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Fluid conservation and entropy jump conditions

Physical scope

Adopted weak balances on piecewise C1 states give Rankine–Hugoniot conservation and physical entropy jump conditions; no smooth-balance theorem through shocks or existence result.

Statement

Assume Countable Choice: every sequence of nonempty sets admits a choice function. Use the restricted complete Euclidean Lebesgue spaces, almost-everywhere identification and signed integrals of the conservative weak-form definition. Let $Ω \subseteq R^{3}$ be a nonempty open spatial region and $I = (0, T_{f})$ a time interval in seconds, with $T_{f} > 0$ . Let $Σ$ be a $C^{1}$ moving interface in $Ω \times I$ that locally is the graph $x_{3} = h (t, x_{1}, x_{2})$ and separates minus and plus sides; here $C^{1}$ means that the graph function and its first coordinate partial derivatives are continuous. Let $n$ be the dimensionless Euclidean unit normal pointing from minus to plus, and let $V_{n}$ be the signed normal speed of $Σ$ in metres per second. For a field with one-sided traces, define $[g] : = g_{+} - g_{-}$ . Let $ρ \geq 0$ be mass density in kilograms per cubic metre, $u$ velocity in metres per second, $σ$ Cauchy stress in pascals, $e$ specific internal energy in joules per kilogram, $q$ outward-positive heat flux in watts per square metre, $T > 0$ absolute temperature in kelvins, $s$ specific entropy in joules per kilogram kelvin, $b$ body acceleration in metres per second squared, and $r$ specific heat supply in watts per kilogram. Set $E : = e + ∣ u ∣^{2} / 2$ in joules per kilogram. Assume these fields are $C^{1}$ on each side and, for every relatively compact interface patch, admit $C^{1}$ one-sided extensions to its closed sides; here $C^{1}$ means that the fields and all their first time and spatial coordinate partial derivatives are continuous on those extensions. In particular, the resulting one-sided traces are finite and continuous. Assume that all displayed products and sources are locally integrable. The sources are ordinary volume functions and have no measure concentrated on $Σ$ . Use traction $t = σ n$ , $(u \otimes u)_{i j} = u_{i} u_{j}$ , and rowwise matrix divergence $(div F)_{i} = \sum_{j} \partial_{x_{j}} F_{i j}$ .

On each side suppose the classical equations hold:

$\partial_{t} ρ + div (ρ u) = 0,$

$\partial_{t} (ρ u) + div (ρ u \otimes u - σ) = ρ b,$

$\partial_{t} (ρ E) + div (ρ E u - σ^{T} u + q) = ρ b \cdot u + ρ r,$

$\partial_{t} (ρ s) + div (ρ s u + q / T) \geq ρ r / T .$

Adopt the separate weak mass, momentum and total-energy identities and physical entropy inequality. Their interior restrictions use compactly supported $C^{1}$ tests in $Ω \times I$ , away from $t = 0$ and $t = T_{f}$ . The full weak forms may also use tests nonzero at $t = 0$ and include initial-data terms; those terms are separate data and are not determined by the interface jumps. The interior weak conservation equations across $Σ$ hold if and only if the side equations hold and

$[ρ u] \cdot n = V_{n} [ρ],$

$[ρ u \otimes u - σ] n = V_{n} [ρ u],$

$[ρ E u - σ^{T} u + q] \cdot n = V_{n} [ρ E] .$

For the entropy jump characterization, assume additionally that the vector field $q / T$ admits $C^{1}$ one-sided extensions to every relatively compact interface patch, with the same meaning of extension as above. This condition holds in particular if the temperature traces $T_{\pm}$ are strictly positive. It does not require positive temperature traces when the quotient itself extends; in that case $(q / T)_{\pm}$ denotes the trace of the quotient, not division of the separate traces. No extension or boundedness of the locally integrable source $ρ r / T$ is assumed. Under this additional flux hypothesis, the interior weak physical entropy inequality holds across the interface if and only if it holds on both sides and

$[ρ s u + q / T] \cdot n - V_{n} [ρ s] \geq 0.$

Equivalently, for any conservative density $U$ and flux $F$ above, the interface coefficient is $[F] n - V_{n} [U]$ . Here $[F] n$ is matrix–vector multiplication for momentum; for a scalar law its vector flux is identified with a one-row matrix, so $[F] n = [F] \cdot n$ . The coefficient must vanish for conservation and be nonnegative for physical entropy. These conditions apply only to the specified piecewise $C^{1}$ fields and traces. They neither establish existence or uniqueness of a multidimensional shock solution nor extend smooth material equations through arbitrary nonsmooth fields.

Statement references

Powers, Lecture Notes on Intermediate Fluid Mechanics, §§4.7.1 and 9.4.1

Mei, MIT OpenCourseWare, Law of Momentum Conservation, §1.5

Mei, MIT OpenCourseWare, Heat and Energy Conservation, §4.1

Facts & Assumptions

Given: The interface, orientation, speed, one-sided traces, local integrability and no-singular-source assumptions in the Statement; the side balance equations; the adopted weak balances and entropy inequality.

[A1]

Integrable weak continuum balance and entropy adoption: the weak branch separately adopts the mass, momentum and total-energy identities and physical entropy inequality, with locally integrable fluxes and sources and the stated initial terms.

[A2]

Adopt the mathematical Countable Choice measure convention of Locally integrable conservative weak and initial forms; it underlies the weak tests and is not an extra physical balance law.

[F1]

Locally integrable conservative weak and initial forms: for $U, F, S \in L_{l o c}^{1}$ , $\partial_{t} U + div F = S$ means the compactly supported $C^{1}$ test identity including the initial term; the physical entropy inequality uses nonnegative scalar tests and the opposite weak sign.

[F2]

Smooth weak equivalence and moving jump residual assumes $C^{1}$ one-sided extensions of the density, flux and source. Its Proof 1.1 gives the integration-by-parts identity with interface coefficient $[F] n - V_{n} [U]$ , using the scalar-law row identification above. With a merely locally integrable source, apply that identity first with source zero, then add the source integral; no source trace is used. The conservation triples here meet the lemma's full regularity hypotheses.

[F3]

Classical frames, primitive quantities and SI units: the velocities, density, energy, heat flux, stress and normals use the adopted classical inertial and SI conventions.

Proof

1.1givenF3algebra

In a local graph chart write $Σ (t)$ as $x_{3} = h (t, x_{1}, x_{2})$ , with the plus side given by $x_{3} > h$ . Then $n = (- \partial_{x_{1}} h, - \partial_{x_{2}} h, 1) / \sqrt{1 + ∣ \nabla_{x^{'}} h ∣^{2}}$ and $V_{n} = \partial_{t} h / \sqrt{1 + ∣ \nabla_{x^{'}} h ∣^{2}}$ . The jump convention is $[g] = g_{+} - g_{-}$ .

2.1givenA1A2F1F2step 1.1algebra

In the graph chart of step 1.1, write a piecewise field as its minus-side value plus its jump times the indicator of $x_{3} > h (t, x_{1}, x_{2})$ . Differentiating this indicator gives the singular part $- h_{t} [U] δ (x_{3} - h) + [F] (- h_{x_{1}}, - h_{x_{2}}, 1) δ (x_{3} - h)$ in $\partial_{t} U + div F - S$ , componentwise for vector or tensor fluxes; the one-dimensional Dirac distribution $δ$ is characterized by $\int f (x_{3}) δ (x_{3} - h) d x_{3} = f (h)$ . Since $(- h_{x_{1}}, - h_{x_{2}}, 1) = n \sqrt{1 + ∣ \nabla_{x^{'}} h ∣^{2}}$ and $h_{t} = V_{n} \sqrt{1 + ∣ \nabla_{x^{'}} h ∣^{2}}$ , and $δ (x_{3} - h) d x = d S / \sqrt{1 + ∣ \nabla_{x^{'}} h ∣^{2}}$ , its action is exactly the interface measure with coefficient $[F] n - V_{n} [U]$ , using the scalar-law row identification from the Statement. The sources have no surface measure, so the interior weak conservation identity requires this coefficient to vanish; when it vanishes, integration by parts leaves only the satisfied side equations.

2.2givenA1A2F1F2step 1.1algebra

For entropy put $H = ρ s$ , $G = ρ s u + q / T$ and $R = ρ r / T$ . The additional flux hypothesis makes $H, G$ admit $C^{1}$ one-sided extensions; $R$ is locally integrable and continuous on each open side, but need not extend to the interface. Apply the integration identity in [F2] with source zero and then add $\int R ζ$ . The weak integral in [F1] becomes the negative of the side integrals of $g ζ$ , where $g = \partial_{t} H + div G - R$ , and the negative interface integral of $K ζ$ , where $K = [G] \cdot n - V_{n} [H]$ and the interface measure is $d S d t$ . Here $g$ is locally integrable across the interface and continuous on each open side, and $K$ is continuous on the interface. Nonnegative $g$ and $K$ therefore suffice for the weak inequality, and tests supported inside either side prove the necessity of its side inequality.

3.1givenA1F2step 2.1algebra

Apply step 2.1 to $(U_{m}, F_{m}, S_{m})$ , $(U_{p}, F_{p}, S_{p})$ and $(U_{E}, F_{E}, S_{E})$ from the Statement. Their zero interface coefficients give the three displayed mass, momentum and energy jump equalities, with no division by $ρ$ .

4.1givenA2F1F2step 2.2algebra∎

If $K$ were negative at an interface point, choose a nonnegative $C^{1}$ graph cutoff supported in a patch where $K$ is bounded above by a negative constant, and nonnegative tests $ζ_{ε}$ with that fixed nonzero trace, uniformly bounded values and support in tubes $A_{ε}$ of width $ε$ . The negative interface integral in step 2.2 is a fixed positive number. The bulk contribution tends to zero: on a fixed compact tube $C$ , for every $M > 0$ , $\int_{A_{ε}} ∣ g ∣ \leq M λ_{4} (A_{ε}) + \int_{C \cap {∣ g ∣ > M}} ∣ g ∣$ . Integrability makes the last term arbitrarily small for large $M$ , while completed-product integration gives $λ_{4} (A_{ε}) = O (ε)$ over the compact graph support. Thus small $ε$ contradicts the weak inequality. Hence $K \geq 0$ , proving both directions of the entropy characterization without assuming bounded entropy sources or positive temperature traces.

Fluid conservation and entropy jump conditions

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources