2016 Computational Mathematics Annual Program Review Arlington, VA (USA), August 8-11, 2016
EXPLICIT SOLUTIONS TO PHASE-CHANGE PROBLEMS AND APPLICATIONS
Domingo Alberto Tarzia
Grant AFOSR – SOARD FA 9550-14-1-0122
CONICET - Depto. Matemática, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina. E-mail: [email protected]
Key Words: Free boundary problems, Lamé-Clapeyron-Stefan problem, Phase-change problems, Heat- Diffusion equation, Fractional diffusion, Caputo fractional derivative, Explicit solutions, Neumann solution, Unknown thermal coefficients, Over-specified boundary condition, Mushy region, Convective boundary condition.
Mathematics Subject Classification 2010: 26A33, 35C05, 35R11, 35R35, 80A22. 2 ABSTRACT
We obtain explicit solutions (in a closed form) for the following one-dimensional phase- change problems: • Two-phase solidification process with a convective boundary condition (T. Thermal Sci (2016), In Press); • One-phase solidification process with a mushy region and a convective boundary condition (T., J. Appl. Math. (2015)); • One-phase solidification process with a mushy region and a heat flux boundary condition (T., J. Appl. Math. (2015)); • Determination of one unknown thermal coefficient through a solidification problem with an over-specified boundary condition (Ceretani-T., Math. Problems in Eng. (2015));
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 3 • Simultaneous determination of two unknown thermal coefficients through a solidification problem with an over-specified boundary condition (Ceretani-T., JP Journal Heat Mass Transfer (2016)); • Two-phase fractional Stefan problem with a temperature boundary condition (Roscani-T., Adv. Math. Sci. Appl. (2014)); • Two-phase fractional Stefan problem with a heat flux boundary condition (Roscani- T., In progress); • Determination of one unknown thermal coefficient through a one-phase fractional Stefan problem with an over-specified boundary condition (T. Appl. Math. (2015));
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 4 INTRODUCTION
Heat and mass transfer problems with a phase-change process (known in the literature as Stefan problem or Lamé-Clapeyron-Stefan problem) begins by modeling the solidification of the Earth planet (1831). Phase-change process such as melting, freezing, solidification, sublimation, desublimation, thawing and drying have been studied in the last century due to their wide scientific and technological applications. Some of them are:
i) Continuous casting of steel (e.g., ingots, slabs); ii) Binary alloy solidification problem (liquidus and solidus curves); iii) Permafrost process (e.g., frozen ground in cold region of the Earth planet); iv) Manufacture of glass and crystal; v) Phase-change process with a mushy region; vi) Thawing in a saturated porous medium with the influence of the pressure on the melting temperature; vii) Coupled heat and mass transfer for a phase-change process (e.g., Luikov systems); viii) Coupled heat and mass transfer during the freezing of the high-water content materials with two free boundaries: the freezing and sublimation fronts (e.g., for food preservation); ix) Ablation process (e.g., entry of a spacecraft to the atmosphere of the Earth planet); x) Welding process;
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 5 xi) Solidification of a supercooled liquid (temperature of the liquid is below of the melting temperature); xii) Phase-change materials to store solar energy as the latent heat of melting (e.g., heating of a house);
and some related free boundary problems as:
xiii) Oxygen diffusion-consumption in a tissue (equivalent to the phase-change problem); xiv) Isothermal diffusion-reaction process of a gas with a solid; xv) Gas flow through a porous medium; xvi) Penetration of solvent in polymers; xvii) Filtration of water through oil in a porous medium; xviii) Nutrient uptake by a growing root system.
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 6 A review of a long bibliography on moving and free boundary problems for phase- change materials (PCM) for the heat equation is given in:
T., "A bibliography on moving-free boundary problems for the heat-diffusion equation. The Stefan and related problems", MAT - Serie A, 2 (2000), 1-297 (with 5869 titles). Available from: http://web.austral.edu.ar/descargas/facultad-cienciasEmpresariales/mat/Tarzia-MAT- SerieA-2(2000).pdf
Obtaining explicit solution we have: i) a tool to get super and sub-solution for general conditions; ii) a benchmark solution for checking on the correctness of a computer code
A review on explicit solutions for heat and mass transfer problems is given in:
T., “Explicit and Approximated Solutions for Heat and Mass Transfer Problems with a Moving Interface”, Chapter 20, In Advanced Topics in Mass Transfer, Mohamed El-Amin (Ed.), InTech Open Access Publisher, Rijeka (2011), pp. 439-484. Available from: http://www.intechopen.com/articles/show/title/explicit-and-approximated-solutions-for-heat- and-mass-transfer-problems-with-a-moving-interface
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 7 OUTLINE I. Classical two-phase Stefan problems Classical Neumann solution for a temperature boundary condition at the fixed face x= 0 Classical Neumann solution with a heat flux boundary condition at the fixed face x= 0 Classical Neumann solution with a convective boundary condition at the fixed face x= 0
II. Classical one-phase Stefan problem with a simple mushy region One-phase with a convective boundary condition One-phase with a heat flux boundary condition Applications: o Determination of one unknown thermal coefficient o Simultaneous determination of two unknown thermal coefficients
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 8 III. Phase-change problems for the fractional heat-diffusion equation Fractional derivatives o Caputo fractional derivative o Wright and Mainardi Functions o Basic properties
Two-phases fractional Stefan problems: o Generalized Neumann solution with a temperature boundary condition o Generalized Neumann solution with a heat flux boundary condition
One-phase fractional Stefan problem: o Determination of one unknown thermal coefficient
IV. Papers which are not reviewed
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 9
I. Classical two-phase Stefan problems
Classical Neumann solution with a temperature boundary condition at the fixed face x= 0 (Weber, Book (1901)).
Classical Neumann solution with a heat flux boundary condition at the fixed face x= 0 (T., Quart. Appl. Math. (1981)).
Classical Neumann solution with a convective boundary condition at the fixed face x= 0 (T. Thermal Science (2016), In Press)
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 10 I.1) CLASSICAL NEUMANN SOLUTION (MELTING PROCESS) WITH A TEMPERATURE BOUNDARY CONDITION
Problem (P1) The two-phase Lamé-Clapeyron-Stefan problem is given by: find the free boundary x= s() t , and the temperature T= T(,) x t defined by: T( x , t )> T if 0 < x < s ( t ), t > 0 f = = > T( x , t ) Tf if x s ( t ), t 0 (T1) < < > Ts( x , t ) T f if s ( t ) x , t 0 such that the following equations and conditions are satisfied: c T− k T =0, x > s ( t ), t > 0, (T2) s st s s xx c T− k T =0, 0 < x < s ( t ), t > 0, (T3) t xx s(0)= 0, (T4) = +∞ = < > > Ts(,0) x T s (,) t T i T f , x 0, t 0, (T5) = > Ts( s ( t ), t ) T f , t 0, (T6) = > Tl( s ( t ), t ) T f , t 0, (T7) kTstt( ( ),) − kTstt( ( ),) = st ( ), t > 0, (T8) s sx x = > > T (0, t ) T0 Tf , t 0. (T9)
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 11
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 12 < < Theorem T1 [Weber, Book (1901)] Let TTTi f 0 be. The solution of the problem (P1) is the classical Neumann explicit solution given by:
TT− = −0 f x ≤ ≤ > TxtT ( , )0 erf , 0 xstt ( ), 0, (T10) erf () 2 t
TT− = +f i x ≤ > TxtTs( , ) i erfc , stxt ( ) , 0, (T11) erfc() 2 st
= =k =2 =ks = 2 = s s( t ) 2 s t , , s s , , (T12) c cs where the dimensionless coefficient > 0 is the unique solution of the following equation:
G( x )= x , x > 0 (T13) with
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 13
= − G()()() x b4 G 2 x b 3 G 1 x , (T14)
2 x erf( x )=∫ exp( − u2 ) du , erfc ( x ) = 1 − erf ( x ), (T15) 0
exp()()− x2 exp − x 2 G(),() x= G x = , (T16) 1erfc()() x 2 erf x
c()() T− T k T − T =s f i > = 0 f > b30, b 4 0. (T17) s
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 14 I.2) CLASSICAL NEUMANN SOLUTION (MELTING PROCESS) WITH A HEAT FLUX BOUNDARY CONDITION
= = Problem (P2 ) Find the free boundary x s() t , and the temperature T T(,) x t defined by: T( x , t )> T if 0 < x < s ( t ), t > 0 f = = > T( x , t ) Tf if x s ( t ), t 0 (F1) < < > Ts( x , t ) T f if s ( t ) x , t 0 such that the following equations and conditions are satisfied:
c T− k T =0, x > s ( t ), t > 0, (F2) s st s s xx c T− k T =0, 0 < x < s ( t ), t > 0, (F3) t xx s(0)= 0, (F4) = +∞ = < > > Ts(,0) x T s (,) t T i T f , x 0, t 0, (F5) = > Ts( s ( t ), t ) T f , t 0, (F6) = > Tl( s ( t ), t ) T f , t 0, (F7) kTstt( ( ),) − kTstt( ( ),) = st ( ), t > 0, (F8) s sx x q k T(0, t )= −0 , t > 0. (F9) x 1 t 2 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 15
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 16 < Theorem F1 [T., Quart. Appl. Math. (1981)] Let TTi f be.
a) If the coefficient q0 satisfies the inequality:
k() T− T > s f i q0 , (F10) s
then there exists an instantaneous phase-change (melting) process and the problem (P2 ) has the unique Neumann explicit solution of a similarity type given by:
q = +0 −x ≤ ≤ > TxtT (,)f erf () F erf ,0 xstt (),0 (F11) k 2 t
TT− = +f i x ≤ > TxtTs( , ) i erfc , stxt ( ) , 0, (F12) erfc()F 2 st
= =k = ks s( t ) 2F s t , , s , (F13) c cs AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 17 > where the dimensionless coefficient F 0 is the unique solution of the following equation:
= > GF ( x ) x , x 0 (F14) with
= −2 2 − GF ( x ) b5 exp( x ) b 3 G 1 ( x ), (F15)
c() T− T 2 =s > =s f i > =q0 > 0,b3 0, b 5 0. (F16) s
b) If the coefficient q0 satisfies the inequalities:
k() T− T < ≤ s f i 0 q0 , (F17) s
then the problem (P2 ) is a classical heat transfer problem for the initial solid phase whose solution is given by: q = +0 s x > > Ts( x , t ) T i erfc , x 0, t 0. (F18) ks 2 st AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 18 I.3) CLASSICAL NEUMANN SOLUTION (SOLIDIFICATION PROCESS) WITH A CONVECTIVE BOUNDARY CONDITION
= = Problem (P3) Find the free boundary x s() t , and the temperature T T(,) x t defined by: T( x , t )< T if 0 < x < s ( t ), t > 0 s f = = > T( x , t ) Tf if x s ( t ), t 0 (C1) > < > T ( x , t ) Tf if s ( t ) x , t 0 such that the following equations and conditions are satisfied:
c T− k T =0, 0 < x < s ( t ), t > 0, (C2) s st s s xx c T− k T =0, x > s ( t ), t > 0, (C3) t xx s(0)= 0, (C4) = +∞ = > > > T(,0) x T (,) t Ti T f , x 0, t 0, (C5) = > Ts( s ( t ), t ) T f , t 0, (C6) = > Tl( s ( t ), t ) T f , t 0, (C7) kTstt( ( ),) − kTstt( ( ),) = st ( ), t > 0, (C8) s sx x −1 2 k T(0,) t= h t( T (0,) t − T∞ ) , t > 0( h > 0). (C9) s sx 0 s 0
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl.
< < 19 Theorem 1 Let TTT∞ f i be. a) If the coefficient h0 satisfies the inequality: − k TTi f h > l , (C10) 0 − TTf ∞ then there exists an instantaneous solidification process and the free boundary problem (P ) has the unique solution of a similarity type given by: 3 h − + 0 s x (Tf T∞ ) 1 erf k 2 t = + s s Ts (,) x t T∞ h 1 + 0 s erf ks s
− x erf erf h () T− T∞ 2 t = − 0 s f s s Tf (C11) k h s 1 + 0 s erf ks s
AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 20 x x erfc erfc 2t 2 t = − − = + − − T ( x , t ) Ti ( T i T f ) T f ( T i T f ) 1 , (C12) erfc()() erfc = s( t ) 2C t , (C13) > and the dimensionless coefficient C 0 satisfies the following equation:
F( x )= x , x > 0, (C14) where the function F and the parameters bi are given by: exp(−bx2 ) F()() x= b − b F x , (C15) 1+ 3 1 1b2 erf ( x b )
h() T− T∞ =0 f > =h0 > b10, b 2 s 0. (C16) ks AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 21 b) If the coefficient h0 satisfies the following inequalities: − k TTi f 0 k Ti x T∞+ +() T − T ∞ erf i − h0 2 t TT∞ k x = = +i + T (,) x t T∞ erf k k 1+ 1 + h0 2 t h0 h 0 TT− ∞ x = −i > > Ti erfc , x 0, t 0. (C18) k 1 + 2 t h0 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 22 II. Classical one-phase Stefan problem with a simple mushy region One-phase with a convective boundary condition (T., J. Appl. Math. (2015)) One-phase with a heat flux boundary condition (T., J. Appl. Math. (2015)) Applications: o Determination of one unknown thermal coefficient (Ceretani-T., Math. Problems in Eng. (2015)) o Simultaneous determination of two unknown thermal coefficients (Ceretani-T., JP Journal Heat Mass Transfer (2016)) AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 23 II.1 Classical one-phase Stefan problem with a simple mushy region and a convective boundary condition at the fixed face x= 0 Solidification process with a convective boundary condition: It consists in finding the free boundaries x= s() t and x= r() t , and the temperature T= T(,) x t such that the following conditions are verified: − = < <( ) > = cTt kT xx 0 , 0 x s t , t 0 ( k / c ) (SC1) T( s( t), t) = 0 , t > 0 (SC2) = + − > kTx ((),) s t t [ s ()(1 t )()], r t t 0 (SC3) − = > > > Tx ((),)(() s t t r t s ()) t 0, t 0 (with 0) (SC4) s(0)= r (0) = 0 (SC5) h0 kT(0, t) =( T (0,) t + D∞) , t > 0(with h > 0, D ∞ > 0). (SC6) x t 0 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 24 Theorem. If the coefficient h0 satisfies the inequality − > 1 (1 ) k h0 (SC7) D∞ 2 then the solution to the problem (SC1)-(SC6) is given by: h D∞ x 0 erf () erf 2 t T( x , t )= −k 1 − , 0 < x < s ( t ), t > 0, (SC8) h erf () 1+ 0 erf ( ) k s( t )= 2 t , t > 0, (SC9) r( t )= 2 t , t > 0, (SC10) k 2 h = +e 1 + 0 erf ( ) , (SC11) 2D∞ h0 k AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 25 where the coefficient is given as the unique solution of the equation: D∞ c = > F3 ( x ) G ( x ), x 0, (SC12) with exp(−x2 ) (1 − ) 1 F( x )= , G ( x ) = x + , x > 0 (SC13) 3 k + erf() x 2D∞ F3 ( x ) h0 . AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 26 II.2 Classical one-phase Stefan problem with a simple mushy region and a heat flux boundary condition at the fixed face x= 0 Solidification process with a heat flux condition: It consists in finding the free boundaries x= s() t and x= r() t , and the temperature T= T(,) x t such that the following conditions are verified: − = < <( ) > = cTt kT xx 0 , 0 x s t , t 0 ( k / c ) (SF1) T( s( t), t) = 0 , t > 0 (SF2) = + − > kTx ((),) s t t [ s ()(1 t )()], r t t 0 (SF3) − = > > > Tx ((),)(() s t t r t s ()) t 0, t 0 (with 0) (SF4) s(0)= r (0) = 0 (SF5) q kT(0, t) =0 , t > 0 (with q > 0). (SF6) x t 0 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 27 Theorem. If the coefficient q0 satisfies the inequality (1− ) k q > (SF7) 0 2 then the solution to the problem (SF1)-(SF6) is given by: x erf q erf () 2 t T( x , t )= −0 1 − , 0 < x < s ( t ), t > 0, (SF8) k erf () s( t )= 2 t , t > 0, (SF9) r( t )= 2 t , t > 0, (SF10) k 2 = + e , (SF11) 2q0 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 28 where the coefficient is given as the unique solution of the equation: =q0 > G3 ( x ) , x 0, (SF12) with − = +(1 )k x2 x 2 > G3 ( x ) x e e , x 0 (SF13) 2q0 . AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 29 III. Determination of unknown thermal coefficients through a one-phase Stefan problem with a simple mushy region and an over-specified condition at the fixed face x= 0 III.1 Determination of one unknown thermal coefficient The problem consists in finding the free boundaries x= s() t and x= r() t , the temperature T= T(,) x t and one unknown termal coefficient among {k,,,,, c } such that the following conditions must be verified: − = < <( ) > = cTt kT xx 0 , 0 x s t , t 0 ( k / c ) (1FC1) T( s( t), t) = 0 , t > 0 (1FC2) = + − > kTx ((),) s t t [ s ()(1 t )()], r t t 0 (1FC3) − = > Tx ((),)(() s t t r t s ()) t , t 0 (1FC4) s(0)= r (0) = 0 (1FC5) q kT(0, t) =0 , t > 0(with q > 0) (1FC6) x t 0 h0 kT(0, t) =( T (0,) t + D∞) , t > 0(with h > 0, D ∞ > 0). (1FC7) x t 0 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 30 The problem can be divided in 6 different cases: Case # One Unknown Coefficient 1 Latent heat of fusion by unit of mass: 2 First parameter of the mushy region: 3 Second parameter of the mushy region: 4 Thermal conductivity: k 5 Density of mass: 6 Specific heat by unit of mass: c AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 31 Theorem. If h0 and q0 are two positive numbers determinated experimentally then the solution of the 6 cases are given by: q x Txt(,)=0 erf − erf (),0 < xstt < (),0 > , (1FC8) k 2 t s( t )= 2 t , t > 0, (1FC9) r( t )= 2 t , t > 0, (1FC10) k 2 = + e . (1FC11) 2q0 and the coefficient and the unknown thermal coefficient are given by the following Table 1. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. Table 1: Formulae for the problem with a convective over-specified boundary condition. Explicit formulae for the unknown thermal coefficient l, γ, , k, ρ or c and the coefficient ξ (or the equation that it must satisfy) and the corresponding restrictions on data that guarantee their validity. Case Thermal coefficient Coefficient ξ that characterizes Restrictions the free boundary x = s(t) on data 2 q c q0 exp (−ξ ) −1 q kρc q 1 l = h √ i ξ = erf D∞ 1 − 0 (R1), (R2) ρk ξ+ γ(1−) kρc exp (ξ2) q0 π h0D∞ 2q0 q q 2 γ = 2q√0 q0 c − ξ exp (ξ2) exp (−2ξ2) ξ = erf−1 D∞ kρc 1 − q0 (R1), (R2) (1−) kρc l ρk q0 π h0D∞ (R3) q q 3 = 1 − √2q0 q0 c − ξ exp (ξ2) exp (−2ξ2) ξ = erf−1 D∞ kρc 1 − q0 (R1), (R2) γ kρc l ρk q0 π h0D∞ (R3), (R4) 2 4 k = π q0 erf(ξ) Unique positive solution (R1) ρc q0 D∞ 1− h0D∞ of the equation (E4) 2 5 ρ = π q0 erf(ξ) Unique positive solution (R1) kc q0 D∞ 1− h0D∞ of the equation (E4) 2 6 c = π q0 erf(ξ) Unique positive solution (R1), (R5) ρk q0 D∞ 1− h0D∞ of the equation (E6) Restrictions on data q 1 − 0 > 0 (R1) h0D∞ r D kρc q ∞ 1 − 0 < 1 (R2) q0 π h0D∞ r !! r −1 D∞ kρc q0 q0 c f2 erf 1 − < (R3) q0 π h0D∞ l ρk √ " r !#2 γ kρc −1 D∞ kρc q0 exp 2 erf 1 − + 2q0 q0 π h0D∞ (R4) r !! r −1 D∞ kρc q0 q0 c f2 erf 1 − > q0 π h0D∞ l ρk q 1 2q2 1 − 0 < 0 − γ(1 − ) (R5) h0D∞ D∞ ρlk Definitions of functions (x > 0) 2 f2(x) = x exp (x ) √ πγ(1 − ) 2 2 f4(x) = x + erf(x) exp (x ) erf(x) exp (x ) 2D 1 − q0 ∞ h0D∞ √ x πγ(1 − ) 2 2 f6(x) = + exp (x ) exp (x ) erf(x) 2D 1 − q0 ∞ h0D∞ Ecuations for ξ cD∞ q0 f4(x) = √ 1 − x > 0 (E4) πl h0D∞ √ 2 πq0 f6(x) = x > 0 (E6) ρlkD 1 − q0 ∞ h0D∞ 32 III.2 Simultaneous Determination of two thermal coefficients We follow the previous section III.1 and we consider that the left boundary of the mushy region is a moving boundary given by the expression: s( t )= 2 t , t > 0, (2FC1) where > 0 is a positive parameter obtained experimentally. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 33 The problem consists in finding the free boundary x= r() t , the temperature T= T(,) x t and two unknown termal coefficients among {k,,,,, c } such that the following conditions are verified: − = < <( ) > = cTt kT xx 0 , 0 x s t , t 0 ( k / c ) (2FC2) T( s( t), t) = 0 , t > 0 (2FC3) = + − > kTx ((),) s t t [ s ()(1 t )()], r t t 0 (2FC4) − = > Tx ((),)(() s t t r t s ()) t , t 0 (2FC5) s(0)= r (0) = 0 (2FC6) q kT(0, t) =0 , t > 0(with q > 0) (2FC7) x t 0 h0 kT(0, t) =( T (0,) t + D∞) , t > 0(with h > 0, D ∞ > 0). (2FC8) x t 0 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 34 The problem can be divided in 15 different cases: Case # Two Unknown Coefficients 1 and 2 and 3 and 4 and k 5 and 6 and c 7 and k 8 and 9 and c 10 and k 11 and 12 and c 13 k and 14 k and c 15 and c AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 35 Theorem. If h0 , q0 and are three positive numbers determinated experimentally then the solution of the 15 cases are given by: q x Txt(,)=0 erf − erf (),0 < xstt < (),0 > , (2FC9) k 2 t 2 k e r( t )= 2 + t , t > 0, (2FC10) q0 and the two unknown thermal coefficients are given by the following Table 2. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. Table 2: Formulae for the problem with a convective over-specified boundary condition. Explicit formulae for the two unknown thermal coefficients chosen among l, γ, , k, ρ or c, the equation for the coefficient ξ and the corresponding restrictions on data that guarantee their validity. Case Thermal coefficients Equation for Restrictions q √σ ρc ξ = α = σ k on data 1 0 < < 1 – (R0), (R1) 2q0σ q0 2 2 γ = k(1−) ρlσ exp (−σ /d) − 1 exp (−σ /d) 2 0 < < 1 – (R0) 2 q0 exp (−σ /d) l = h i ρσ 1+ γk(1−) exp (σ2/d) 2q0σ 3 γ > 0 – (R0) 2 q0 exp (−σ /d) l = h i ρσ 1+ γk(1−) exp (σ2/d) 2q0σ 4 = 1 − F4(ξ) (E4) (R2), (R3), (R4), 2 σ k = ρc ξ (R5), (R6) or: (R2), (R3), (R4), (R7), (R8) or: (R2), (R3), (R4), (R9) 5 = 1 − F5(ξ) (E5) (R12) 2 k ξ ρ = c σ 6 = 1 − F6(ξ) (E5) (R13), (R14), (R15) 2 k ξ c = ρ σ or: (R13), (R16), (R17) 2q0 q0 2 2 2 7 γ = σρc(1−) ρlσ exp (−ξ ) − 1 ξ exp (−ξ ) (E4) (R2), (R3), (R4) 2 σ k = ρc ξ Case Thermal coefficients Equation for Restrictions q ξ = √σ = σ ρc on data d k 2 2q0σ q0cσ exp (−ξ ) 2 8 γ = k(1−) lk ξ − 1 exp (−ξ ) (E5) (R2), (R17), (R18) 2 k ξ ρ = c σ 2q0σ q0 2 2 9 γ = k(1−) ρlσ exp (−ξ ) − 1 exp (−ξ ) (E5) (R3), (R13), (R17) 2 k ξ c = ρ σ 2 q0 1 exp (ξ ) 2 10 l = 2 exp (−ξ ) (E4) (R2) ρσ 1+ ρcσγ(1−) ξ 2q0 2 σ k = ρc ξ 2 11 l = q0cσ 1 exp (−ξ ) (E5) (R2), (R17) k 1+ γk(1−) exp (ξ2) ξ 2q0σ 2 k ξ ρ = c σ 12 l = q0 1 exp (−ξ2) (E5) (R2), (R17) ρσ 1+ γk(1−) exp (ξ2) 2q0σ 2 k ξ c = ρ σ √ 13 k = πq0σ G (ξ) (E13) (R2) q0 5 D∞ 1− √ h0D∞ ρ = πq0 G (ξ) cσD∞ 4 √ 14 k = πq0σ G (ξ) (E14) (R2), (R3), (R19) q0 5 D∞ 1− √ h0D∞ c = πq0 G (ξ) q0 4 σρD∞ 1− h0D∞ 2 15 ρ = q0 exp (−ξ ) (E5) (R2), (R17) lσ 1+ γk(1−) exp (ξ2) h 2q0σ i c = kl 1 + γk(1−) exp (ξ2) ξ2 exp (ξ2) σq0 2q0σ Restrictions on data σ kD q erf √ = √∞ 1 − 0 (R0) α q0 πα h0D∞ q 0 < 0 exp (−σ2/d) − 1 (R1) ρlσ q 0 < 1 − 0 (R2) h0D∞ q 0 < 0 − 1 (R3) ρlσ √ s ! q0 πq0 q0 1 − < g4 ln (R4) h0D∞ σρcD∞ ρlσ F4(η) > 1 (R5) where η is the only one positive solution to the equation: q 0 (1 − 2x2) = (1 − x2) exp(x2), (1) ρlσ √ √ q0 πq0 q0 πq0 1 − < G4(ζ1) o 1 − > G4(ζ2) (R6) h0D∞ σρcD∞ h0D∞ σρcD∞ where ζ1 and ζ2 are the only two positive solutions to the equation: F4(x) = 1, F4(η) = 1 (R7) where η is the only one positive solution to the equation (1). √ q0 πq0 1 − 6= G4(η) (R8) h0D∞ σρcD∞ where η is the only one positive solution to the equation (1). F4(η) < 1 (R9) where η is the only one positive solution to the equation (1). √ s ! √ s ! πq0 1 q0 πq0 q0 G4 ln < 1 − < G4 ln (R10) σρcD∞ ν4 h0D∞ σρcD∞ ρlσ where: v ρlσ q0 u 2γc ν4 = ln 1 + u1 + 2q0 ρlσ t q0 l ln ρlσ 2q q q 0 < 0 ln 0 0 − 1 − 1 (R11) ργcσ ρlσ ρlσ √ √ πq0σ q0 2q0σ πq0σ G5(ζ1) < 1 − < min , G5(ζ2) (R12) kD∞ h0D∞ kD∞ kD∞ where ζ1 and ζ2 are the only one positive solutions to the equations: q σc 0 exp (−x2) = x2 lk y: q σc γk 0 exp (−x2) = exp (x2) + 1 x2, lk 2q0σ respectively. √ s ! πq0σ q0 q0 G5 ln < 1 − (R13) kD∞ ρlσ h0D∞ q γk 0 ≥ + 1 (R14) ρlσ 2q0σ ( √ s !) q0 2q0σ πq0σ 1 1 − < min , G5 ln (R15) h0D∞ kD∞ kD∞ ν6 where: " s # ρlσ 2γk ν6 = 1 + 1 + 2 2q0 σ ρl q γk 1 < 0 < + 1 (R16) ρlσ 2q0σ q 2q σ 1 − 0 < 0 (R17) h0D∞ kD∞ kD∞ q0 G5(η) < √ 1 − (R18) πq0σ h0D∞ where η is the only one positive solution to the equation: exp(−x2) lk = , x q0cσ G14(η) > H14(η) (R19) where η is the only one positive solution to the equation: q 0 (1 − 2x2) = exp (x2). ρlσ Definition of functions (x > 0) 2q q F (x) = 0 0 exp (−x2) − 1 x2 exp (−x2) 4 γρcσ ρlσ G4(x) = x erf (x) 2q σ q cσ exp (−x2) F (x) = 0 0 − 1 exp (−x2) 5 γk lk x2 erf (x) G (x) = 5 x 2q σ q f (x) = 0 0 exp (−x2) exp (−x2) 6 γk ρlσ exp (−x2) G (x) = 13 erf (x) √ 2 πγ(1 − ) H13(x) = x + b13 exp (x ) erf (x), con b13 = 2D 1 − q0 ∞ h0D∞ q G (x) = 0 exp (−x2) − 1 x 14 ρlσ 2 H14(x) = erf (x) exp (x ) Equations for ξ σρcD∞ q0 G4(x) = √ 1 − x > 0 (E4) πq0 h0D∞ kD q G (x) = √ ∞ 1 − 0 x > 0 (E5) 5 πq σ h D 0 0 ∞ √ πl a13G13(x) = H13(x), with a13 = x > 0 (E13) cD 1 − q0 ∞ h0D∞ 2D∞ q0 a14G14(x) = H14(x), with a14 = √ 1 − x > 0 (E14) πγ(1 − ) h0D∞ 36 III. Phase-change problems for the fractional heat-diffusion equation Fractional derivatives o Caputo fractional derivative o Wright and Mainardi Functions o Basic properties Two-phase fractional Stefan problems: o Generalized Neumann solution with a temperature boundary condition o Generalized Neumann solution with a heat flux boundary condition (In progress) One-phase fractional Stefan problem: o Determination of one unknown thermal coefficient AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 37 III.1 Fractional derivatives Definition 1. Caputo fractional derivative (Caputo, Geophys. J. R. Astr. Soc. (1967)): t ' α 1 f (τ ) D f (t)= d τ for 0 < α < 1 Γ − α ∫ α (1 ) 0 (t − τ) (D1) =f' (t) for α = 1 where +∞ Γ(x) =∫ tx− 1 exp( − t) dt (Gamma function). (D2) 0 Definition 2. Wright function (Wright, J. London Math. Soc. (1933)): +∞ zn W(z;,)α β =∑ ,z, ∈ α > − 1, β∈ . (D3) Γ α + β n= 0 n! (n ) AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 38 Definition 3. Mainardi function (Mainardi, Fract. Calc. Appl. Anal. (1999)): +∞ (− z)n M(z)W(z;,1)= − −υ − υ =∑ ,z ∈ , υ < 1 (D4) υ Γ − υ + − υ n= 0 n! ( n 1 ) which is a particular case of the Wright function. Basic Properties: ∂W (z;,)α β = W(z;, α α + β ). (D5) ∂z 1 x 1 x W(− x; − ,1) = erfc , 1 − W( − x; − ,1) = erf . (D6) 2 2 2 2 α βΓ(1 + β ) β−α D( t) = t (D7) Γ(1 + β − α ) AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 39 III.2 Generalized Neumann solutions III.2.1. For Temperature boundary condition at the fixed face x= 0 (Roscani – T. (2014)). III.2.2. For Heat Flux boundary condition at the fixed face x= 0 (Roscani – T., In progress). AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 40 III.2.1) GENERALIZED NEUMANN SOLUTION (MELTING PROCESS) WITH A TEMPERATURE BOUNDARY CONDITION = = Problem (FP1) Find the free boundary x s() t , and the temperature T T(,) x t such that the following equations and conditions are satisfied (0< α < 1): D T− 2 T =0, x > s ( t ), t > 0, (FT1) s s sxx D T− 2 T =0, 0 < x < s ( t ), t > 0, (FT2) xx s(0)= 0, (FT3) = +∞ = < > > Ts(,0) x T s (,) t T i T f , x 0, t 0, (FT4) = > Ts( s ( t ), t ) T f , t 0, (FT5) = > Tl( s ( t ), t ) T f , t 0, (FT6) kTstt( ( ),) − kTstt( ( ),) = Dst ( ), t > 0, (FT7) s sx x = > > T (0, t ) T0 Tf , t 0, (FT8) 2 ks 2 k = , = . where s cs c AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 41 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 42 < < Theorem FT1 Let TTTi f 0 be. A solution of the problem (FP1) is the generalized Neumann explicit solution given by: − −x − 1W ; ,1 t 2 2 T( x , t )= T − ( T − T ) , =s > 0, (FT9) 0 0 f 1−W − ; − ,1 2 −x − W ; ,1 t 2 2 T(,)() x t= T + T − T s , (FT10) s i f i W − ; − ,1 2 = 2 s() t s t , (FT11) = > where the coefficient 0 is the solution of the following equation: AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 43 Γ(1 + ) F( x )=2 x , x > 0 (FT12) Γ( − ) 1 2 with k()() T− T k T − T = 0 f − s f i , (FT13) F()()() x F1 x2 F 2 x s s and M()() x M x F(),() x=2 F x = 2 , (FT14) 1 2 1−W − x ; − ,1 W − x ; − ,1 2 2 where W and M α are the Wright and Mainardi functions respectively. 2 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 44 III.F2) GENERALIZED NEUMANN SOLUTION (MELTING PROCESS) WITH A HEAT FLUX BOUNDARY CONDITION = = Problem (FP2 ) Find the free boundary x s() t , and the temperature T T(,) x t such that the following equations and conditions are satisfied (0< α < 1): D T− 2 T =0, x > s ( t ), t > 0, (FF1) s s sxx D T− 2 T =0, 0 < x < s ( t ), t > 0, (FF2) xx s(0)= 0, (FF3) = +∞ = < > > Ts(,0) x T s (,) t T i T f , x 0, t 0, (FF4) = > Ts( s ( t ), t ) T f , t 0, (FF5) = > Tl( s ( t ), t ) T f , t 0, (FF6) kTstt( ( ),) − kTstt( ( ),) = Dst ( ), t > 0, (FF7) s sx x q k T(0, t )= −0 , t > 0, (FF8) x t 2 2 ks 2 k = , = . where s cs c AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 45 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 46 NECESSARY CONDITION TO HAVE A FRACTIONAL TWO-PHASE PROBLEM (MELTING PROCESS) WITH A HEAT FLUX BOUNDARY CONDITION = Problem (FDEF2 ) Find the temperature T T(,) x t such that the following equation and conditions are satisfied (0< α < 1): −2 = > > D T Txx 0, x 0, t 0, (FDEF1) = +∞ = < > > T(,0)(,) x T t Ti T f , x 0, t 0, (FDEF2) = −q0 > kTx (0, t ) , t 0, (FDEF3) t 2 2 k where = . c AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 47 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 48 < Theorem FDEF1 Let TTi f be. a) The solution of the problem (FDEF1)- (FDEF3) is given by: q Γ(1 − ) = +0 2 −x − > > T( x , t ) Ti W ; ,1 , x 0, t 0. (FDEF4) k t 2 2 b) The temperature at the boundary x = 0 is given by: Γ − q0 (1 ) T(0, t )= T +2 = Const ., ∀ t > 0. (FDEF5) i k c) The necessary and sufficient condition in order to have an instantaneous fractional phase- change is that the coefficient q0 satisfies the inequality: k() T− T q > f i . (FDEF6) 0 Γ − (12 ) AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 49 < Theorem FF1 Let TTi f be. a) If the coefficient q0 satisfies the inequality: k() T− T > s f i q0 , (FF9) Γ(1 − ) s 2 then there exists an instantaneous phase-change (melting) process and the problem (FP2 ) has the generalized Neumann explicit solution given by: Γ − q0 (1 ) x = +2 − − − − − , (FF10) T ( x , t ) Tf W ; ,1 W F ; ,1 k 2 2 2 t −x − W ; ,1 t 2 2 T(,)() x t= T + T − T s , (FF11) s i f i W − ; − ,1 F 2 = 2 s() tF s t , (FF12) AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 50 = > where the coefficient F 0 is the solution of the following equation: Γ(1 + ) F( x )=2 x , x > 0 (FF13) F Γ( − ) 1 2 with q Γ(1 − ) k() T− T =0 2 − s f i . (FF14) FF ()()() x M x2 F2 x 2 s s b) If the coefficient q0 satisfies the inequalities: k() T− T < ≤ s f i 0 q0 , (FF15) Γ(1 − ) s 2 then the problem (FP2 ) is a fractional heat transfer problem for the initial solid phase whose solution is given by: Γ − q0s (1 ) x = +2 − − > > . (FF16) Ts( x , t ) T i W ; ,1 , x 0, t 0 k 2 2 s st AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 51 < Theorem FF2 Let TTi f be. If the coefficient q0 verifies the inequality (FF9) then the α → − solution of the problem (FP2 ) converges to the solution of the problem (P2 ) when 1 and α = we recover the solution and the inequality for the coefficient q0 obtained for 1 in [T - Quart. Appl. Math., 1981]. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 52 III.3 One-phase fractional Stefan problem III.3.1 Determination of one unknown thermal coefficient through a one-phase fractional Stefan problem with an over-specified condition on the fixed face x= 0 (T. Appl. Math., 2015) The method for determining unknown thermal coefficients through a one-phase fractional Lamé-Clapeyron-Stefan with an over-specified condition on the fixed face x = 0 is given by: Find the free boundary x= s() t , defined for t > 0, and the temperature T= T(,) x t , defined for x > 0 and t > 0, such that the following equation and conditions are satisfied: AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 53 =2 < < > D T Txx , 0 x s ( t ), t 0, (FC1) s(0)= 0, (FC2) = +∞ = > > T(,0) x T (,) t Tm , x 0, t 0, (FC3) = > T( s ( t ), t ) Tm , t 0 , (FC4) −( ) = > kTx s( t ), t D s ( t ), t 0, (FC5) = > > T(0, t ) T0 Tm , t 0, (FC6) ( ) = −q0 > kTx 0, t , t 0, (FC7) t 2 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 54 where: : density of mass, k: thermal conductivity, c: specific heat by unit of mass, : latent heat of fusion by unit of mass, k 2 = > 0: difussion coefficient, c > = TT0 ()m : temperature at the fixed face x 0. > = q0 0: coefficient that characterized the heat flux at the heat flux x 0, which must be simultaneous experimentally determined with the temperature T0 . The unknown thermal coefficient can be chosen among: k, , c and . AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 55 Theorem 1 The solution to problem (FC1)-(FC7) with 0< α < 1 and one unknown thermal coefficient is given by: s( t )= t 2 , > 0 , (FC8) TT− x = −0 m − − − , (FC9) T( x , t ) T0 1 W ; ,1 2 2 1−W − ; − ,1 t 2 where the coefficient > 0 and the unknown termal coefficient must satisfy the following system of equations: k( T− T ) 0 m =1 −W − ; − ,1 , (FC10) 2 q Γ1 − 0 2 c( T− T) Γ1 − 1 − W − ; − ,1 0 m 2= 2 . (FC11) Γ + M () 1 2 2 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 56 We have 4 cases: Case 1: determination of {,} k ; Case 2: determination of {,} ; Case 3: determination of {,} c ; Case 4: determination of {,} . We show only the result for case # 3, and the Table 1 with the results of the 4 cases. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 57 Theorem 2 (Case 3: Determination of thermal coefficient c) If data satisfy the condition: k( T− T ) Γ1 + 0 m 2 < 1, (FC12) q2 Γ1 − 0 2 then the solution to case 3 (problem (FC1)-(FC7) with 0< α < 1 and the thermal unknown coefficient c) is given by: Γ1 + 2 c= c = 1 − W − ; − ,1 , (FC13) − Γ − M( ) 2 (TT ) 1 2 0 m 2 = = − > where the coefficient ( ,k , , , q0 , T 0 Tm ) 0 is the unique solution of the equation: 1−W − x ; − ,1 k ( T − T ) Γ 1 + 2 0 m 2 1 =,x > 0. (FC14) x2Γ 3 − M () x q 1 2 0 2 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 58 = = Moreover, the temperature T(,)(,) x t T x t and the free boundary s()() t s t are given by the following expressions (0< α < 1): TT− x = = −0 m − − − , (FC15) T( x , t ) T ( x , t ) T0 1 W ; ,1 2 2 1−W − ; − ,1 t 2 = = 2 s()() t s t t , (FC16) = = − > where the coefficient ( ,k , , , q0 , T 0 Tm ) 0 is the unique solution of the equation 2= 2 (FC14) and the difusión coefficient is given by: ( −) Γ − k T0 Tm 1 k M() 2= 2 = = 2 2 . (24) c Γ1 + 1−W − ; − ,1 2 2 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 59 Theorem 3 If the parameter α → 1− then, under the hipothesis (FC12), the solution to case 3, coincides with the one given in [T., Adv. Appl. Math. (1982)], that is: 1 =2 > s1( t ) 2 1 1 t , 1 0 , (25) TT− x = − 0 m , (26) T1(,) x t T 0 erf 1 erf ( ) 2 1 21t q2 k k() T− T c=0 erf 2 (), = = 0 m , (27) 1− 2 1 1 ( ) k() T0 Tm c 1 q0 erf 1 > and the coefficient 1 0 is the unique solution of the following equation: q2 erf() x 2 =0 > . (28) exp(x )( − ) , x 0 k T0 Tm x In particular, the inequality (FC12) is transformed into the following one: k ( T− T ) 0 m < . (29) 2 1 2q0 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 60 Explicit formulae for the Case Parameter as the unique solution Restriction on unknown thermal # of the equation data coefficient 2 2 c( T− T ) Γ1 − q Γ1 − 2 0 m 0 =2 > =2 − − − F4 ( x ) , x 0 1 k2 1 W ; ,1 ------c( T− T ) 2 Γ1 + 0 m 2 ( −) Γ − c T0 Tm 1 ck( T− T ) 2 =0 m > = G3 ( x ) , x 0 2 Γ − R2 Γ + q0 1 1 F4 ( ) 2 2 Γ + 1−W − x ; − ,1 k ( T − T ) Γ 1 + 1 0 m R1 = 2 2= 2 1 > c F4 () ,x 0 3 (TT−) Γ1 − x2Γ 3 − M () x 0 m q 1 2 2 0 2 2 2 c( T− T ) Γ1 − q Γ1 − 2 0 m 0 =2 > =2 − − − F4 ( x ) , x 0 4 2 1W ; ,1 ------kc( T− T ) 2 Γ1 + 0 m 2 Table 1. Summary of the determination of one unknown thermal coefficient through a one- phase fractional Lamé-Clapeyron-Stefan with an over-specified condition on the fixed face (4 cases) AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 61 Restrictions on data: k( T− T ) Γ1 + 0 m 2 R1: < 1; q2 Γ1 − 0 2 ck( T− T ) R2: 0 m < 1 q Γ1 − 0 2 Real functions: x1− W − x ; − ,1 =2 > F4 ( x ) , x 0 M () x 2 = − − − > G ( x ) 1 W x ; ,1 , x 0 3 2 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 62 Case Explicit formulae for the Parameter as the unique solution Restriction on # unknown coefficient of the equation data q2 c( T− T ) k= 0 erf 2 () x x2 erf x=0 m x > 1 ( − )2 exp( ) ( ) , 0 ------c T0 Tm c exp(− 2 ) erf() x (TT− ) ck (TT− ) ck = q =0 m ,x > 0 0 m < 1 2 0 k x q0 q0 q2 erf() x k ( T− T ) k( T− T ) = 0 2 =0 m 2 > 0 m < c2 erf () exp(x ), x 0 1 3 k( T− T ) 2 2 q2 0 m x q0 0 q2 c( T− T ) = 0 erf 2 () xexp( x2 ) erf ( x )=0 m , x > 0 4 ( − )2 ------kc T0 Tm Tabla 2. Summary of the determination of one unknown thermal coefficient through a one- phase classical Lamé-Clapeyron-Stefan ( = 1) with an over-specified condition on the fixed face (4 cases). These results can be obtained by taking the limit α → 1− in Table 1 (see T., Adv. Appl. Math. (1982). AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 63 IV. Papers which are not reviewed • Blengino – Reginto – T., Applied Math. Modelling (2015): We propose a coupled water uptake-root growth model by using a moving boundary problem for which the root length is prescribed as a function of time. The solution is obtained by front-fixing and finite element methods. • Gonzalez – Reginato - T., J. Biological Systems (2015): We find a correct approximated solution using a polynomial of sixth degree for the free boundary problem corresponding to the diffuion of oxygen in a spherical medium with simultaneous absorption at a constant rate, and we show some mistakes in previously published solutions. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 64 Books on free boundary problems for the heat-diffusion equation V. Alexiades, A.D. Solomon, Mathematical modeling of melting and freezing processes, Hemisphere-Taylor & Francis, Washington, 1996. J.R. Cannon, The one-dimensional heat equation, Addison-Wesley, Menlo Park, California, 1984. H.S. Carslaw, C.J. Jaeger, Conduction of heat in solids, Clarendon Press, Oxford, 1959. J. Crank, Free and moving boundary problem, Clarendon Press, Oxford, 1984. C.M. Elliott, J.R. Ockendon, Weak and variational methods for moving boundary problems, Research Notes in Math. #59, Pitman, London, 1982. A. Fasano, Mathematical models of some diffusive processes with free boundary, MAT – Serie A, 11 (2005), 1- 128. S.C. Gupta, The classical Stefan problem. Basic concepts, modelling and analysis, Elsevier, Amsterdam, 2003. V.J. Lunardini, Heat transfer with freezing and thawing, Elsevier, London, 1991. L.I. Rubinstein, The Stefan problem, American Mathematical Society, Providence, 1971. D.A. Tarzia, A bibliography on moving-free boundary problems for heat diffusion equation. The Stefan problem, MAT - Serie A, 2 (2000), 1-297. A.B. Tayler, Mathematical models in applied mechanics, Clarendon Press, Oxford, 1986. H. Weber, Die partiellen Differential-Gleinchungen der Mathematischen Physik, nach Riemann's Vorlesungen, t. II, Braunwschweig, 1901. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 65 Articles with explicit solutions on free boundary problems for the heat-diffusion equation P. Boadbridge, Solution of a nonlinear absorption model of mixed saturated-unsaturated flow, Water Resources Research, 26 (1990), 2435-2443. C. Briozzo, M.F. Natale, D.A. Tarzia, Explicit solutions for a two-phase unidimensional Lamé-Clapeyron-Stefan problem with source terms in both phases, J. Math. Anal. Appl., 329 (2007), 145-162. A.C. Briozzo, D.A. Tarzia, Explicit solution of a free-boundary problem for a nonlinear absorption model of mixed saturated-unsaturated flow, Adv. Water Resources, 21 (1998), 713-721. A.N. Ceretani, D.A. Tarzia, Similarity solutions for thawing processes with a convective boundary condition, Rendiconti del’Istituto di Matematica dell’Università di Trieste, 46 (2014), 137-155. A.N. Ceretani, D.A. Tarzia, Determination of the one unknown thermal coefficient through a mushy zone model with a convective overspecified boundary condition, Mathematical Problems in Engineering, Vol. 2015 Art ID 637852 (2015), 1-8. A.N. Ceretani, D.A. Tarzia, Simultaneous determination of the two unknown thermal coefficients through a mushy zone model with an over-specified convective boundary condition, JP Journal of Heat and Mass Transfer, 13 No. 2 (2016), 277-301. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 66 R. Grzymkowsi, E. Hetmaniok, M. Pleszczynski, D. Slota, A certain analytical method used for solving the Stefan problem, Thermal Science, 17 (2013), 635-642. G. Lamé, B.P. Clapeyron, Memoire sur la solidification par refroidissement d'un globe liquide, Annales Chimie Physique, 47 (1831), 250-256. M.F. Natale, D.A. Tarzia, Explicit solutions to the two-phase Stefan problem for Storm's type materials, Journal of Physics A: Mathematical and General, 33 (2000), 395-404. N.N. Salva, D.A. Tarzia, Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions, J. Math. Anal. Appl., 379 (2011), 240-244. A.D. Solomon, D.G. Wilson, V. Alexiades, Explicit solutions to change problems, Quart. Appl. Math., 41 (1983), 237-243. J. Stefan, Über einge probleme der theorie der Wärmeleitung, Zitzungberichte der Kaiserlichen Akademie der Wissemschaften Mathematisch-Naturwissemschafthiche classe, 98 (1889), 473-484. D.A. Tarzia, An inequality for the coefficient of the free boundary s( t )= 2 t of the Neumann solution for the two-phase Stefan problem, Quart. Appl. Math., 39 (1981), 491-497. D.A. Tarzia, An explicit solution for a two-phase unidimensional Stefan problem with a convective boundary condition at the fixed face, MAT – Serie A, 8 (2004), 21-27. D.A. Tarzia, Explicit and approximated solutions for heat and mass transfer problems with a moving interface, Chapter 20, in Advanced Topics in Mass Transfer, InTech Open Access Publisher, Rijeka, 2011, 439-484. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 67 D.A. Tarzia, Explicit solutions for the Solomon-Wilson-Alexiades’s mushy zone model with convective or heat flux boundary conditions, J. Appl. Math., 2015 Art ID 375930 (2015), 1-9. D.A. Tarzia, Determination of the one unknown thermal coefficient through the one-phase fractional Lamé- Clapeyron-Stefan problem, Appl. Math., 6 (2015), 2182-2191. D.A. Tarzia, elationship between Neumann solutions for two-phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions, Thermal Science, In Press (2016). See arXiv:1406.0552 V.R. Voller, F. Falcini, Two exact solutions of a Stefan problem with varying diffusivity, Int. J. Heat Mass Transfer, 58 (2013), 80-85. V.R. Voller, J.B. Swenson, C. Paola, An analytical solution for a Stefan problem with variable latent heat, Int. J. Heat Mass Transfer, 47 (2004), 5387-5390. S.M. Zubair, M.A. Chaudhry, Exact solutions of solid-liquid phase-change heat transfer when subjected to convective boundary conditions, Heat Mass Transfer, 30 (1994), 77-81. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 68 Books on fractional derivatives Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, 2006. F. Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, 2010. Podlubny, Fractional Differential Equations, Vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, 1999. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 69 Articles on fractional derivatives M. Caputo, Linear model of dissipation whose Q is almost frequency independent - II, Geophys. J. R. Astr. Soc., 13 (1967), 529-539. R. Gorenflo, Y. Luchko F. Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal., 2 (1999), 383-414. Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Computer and Mathematics with Applications, 59 (2010), 1766-1772. F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192. F. Mainardi, A. Mura, G. Pagnini, The M-Wright function in time-fractional diffusion processes: a tutorial survey, International Journal of Differential Equations, Vol. 2010, Article ID 104505, 1-29. E.M. Wright, On the coefficients of power series having exponential singularities, J. London Math. Soc., 8 (1933), 71-79. E.M. Wright, The asymptotic expansion of the generalized Bessel function, J. London Math. Soc., 10 (1935), 287- 293. E.M. Wright, The generalized Bessel function of order greater than one, Quart. J. Math., 11 (1940), 36-48. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 70 Articles with fractional derivatives on the Lamé-Clapeyron-Stefan problem C. Atkinson, Moving boundary problems for time fractional and composition dependent diffusion, Fract. Calc. Appl. Anal., 15 (2012), 207-221. F. Falcini, R. Garra, V.R. Voller, Fractional Stefan problems exhibing lumped and distributed latent-heat memory effects, Physical Review E, 87 (2013), 042401, 1-6. L. Jinyi, X. Mingyu, Some exact solutions to Stefan problems with fractional differential equations, J. Math. Anal. Appl., 351 (2009), 536-542. L.P. Kholpanov, Z.E. Zaklev, V.A. Fedotov, Neumann-Lamé-Clapeyron-Stefan Problem and its solution using Fractional Differential-Integral Calculus, Theoretical Fundations of Chemical Engineering, 37 (2003), 113-121. S. Roscani, E.A. Santillan Marcus, Two equivalent Stefan's problems for the time-fractional diffusion equation, Fract. Calc. Appl. Anal., 16 (2013), 802-815. S. Roscani, E.A. Santillan Marcus, A new equivalence of Stefan's problems for the time-fractional diffusion equation, Fract. Calc. Appl. Anal., 17 (2014), 371-381. S. Roscani, D.A. Tarzia, A generalized Neumann solution for the two-phase fractional Lamé-Clapeyron-Stefan problem, Adv. Math. Sci. Appl. 24 (2014), 237-249. D.A. Tarzia, Determination of the one unknown thermal coefficient through the one-phase fractional Lamé- Clapeyron-Stefan problem, App. Math.s, 6 (2015), 2182-2191. V.R. Voller, An exact solution of a limit case Stefan problem governed by a fractional diffusion equation, Int. J. Heat and Mass Transfer, 53 (2010), 5622-5625. V.R. Voller, Fractional Stefan problems, Int. J. Heat and Mass Transfer, 74 (2014), 269-277. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 71 OUR RECENT PUBLICATIONS ON JOURNALS J. L. Blengino Albrieu, J.C. Reginato, D.A. Tarzia, Modeling water uptake by a root system growing in a fixed soil volume, Applied Mathematical Modelling, 39 (2015), 3434-3447. A.C. Briozzo, D.A. Tarzia, Convergence of the solution of the one-phase Stefan problem with respect two parameters, MAT – Serie A, 20 (2015), 31-38. A.N. Ceretani, D.A. Tarzia, Similarity solutions for thawing processes with a convective boundary condition, Rendiconti del’Istituto di Matematica dell’Università di Trieste, 46 (2014), 137-155. A.N. Ceretani, D.A. Tarzia, L.T. Villa, Explicit solutions for a non-classical heat conduction problem for a semi- infinite strip with a non-uniform heat source, Boundary Value Problems, 2015 No. 156 (2015), 1-26. A.N. Ceretani, D.A. Tarzia, Determination of the one unknown thermal coefficient through a mushy zone model with a convective overspecified boundary condition, Mathematical Problems in Engineering, Vol. 2015 Art ID 637852 (2015), 1-8. A.N. Ceretani, D.A. Tarzia, Simultaneous determination of the two unknown thermal coefficients through a mushy zone model with an over-specified convective boundary condition, JP Journal of Heat and Mass Transfer, 13 No. 2 (2016), 277-301. A.M. Gonzalez, J.C. Reginato, D.A. Tarzia, A free boundary problem for oxygen diffusion in a sphere, Journal of Biological Systems, 23 Supp 01 (2015), S67-S76. S.D. Roscani, D.A. Tarzia, A generalized Neumann solution for the two-phase fractional Lamé-Clapeyron-Stefan problem, Advances in Mathematical Sciences and Applications, 24 No. 2 (2014), 237-249. D.A. Tarzia, Explicit solutions for the Solomon-Wilson-Alexiades’s mushy zone model with convective or heat flux boundary conditions, Journal of Applied Mathematics, 2015 Art ID 375930 (2015), 1-9. D.A. Tarzia, Determination of the one unknown thermal coefficient through the one-phase fractional Lamé- Clapeyron-Stefan problem, Applied Mathematics, 6 (2015), 2182-2191. D.A. Tarzia, Relationship between Neumann solutions for two-phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions, Thermal Science, In Press (2016). See arXiv:1406.0552 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 72 WORK IN PROGRESS 1) J. BOLLATI – A.N. CERETANI – S.D. ROSCANI - D.A. TARZIA, “The two-phase fractional Lamé-Clapeyron- Stefan problem with a convective boundary condition”. 2) J. BOLLATI – D.A. TARZIA, “Explicit solution for the one-phase Stefan problem with latent heat depending on the position and a convective boundary condition by using Kummer functions”. 3) A.N. CERETANI – N. SALVA - D.A. TARZIA, “The one phase Stefan problem with a temperature-depended thermal conductivity and a convective boundary condition”. 4) A.N. CERETANI – D.A. TARZIA, "Simultaneous determination of the two unknown thermal coefficients through the one-phase fractional Lamé-Clapeyron-Stefan problem”. 5) A.N. CERETANI – D.A. TARZIA, "Similarity solution for a two-phase Stefan problem with a convective boundary condition and a mushy model”. 6) J.C. REGINATO – J. L. BLENGINO ALBRIEU – D.A. TARZIA, "Analysis and use of cumulative nutrient uptake formulas in plant nutrition and the temporal weight averaged influx”. 7) S.D. ROSCANI - D.A. TARZIA, “The two-phase fractional Lamé-Clapeyron-Stefan problem with a heat flux boundary condition”. 8) S.D. ROSCANI - D.A. TARZIA, “Integral relation between temperature and the free boundary in the one-phase fractional Lamé-Clapeyron-Stefan problem with a temperature boundary condition”. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 73 Aknowledgements: The present work has been partially sponsored by the Project PIP No 0534 from CONICET – Univ. Austral, Rosario, Argentina, and Grant AFOSR-SOARD FA9550-14-1- 0122. Collaborators: • Dr. Andrea Ceretani (CONICET & Univ. Austral), Rosario, Argentina. Postdoc Fellowship; • Dr. Juan C. Reginato (Univ. Nac. Río Cuarto), Río Cuarto, Argentina; • Dr. Sabrina Roscani (CONICET & Univ. Austral), Rosario, Argentina. Postdoc Fellowship; • Julieta Bollati (CONICET & Univ. Austral), Rosario, Argentina. Doc Fellowship, PhD. Student (2nd year). THANK YOU VERY MUCH FOR YOUR ATTENTION AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl.