Nonlinear Analysis: Real World Applications ( ) – Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa Remarks on traveling waves and equilibria in fluid dynamics with viscosity, capillarity, and heat conduction Mai Duc Thanh ∗ Department of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam article info Article history: Received April 2012 Accepted September 2013 abstract Heat conduction causes a tough obstacle in studying traveling waves in fluid dynamics In this note we consider the fluid dynamics equations where viscosity, capillarity and heat conduction coefficients are present First we transform the model into the one with an equation for the entropy as the conservation of energy Then, given any traveling wave of the viscous–capillary–heat conductive model connecting two given states, we derive a corresponding system of differential equations Then we show that this system of differential equations possesses the equilibria which correspond to the two states of the given traveling wave This work may therefore motivate future study to solve challenging open questions on the stability of these equilibria and the existence of the traveling waves in fluid dynamics with heat conduction © 2013 Published by Elsevier Ltd Introduction We are interested in the traveling waves of the following model of fluid dynamics equations with viscosity, capillarity, and heat conduction v t − ux = 0, µ λ v ut + p x = ux − (µvx )xx + vx , v x x κ µv λ uux + uvx − u(µvx )x + (µux vx )x + Tx , Et + (up)x = v v x x x (1.1) for x ∈ R and t > Here, v, S , p, ε, T denote the specific volume, entropy, pressure, internal energy, temperature, respectively; u is the velocity, and E =ε+ u2 + µ vx2 (1.2) is the total energy The non-negative quantities λ, µ, κ represent the viscosity, capillarity, and the heat conduction, respectively In general, these quantities can be considered as functions of the thermodynamic variables One may therefore write λ = λ(v, S ), µ = µ(v, S ) and κ = κ(v, S ) when choosing v, S as the independent thermodynamic variables Here, the Lagrangian coordinates are chosen so that the calculations are simple only, since similar results hold for the Eulerian coordinates ∗ Tel.: +84 2211 6965; fax: +84 3724 4271 E-mail addresses: mdthanh@hcmiu.edu.vn, mdthanh1@gmail.com 1468-1218/$ – see front matter © 2013 Published by Elsevier Ltd http://dx.doi.org/10.1016/j.nonrwa.2013.09.004 M.D Thanh / Nonlinear Analysis: Real World Applications ( ) – A shock wave connecting a left-hand state U− and a right-hand states U+ can be seen as admissible if it is obtained as the point-wise limit by vanishing additional terms involving higher-order derivatives such as viscosity, capillarity, and heat conduction, see [1] Therefore, traveling waves can be used to justify an admissibility criterion for shock waves of different types, such as the Lax shocks [2] and nonclassical shocks [3] However, the study of traveling waves, though it has attracted many authors, has been restricted mostly to the case where additional terms are merely viscosity and capillarity Heat conduction is commonly ignored The obstacle for the study of traveling waves when heat conduction is present is that this quantity causes serious inconvenience to establish a system of ordinary differential equations whose equilibria are used to characterize the traveling waves In this work, we propose a ‘‘complete’’ model that contains not only the viscosity and capillarity, but also the heat conduction It turns out that the capillarity coefficient is useful to deal with the heat conductive term First, we transform system (1.1) into a new one where the conservation of energy is expressed by an equation of the entropy Second, this new system is suitable to establish a system of ordinary differential equations whose equilibria correspond to the left-hand and right-hand states of a given traveling wave Note that these two states are also the states of a shock wave approximated by the traveling wave It is interesting to note also that unlike most results known in the literature, the system of ordinary differential equations depends on the type of fluid This explains the presence of the heat conduction, where the temperature is characterized by equations of state of the fluid under consideration and reflects the type of the fluid We choose the three most popular types of fluid to establish the corresponding system of ordinary differential equations: the ideal fluids, the stiffened gases, and the van der Waals fluids Nevertheless, the corresponding system of differential equations for fluids of other types can be derived in a similar manner This work thus raises open questions for future study on the stability of the equilibria and the existence of the traveling waves There have been many studies on traveling waves in the literature A pioneering work on the shock layers of the gas dynamics equations with viscosity and heat conduction effects (with zero capillarity) was presented in [4] Traveling waves for diffusive–dispersive scalar equations were earlier considered in [5,6] Traveling waves of the hyperbolic–elliptic model of phase transition dynamics were also studied in [7–11] Traveling waves in Korteweg models in the isothermal case, the Eulerian and Lagrangian capillarity models were also studied in [12–14] The existence of traveling waves associated with Lax shocks for viscous–capillary models was considered in our recent works [15–19] These works developed the method of estimating the attraction domain of the asymptotically stable equilibrium point to establish the existence of traveling waves The existence of traveling waves corresponding to nonclassical shocks for viscous–capillary models was considered in [3,20–25] See also the references therein The organization of this paper is as follows Section provides basic concepts and properties of the fluid dynamics equations In Section we present the derivation of an equivalent system to (1.1) where the conservation of energy is given as an equation of the entropy In Section we establish the autonomous system of ordinary differential equations for each given traveling wave It is shown that the equilibria of this system correspond to the left-hand and right-hand states of the given traveling wave Preliminaries Consider the fluid dynamics equations in the Lagrangian coordinates vt − ux = 0, ut + px = 0, Et + (up)x = 0, x ∈ R, t > (2.1) A shock wave of (2.1) is a weak solution U of the form U (x, t ) = U− , U+ , if x < st , if x > st , (2.2) where U− and U+ are constant, known as the left-hand and right-hand states, and a constant s, known as the shock speed These values must satisfy the Rankine–Hugoniot relations s(v+ − v− ) + (u+ − u− ) = 0, −s(u+ − u− ) + p+ − p− = 0, p+ + p− (v+ − v− ) = ε+ − ε− + (2.3) The last equation of (2.3) is also known as the Hugoniot equation The shock speed s can be given from (2.3) as s = s(U− , U+ ) = ± − provided p+ − p− ≤ v+ − v− p+ − p− , v+ − v− (2.4) M.D Thanh / Nonlinear Analysis: Real World Applications ( ) – In (2.4), the minus sign corresponds to 1-shocks, and the plus sign corresponds to 3-shocks Recall that an i-shock is a shock associated with the ith characteristic field, i = 1, 3, which becomes clear just below First, choosing U = (v, u, S ), we can re-write (2.1) as vt − ux = 0, ut + pv (v, S )vx + pS (v, S )Sx = 0, St = (2.5) The Jacobian matrix of the system (2.5) is given by A= −1 pv 0 0 pS Providing that pv (v, S ) < 0, the Jacobian matrix A admits three distinct real eigenvalues λ1 = − −pv (v, S ) < λ2 = < λ3 = −pv (v, S ), so that the system (1.2) is strictly hyperbolic Recall also that the first and the third characteristic fields are genuinely nonlinear, while the second characteristic field is linearly degenerate An i-Lax shock of the hyperbolic system (2.1) is a shock wave of the form (2.2) that satisfies the following Lax shock inequalities λi (U+ ) < s(U− , U+ ) < λi (U− ), i = 1, 3, (2.6) where s(U− , U+ ) is the shock speed given by (2.4) Model with equation of the specific entropy as conservation of energy Let us express the equation of conservation of energy in the form of an equation of the specific entropy S The left-hand side of the equation of conservation of energy of (1.1) can be re-written as LHS := Et + (up)x = εt + uut + upx + pux + µ µ vx2 t v t µ = TSt − p(vt − ux ) + u(ut + px ) + vx = TSt − pvt + uut + upx + pux + x t The second term of the last equation vanishes due to the conservation of mass The third term can be substituted by the equation of conservation of momentum in (1.1) Thus, it holds that LHS = TSt + u = TSt + u λ ux v λ ux v − (µvx )xx + x − (µvx )xx + x µ v µ v v x vx2 + µ x + x vx2 t 1 (µv vt + µS St )vx2 + 2µvx vxt (3.1) The right-hand side of the equation of conservation of energy of (1.1) is given by µ κ λ v RHS := uux + uvx2 − u(µvx )x + (µux vx )x + Tx v v x x x κ λ µv µv λ = u ux + u2x + u vx + ux vx2 − ux (µvx )x − u(µvx )xx + (µux vx )x + Tx v v 2 v x x x Equating LHS = RHS from (3.1) and (3.2), and eliminating the terms, one obtains, since vt = ux , TSt + µS St vx2 = (µux vx ) − [µvx vxt + ux (µvx )x ] + λ κ u + Tx v x v x (3.2) M.D Thanh / Nonlinear Analysis: Real World Applications ( ) – Since vxt = vtx = uxx , it is derived from the last equation that TSt + µS St vx2 = (µux vx ) − [µvx uxx + ux (µvx )x ] + λ κ u + Tx v x v x The first and the second terms on the right-hand side of the last equation cancel each other, so one gets the following equation for the entropy T+ µS κ λ vx2 St = u2x + Tx v v x (3.3) Thus, we arrive at the following result Lemma 3.1 (Conservation of Energy as Entropy Equation) The viscous–capillary–heat conductive system (1.1) is equivalent to the following system v t − ux = 0, µ λ v ut + px = ux − (µvx )xx + vx , v x x κ λ µS vx St = ux + Tx T+ v v x (3.4) Traveling waves and equilibria 4.1 Traveling waves with heat conduction A traveling wave of (1.1) connecting the left-hand state U− and the right-hand state U+ is a smooth solution of (1.1) depending on the variable U = U (y) = (v(y), u(y), S (y)), y = x − st , where s is a constant, and satisfying the boundary conditions lim U (y) = U± y→±∞ lim y→±∞ d dy U (y) = lim y→±∞ d2 dy2 (4.1) U (y) = Substituting U = U (y), y = (x − st ), into (3.4), we get −s v ′ − u ′ = 0, µ ′ λ ′ ′ v ′ ′ −su + p = u − (µv ′ )′′ + (v ′ )2 , (4.2) v λ ′ κ ′ ′ µS ′ ′ (v ) S = (u ) + T −s T + v v where (.)′ = d(.)/dy Eliminating u′ in the second and the third equations of (4.2) by substituting u′ from the first equation of (4.2), we obtain λ ′ v v ′ µ ′ (v ′ )2 , κ2 ′ µS ′ ′ 2λ ′ (v ) S = s (v ) + T′ −s T + v v Integrate the first equation of (4.3) on the interval (−∞, y) and use the boundary conditions (4.1) to get s2 v ′ + p′ = −s − (µv ′ )′′ + v λ v s2 (v − v− ) + p(v, S ) − p(v− , S− ) = −s v ′ − (µv ′ )′ + µv (v ′ )2 (4.3) (4.4) Re-arranging terms of (4.4) we can now re-write the first equation in (4.3) as λ µv ′ (µv ′ )′ = −s v ′ + (v ) − (p(v, S ) − p(v− , S− )) − s2 (v − v− ) v (4.5) We can also simplify the second equation in (4.3) Indeed, multiplying both sides of the equation (4.4) by sv ′ one gets λ v s3 (v − v− )v ′ + s(p − p− )v ′ = −s2 (v ′ )2 − s(µv ′ )′ v ′ + s µv v ′ (v ′ )2 (4.6) M.D Thanh / Nonlinear Analysis: Real World Applications ( ) – Add (4.6) to the second equation in (4.3) side-by-side to get s3 (v − v− )v ′ + s(p − p− )v ′ − s T + s µv v ′ ′ κ ′ ′ (v ′ )2 S ′ = − s(µv ′ )′ v ′ + (v ) + T 2 v µS Simplifying terms in the last equations yields κ ′ ′ (µS S ′ + µv v ′ )(v ′ )2 + 2µv ′ v ′′ + T , v or ′ κ ′ ′ s s3 (v − v− )v ′ − sp− v ′ − s(−pv ′ + TS ′ ) = − µ(v ′ )2 + T v Since ε(v, S )′ = −pv ′ + TS ′ , the last equation gives κ ′ ′ s µ(v ′ )2 = sε(v, S )′ + sp− v ′ − s3 (v − v− )v ′ + T′ v Integrating the last equation in (−∞, y), we have s3 (v − v− )v ′ + s(p − p− )v ′ − sTS ′ = − s s µ(v ′ )2 = s(ε(v, S ) − ε− ) + sp− (v − v− ) − s3 (v − v− )2 + κ ′ T v (4.7) 4.2 Equations for traveling waves for ideal fluid An ideal fluid has equations of state of the form pv ε= = cT , γ −1 where γ and c are constant Substituting T from (4.8) into (4.7) and using the thermodynamic identity (4.8) dε = TdS − pdv, one obtains s ′ µ(v )2 = s(ε(v, S ) − ε− ) + sp− (v − v− ) − Re-arranging terms in the last equation yields S′ = p ′ csv v + T κT s3 (v − v− )2 + µ(v ′ )2 − (ε(v, S ) − ε− ) − p− (v − v− ) + κ cv s2 (TS ′ − pv ′ ) (v − v− )2 , or (γ − 1)c ′ c s(γ − 1) s2 v + µ(v ′ )2 − (ε(v, S ) − ε− ) − p− (v − v− ) + (v − v− )2 (4.9) v κp 2 By setting w = µv ′ , we can see from (4.5) and (4.9) that a traveling wave of (1.1) satisfies the following system of ordinary S′ = differential equations v′ = w , µ µv w − p(v, S ) − p(v− , S− ) + s2 (v − v− ) , µv 2µ (γ − ) c c s(γ − 1) s2 S′ = w+ w − ε(v, S ) − ε− + p− (v − v− ) − (v − v− )2 µv κ p(v, S ) 2µ w′ = − sλ w+ (4.10) By letting the right-hand side of the system (4.10) vanish, one obtains the equilibria of the systems Proposition 4.1 (Equilibria for Ideal Fluids) A right-hand state (v+ , u+ , S+ ) can be connected to the left-hand state (v− , u− , S− ) by a traveling wave of the viscous–capillary–heat conductive model (1.1) only if the point (v+ , w+ , S+ ) is an equilibrium point of the systems of differential equations (4.10) for an ideal fluid, i.e., w+ = , p(v+ , S+ ) − p(v− , S− ) + s2 (v+ − v− ) = 0, ε(v+ , S+ ) − ε− + p− (v+ − v− ) − s2 (v+ − v− )2 = Remark 4.2 Observe from (4.2) that −su′ = v ′ , so the last two equations of (4.11) in Proposition 4.1 yield the Rankine– Hugoniot relations (2.3) Consequently, the right-hand state (v+ , u+ , S+ ) can be connected to the left-hand state (v− , u− , S− ) by a shock wave of the fluid dynamics equations (2.1) 6 M.D Thanh / Nonlinear Analysis: Real World Applications ( ) – 4.3 Equations for traveling waves for stiffened gas Equations of state of a stiffened gas are given by p = (γ − 1) ε − ε∗ − γ p∞ , v ε = cT + p∞ v + ε∗ , (4.11) where γ , ε∗ , c and p∞ are constant Substituting T = (ε − p∞ v − ε∗ )/c from (4.11) into (4.7) and using the thermodynamic identity dε = TdS − pdv, one obtains s µ(v ′ )2 = s(ε(v, S ) − ε− ) + sp− (v − v− ) − Re-arranging terms in the last equation yields S′ = ( p + p∞ ) T v′ + csv κT s3 (v − v− )2 + κ cv (TS ′ − (p + p∞ )v ′ ) µ(v ′ )2 − (ε(v, S ) − ε− ) − p− (v − v− ) + s2 (v − v− )2 , or S′ = (γ − 1)c ′ c s(γ − 1) s2 v + µ(v ′ )2 − (ε(v, S ) − ε− ) − p− (v − v− ) + (v − v− )2 v κ(p + p∞ ) 2 (4.12) By setting w = µv ′ , we can see from (4.5) and (4.12) that a traveling wave of (1.1) for a stiffened gas satisfies the following system of ordinary differential equations v′ = w , µ µv sλ w+ w − p(v, S ) − p(v− , S− ) + s2 (v − v− ) , µv µ2 (γ − 1)c c s(γ − 1) s2 S′ = w+ w − ε(v, S ) − ε− + p− (v − v− ) − (v − v− )2 µv κ(p(v, S ) + p∞ ) 2µ w′ = − (4.13) Similar to the case of an ideal fluid, by letting the right-hand side of the system (4.13) vanish, one obtains the equilibria of the systems Proposition 4.3 (Equilibria for Stiffened Gases) A right-hand state (v+ , u+ , S+ ) can be connected to the left-hand state (v− , u− , S− ) by a traveling wave of the viscous–capillary–heat conductive model (1.1) only if the point (v+ , w+ , S+ ) is an equilibrium point of the systems of differential equations (4.13) for a stiffened gas In other words, the Rankine–Hugoniot relations (2.3) hold, and so the right-hand state (v+ , u+ , S+ ) can be connected to the left-hand state (v− , u− , S− ) by a shock wave of the fluid dynamics equations (2.1) 4.4 Equations for traveling waves for van der Waals fluids For a fluid of van der Waals type, equations of state are given by p+ a v2 (v − b) = RT , (4.14) where a, b and R are constant Let A = ε − TS be the Helmholtz energy that determines the thermodynamical system of the fluid Then, it follows from the thermodynamical identity dε = TdS − pdV that dA = −SdT − pdv The last equation implies the Maxwell relation Sv (v, T ) = pT (v, T ) = −Av T It follows from the last Maxwell relation and (4.14) that TdS = cv dT + TpT (v, T )dv = cv dT + RT v−b dv, where cv = TST (v, T ) is the heat capacity at constant volume One deduces from (4.15) that TS ′ = cv T ′ + RT v−b v′ (4.15) M.D Thanh / Nonlinear Analysis: Real World Applications ( ) – which yields T′ = TS ′ cv RT v ′ − cv (v − b) (4.16) Substituting T ′ from (4.16) into (4.7), one obtains κT µ(v ) = s(ε(v, S ) − ε− ) + sp− (v − v− ) − (v − v− ) + 2 cv v s s3 ′ 2 ′ S − Rv ′ v−b Re-arranging terms in the last equation yields S′ = Rv ′ v−b + cv sv κT µ(v ′ )2 − (ε(v, S ) − ε− ) − p− (v − v− ) + s2 (v − v− ) (4.17) By setting w = µv ′ , we can see from (4.5) and (4.17) that a traveling wave of (1.1) for a fluid of van der Waals type satisfies the following system of ordinary differential equations v′ = w , µ µv sλ w+ w − p(v, S ) − p(v− , S− ) + s2 (v − v− ) , µv 2µ2 Rw cv sv s2 S′ = + w − ε(v, S ) − ε− + p− (v − v− ) − (v − v− )2 µ(v − b) κ T (v, S ) 2µ w′ = − (4.18) Similarly, by letting the right-hand side of the system (4.18) vanish, one obtains the equilibria of the systems Proposition 4.4 (Equilibria for van der Waals Fluids) A right-hand state (v+ , u+ , S+ ) can be connected to the left-hand state (v− , u− , S− ) by a traveling wave of the viscous–capillary–heat conductive model (1.1) only if the point (v+ , w+ , S+ ) is an equilibrium point of the systems of differential equations (4.18) for van der Waals fluids In other words, the Rankine–Hugoniot relations (2.3) hold, and so the right-hand state (v+ , u+ , S+ ) can be connected to the left-hand state (v− , u− , S− ) by a shock wave of the fluid dynamics equations (2.1) Remark 4.5 Whenever there is a trajectory connecting the equilibrium point (v− , w− , S− ) at −∞ to the equilibrium point (v+ , w+ , S+ ) at ∞, there is a traveling wave connecting the left-hand state (v− , u− , S− ) and the right-hand state (v+ , u+ , S+ ) The existence of the traveling waves is therefore reduced to establishing a connection between these two equilibria Remark 4.6 As well-known, the stability of the equilibria corresponding to the given traveling wave connecting a lefthand state U− to a right-hand state U+ for viscous–capillary models depends on the type of the associated shock wave If the shock is a Lax shock, then one expects a saddle-to-stable and/or stable-to-saddle connection, see [15–19] If the shock is nonclassical, one may expect a saddle-to-saddle connection, see [3,20–25] Different approaches in future study for the existence of traveling waves for the viscous–capillary–heat conductive model (1.1) are therefore expected, depending on the type of associated shock wave in which one is interested Acknowledgment This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) References [1] P.G LeFloch, Hyperbolic systems of conservation laws, in: The Theory of 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(3.4) Traveling waves and equilibria 4.1 Traveling waves with heat conduction A traveling wave of (1.1) connecting the left-hand state U− and the right-hand state U+ is a smooth solution of (1.1)... gas dynamics equations with viscosity and heat conduction effects (with zero capillarity) was presented in [4] Traveling waves for diffusive–dispersive scalar equations were earlier considered in. .. of traveling wave for general flux functions, Nonlinear Anal.: TMA 72 (2010) 231–239 [16] M.D Thanh, Attractor and traveling waves of a fluid with nonlinear diffusion and dispersion, Nonlinear