Reynolds transport theorem tells us that this can be rewritten as:

$${\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} ,dV=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}},dV+\int _{\partial \Omega (t)}\left(\mathbf {v} _{b}\cdot \mathbf {n} \right)\mathbf {f} ,dA$$

Let’s break down what these symbols are doing. Here, n ( x, t) is the outward-pointing unit normal vector at a point x on the boundary ∂Ω(t) at time t. Think of it as a tiny arrow sticking straight out from the surface. dV and dA are the infinitesimal volume and surface elements, respectively, at that point x. The crucial new player here is v_b ( x, t), which represents the velocity of the boundary element itself. It’s important to note that this is not necessarily the flow velocity of the substance within the region; it’s the velocity of the surface that defines the boundary of our region of integration. The function f itself can be a scalar, a vector, or even a tensor-valued quantity. The integral on the left-hand side, being a function solely of time, warrants the use of the total derivative, ’d/dt'.

Form for a Material Element

In the realm of continuum mechanics , a particularly common application of this theorem is when we consider material elements . These are conceptual parcels of fluid or solid that are followed through space and time. The defining characteristic of a material element is that no material enters or leaves it. If Ω(t) represents such a material element, then there exists a velocity field v = v ( x, t) that describes the motion of every point within the element. For such an element, the velocity of its boundary elements, v_b, is precisely equal to the flow velocity v at the boundary. That is:

$$\mathbf {v} _{b}\cdot \mathbf {n} =\mathbf {v} \cdot \mathbf {n} $$

Substituting this condition into the general form of the theorem yields a slightly simplified, yet profoundly useful, version:

$${\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} ,dV\right)=\int _{\Omega (t)}{\frac {\partial \mathbf {f} }{\partial t}},dV+\int _{\partial \Omega (t)}(\mathbf {v} \cdot \mathbf {n} )\mathbf {f} ,dA$$

This form is frequently encountered when analyzing conserved quantities within a moving fluid or solid.

Proof for a Material Element

To truly appreciate the theorem, let’s walk through its derivation for a material element. We start by defining a reference configuration, Ω₀, which is a fixed, time-independent representation of the region Ω(t). The motion of the material is described by a mapping x = φ( X, t), where X are the material coordinates (fixed for a given particle) and x are the spatial coordinates (which change with time). The deformation gradient , F( X, t), is given by the gradient of this mapping: F = ∇_X φ. The determinant of this deformation gradient, J( X, t) = det F( X, t), represents the local volume ratio between the deformed and undeformed states.

We can then define a “materially transported” version of our function f, denoted as f̂( X, t), which is f evaluated at the spatial position corresponding to the material point X at time t: f̂( X, t) = f(φ( X, t), t). The relationship between the integral of f over the spatial region Ω(t) and the integral over the fixed reference configuration Ω₀ is given by:

$${\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t),dV=\int _{\Omega {0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t),J(\mathbf {X} ,t),dV{0}}$$

This equation essentially states that the total quantity of f within the moving region Ω(t) can be found by integrating over the fixed reference configuration, accounting for the volume changes via the Jacobian J.

Now, we consider the time derivative of the integral over the moving region:

$${\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t),dV=\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\left(\int _{\Omega (t+\Delta t)}\mathbf {f} (\mathbf {x} ,,t{+}\Delta t),dV-\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t),dV\right)$$

By converting both integrals to the reference configuration, we get:

$${\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t),dV=\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\left(\int _{\Omega {0}}{\hat {\mathbf {f} }}(\mathbf {X} ,,t{+}\Delta t),J(\mathbf {X} ,,t{+}\Delta t),dV{0}-\int _{\Omega {0}}{\hat {\mathbf {f} }}(\mathbf {X} ,t),J(\mathbf {X} ,t),dV{0}\right)$$

Since Ω₀ is fixed in time, we can bring the limit inside the integral:

$${\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t),dV=\int _{\Omega {0}}{\frac {\partial }{\partial t}}\left({\hat {\mathbf {f} }}(\mathbf {X} ,t),J(\mathbf {X} ,t)\right),dV{0}$$

Expanding the derivative using the product rule:

$${\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} (\mathbf {x} ,t),dV=\int _{\Omega {0}}\left({\frac {\partial }{\partial t}}{\big (}{\hat {\mathbf {f} }}(\mathbf {X} ,t){\big )},J(\mathbf {X} ,t)+{\hat {\mathbf {f} }}(\mathbf {X} ,t),{\frac {\partial }{\partial t}}{\big (}J(\mathbf {X} ,t){\big )}\right),dV{0}$$

A crucial result from tensor calculus relates the time derivative of the Jacobian to the divergence of the velocity field:

$${\frac {\partial }{\partial t}}J(\mathbf {X} ,t) = J(\mathbf {X} ,t),{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)$$

where v is the spatial velocity field. Substituting this back into the equation for the time derivative of the integral, and then converting back to the spatial configuration, we arrive at:

$${\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} ,dV=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {f} \cdot \mathbf {v} +\mathbf {f} ,{\boldsymbol {\nabla }}\cdot \mathbf {v} \right),dV$$

The terms ${\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\mathbf {f} \cdot \mathbf {v}$ constitute the material time derivative of f, denoted as ḟ. Therefore, the equation can be succinctly written as:

$${\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} ,dV=\int _{\Omega (t)}\left({\dot {\mathbf {f} }}(\mathbf {x} ,t)+\mathbf {f} (\mathbf {x} ,t),{\boldsymbol {\nabla }}\cdot \mathbf {v} (\mathbf {x} ,t)\right),dV$$

Using the vector identity ${\boldsymbol {\nabla }}\cdot (\mathbf {v} \otimes \mathbf {w} )=\mathbf {v} ({\boldsymbol {\nabla }}\cdot \mathbf {w} )+{\boldsymbol {\nabla }}\mathbf {v} \cdot \mathbf {w} $, and noting that f is a vector (or tensor) field, we can rewrite the integrand:

$${\frac {d}{dt}}\left(\int _{\Omega (t)}\mathbf {f} ,dV\right)=\int _{\Omega (t)}\left({\frac {\partial \mathbf {f} }{\partial t}}+{\boldsymbol {\nabla }}\cdot (\mathbf {f} \otimes \mathbf {v} )\right),dV$$

Finally, applying the divergence theorem to the second term and using the identity ( a ⊗ b ) · n = ( b · n ) a, we recover the original form of the theorem for a material element:

Q.E.D.

A Special Case

It’s always satisfying when things simplify, isn’t it? If our region of integration, Ω, happens to be fixed in time, meaning its boundaries aren’t moving, then the boundary velocity v_b is zero. In this scenario, the second integral in the Reynolds transport theorem vanishes, and we’re left with a much simpler expression:

$${\frac {d}{dt}}\int _{\Omega }f,dV=\int _{\Omega }{\frac {\partial f}{\partial t}},dV$$

This result is exactly what one would intuitively expect: if the domain isn’t changing, the time rate of change of the integral is simply the integral of the time rate of change of the integrand. It’s a good sanity check, and a reminder that the complexities of the theorem arise precisely from the movement of the integration domain.

Interpretation and Reduction to One Dimension

The Reynolds transport theorem is, in essence, the multidimensional cousin of differentiation under the integral sign . It elegantly extends that concept to higher dimensions and dynamic boundaries. Consider a simplified scenario where our function f only depends on one spatial variable, say x, and time t, so f(x, t). Let our region of integration be a one-dimensional interval with time-dependent limits, a(t) and b(t). In this case, the Reynolds transport theorem reduces to:

$${\frac {d}{dt}}\int _{a(t)}^{b(t)}f(x,t),dx=\int _{a(t)}^{b(t)}{\frac {\partial f}{\partial t}},dx+{\frac {\partial b(t)}{\partial t}}f{\big (}b(t),t{\big )}-{\frac {\partial a(t)}{\partial t}}f{\big (}a(t),t{\big )}$$

This formula is precisely the standard rule for differentiating an integral with variable limits. It shows how the overall change in the integral is composed of the change due to the integrand varying with time within the interval, plus the contributions from the upper and lower limits moving. It’s a beautiful illustration of how a general theorem can simplify to familiar forms in specific cases.

General Form

Form for a Material Element

Proof for a Material Element

A Special Case

Interpretation and Reduction to One Dimension

See also