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Correct Expression of Material Derivative and Application to the Navier-Stokes Equation —– the Solution Existence Condition of Navier-Stokes Equation

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Correct Expression of Material Derivative and Application to the Navier-Stokes Equation —– the Solution Existence Condition of Navier-Stokes Equation Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 June 2020 doi:10.20944/preprints202003.0030.v4 Correct Expression of Material Derivative and Application to the Navier-Stokes Equation |{ The solution existence condition of Navier-Stokes equation Bo-Hua Sun1 1 School of Civil Engineering & Institute of Mechanics and Technology Xi'an University of Architecture and Technology, Xi'an 710055, China http://imt.xauat.edu.cn email: [email protected] (Dated: June 15, 2020) The material derivative is important in continuum physics. This paper shows that the expression d @ dt = @t + (v · r), used in most literature and textbooks, is incorrect. This article presents correct d(:) @ expression of the material derivative, namely dt = @t (:) + v · [r(:)]. As an application, the form- solution of the Navier-Stokes equation is proposed. The form-solution reveals that the solution existence condition of the Navier-Stokes equation is that "The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely det(rv) 6= 0." Keywords: material derivative, velocity gradient, tensor calculus, tensor determinant, Navier-Stokes equa- tions, solution existence condition In continuum physics, there are two ways of describ- derivative as in Ref. 1. Therefore, to revive the great ing continuous media or flows, the Lagrangian descrip- influence of Landau in physics and fluid mechanics, we at- tion and the Eulerian description. In the Eulerian de- tempt to address this issue in this dedicated paper, where scription, the material derivative with respect to time we revisit the material derivative to show why the oper- d @ dv @v must be defined. For mass density ρ(x; t), flow velocity ators dt = @t + (v · r) and expression dt = @t + (vr)v v(x; t) = vkek, and stress tensor σ = σijei ⊗ ej, the are incorrect, and derive a correct expression by using dρ @ρ @ρ standard tensor calculus [21{24]. material derivatives are given by: dt = @t + v1 @x1 + @ρ @ρ dv @v @v @v @v In Landau & Lifshitz,[1] the material derivative is given v2 @x2 + v3 @x3 ;, dt = @t + v1 @x1 + v2 @x2 + v3 @x3 , and dσ @σ @σ @σ @σ as: dt = @t + v1 @x1 + v2 @x2 + v3 @x3 , respectively, where t is time, x = xkek is position, ek is a base vector and @v @v @v @xk dx + dy + dz = (dr · grad)v: (1) vk = @t is the flow velocity. @x @y @z In most popular textbooks, handbooks and encyclope- dia, such as Refs. 1{12, the above material derivatives Although Landau's fluid mechanics is well known world- dρ @ρ dv @v wide, the above expression is incorrect. are expressed as: dt = @t +(v ·r)ρ, dt = @t +(v ·r)v, and dσ = @σ + (v · r)v, where the gradient operator dt @t Proof: Since v · grad = v · r = (viei) · @ r = ek . @xk (@jej) = vi;jei · ej = vi;jδij = divv, and To simplify the aforementioned equations further, a d- (v · grad)v = (v · r)v = (vj;iei · ej)(vkek) = d @ ifferential operator dt = @t + (v · r) is introduced. This δijvj;ivkek = vi;ivkek = (divv)v, where the operator is used by most fluid mechanics textbooks, in- @v1 @v2 @v3 divergence is divv = @x1 + @x2 + @x3 , so cluding well-known graduate textbooks such as Landau (divv)v = ( @v1 + @v2 + @v3 )v. Thus: & Lifshitz,[1, 2] Prandtl,[3, 4] Anderson,[5] Pope,[6] Cen- @x1 @x2 @x3 gel & Cimbala,[7], Kundu et al.[8], Woan [9], wikipedia @v @v @v (v · grad)v 6= v + v + v : (2) [10, 11], Britannica [12] and even mathematical paper 1 @x1 2 @x2 3 @x3 [13]. In the latest version of Landau & Lifshitz,[14] the material derivative of flow velocity is given in another In the latest version of Landau & Lifshitz,[14] for rea- dv @v form: dt = @t + (vr)v. sons which are unclear, the material derivative expression Besides the books mentioned above, there is a large dv @v has been changed to dt = @t + (vr)v; this is also incor- amount of literature in fluid mechanics taking the same rect. expressions as Refs. 1 and 14. This makes it difficult for readers to identify which expression for the materi- Proof: Since vr = (viei)(@jej) = vi;jeiej, al derivative is correct, causing great confusion to both students and scholars. (vr)v = (vj;ieiej)(vkek) = (vj;ieiej)(vkek); Although some authors, such as Lighthill,[15] which indicates that vr is a 3rd-order tensor Batchelor,[16] Frisch,[17], Nhan & Nam [18], Xie,[19] and rather than a vector. Thus: Zhao [20] have used the correct expression, it seems that @v @v @v most fluid mechanics textbooks and academic literature (vr)v 6= v + v + v : (3) have adopted an incorrect expression for the material 1 @x1 2 @x2 3 @x3 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 June 2020 doi:10.20944/preprints202003.0030.v4 2 T @σ To obtain the correct formulation of the material (rσ) , hence dσ = @t dt + (σr) · vdt, or dividing both derivative, we perform some basic tensor calculus [21, 22]. dσ @σ sides by dt leads to dt = @t +(σr)·v. Since (σr)·v = The del operator is a vector differential operator, and de- v · (rσ), @ fined by r = ei . An important note concerning the @xi dσ @σ del operator r is in order. Two types of gradients are = + v · (rσ): (6) used in continuum physics: left and right gradients. The dt @t left gradient is the usual gradient and the right gradient If an operator must be introduced for the material is the transpose of the forward gradient operator. derivative, it should be in the following form: To see the difference between the left and right of gradients, consider a vector function A = Ai(x)ei. d(:) @ @ = (:) + v · [r(:)]: (7) The left gradient of a vector A is r ⊗ A ≡ ej ⊗ @xj dt @t @Ai (Aiei) = ej ⊗ ei = Ai;jej ⊗ ei, or written as rA ≡ @xj As an application, the Navier-Stokes equations of in- @ @Ai ej (Aiei) = ejei = Ai;jejei. The right gradient compressible flow can be expressed as follows: @xj @xj @ @Ai of a vector A is A ⊗ r ≡ (Aiei) ⊗ ej = ei ⊗ @xj @xj @v 1 2 @ + v · (rv) = − rp + νr v; (8) ej = Ai;jei ⊗ ej, or written as Ar ≡ (Aiei)ej = @t ρ @xj @Ai @Ai eiej = Ai;jeiej, where Ai;j = [21{24]. r · v = 0: (9) @xj xj The gradient of a scalar function is a vector, the diver- The Eq. 8 is momentums equation and Eq. 9 is mass gence of a vector-valued function is a scalar r · A, and conservation equation. In which, v(x; t) is flow velocity the gradient of a vector-valued function is a second-order field, ρ is constant mass density, p(x; t) is flow pressure, tensor rA. Although the del operator has some of the ν is kinematical viscosity, t is time, x = xke is position properties of a vector, it does not have them all, because k coordinates, e is a base vector and v is flow velocity, it is an operator. For instance, r · A is a scalar (called k r = e @ is gradient operator, and r2 = r · r. the divergence of A) whereas A·r is a scalar differential k @xk Applying the divergence operation to both sides of the operator, where A is a vector. Thus the del operator r momentum equation Eq.8 and use the mass conservation does not commute in this sense [21{24]. leads to a pressure equation: r2 ·(p1) = −ρr·(v·rv) = It worth to emphasise that the right gradient of a vec- −ρf(rv)2 + [r(vr)] · vg, where 1 = e e is an identity tor Ar is a more natural one, is often used in defining k k tensor and vr = (rv)T . the deformation gradient tensor, displacement gradien- The NavierõStokes existence and smoothness prob- t tensor, and velocity gradient tensor. It is clear that lem is an open problem in mathematics [25], regardless of Ar = (rA)T . numerous abstract studies that have been done by math- With the above-mentioned understanding of tensor ematicians. Now the question is that, is it possible to calculus, we material derivative expression for scalar, vec- propose a simple criteria on the solution existence of the tor and the 2nd order tensor as follows: Navier-Stokes equation without complicated mathemat- 1. Mass density ρ = ρ(x; t) is a scalar-valued function ics. of x and t and its differential is dρ = @ρ dt + @ρ · @x dt. @t @x @t In order to find some useful information from the @ρ @ρ For a scalar @x = ρr = rρ, so dρ = @t dt+(ρr)·vdt = Navier-Stokes equation, let's have a look at the meaning @ρ @t dt + v · (rρ)dt, or, dividing both sides by dt, of v ·(rv) in Eq. 1. This term is called convective accel- eration that is caused by the flow velocity gradient. It is dρ @ρ @ρ = + (ρr) · v = + v · (rρ): (4) obvious that the convective acceleration v · (rv) is the dt @t @t central point of the Navier-Stokes equations. Without the convective acceleration, the solution existence would 2. Flow velocity v = v(x; t) is a vector-valued function @v @v @x not be a problem at all. The understanding on the con- of x and t and its differential is dv = @t dt + @x · @t dt. @v T vective term is quite important for the study on the so- For a vector @x = v ⊗ r = vr = (rv) .
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