Appendix A Experimental approximation
Consider a vertical slot that is filled with an incompressible fluid, the upper and lower parts of which are maintained at temperatures T2 and T1>T2, respectively, by top and bottom heaters. The two heaters are separated by thin thermally insulated barriers and the two fluid masses are separated by a retractable thermally insulated membrane at z=Z0. The mixing of the cold and warm fluids is achieved by gently pulling the membrane. We consider a membrane of thickness 2ϵ˜ and thermal conductance Km that is centrally located and thermal barriers of thickness δ˜−ϵ˜ and thermal conductance Kb. We have TˆB=T1 for z>δ˜ and TˆB=T2 for z<δ˜. The temperatures Tu and Tb, of the fluids within the upper and lower thermal barrier regions, respectively, and the temperature Tm of the membrane are given by
Tu(z)=BIT2−T12(ϵ˜−BI(ϵ˜−δ˜))(z−δ˜)+T2,TB(z)=BIT2−T12(ϵ˜−BI(ϵ˜−δ˜))(z+δ˜)+T1,Tm(z)=T2−T12(ϵ˜−BI(ϵ˜−δ˜))z+T1+T22,
where BI=Km/KL is the Biot number, KL being the thermal conductance of the fluid. While our theoretical analysis pertains to the limiting case δ˜→0 and BI→0, our results will be well approximated by experiments having ϵ˜<δ˜≪1 and BI≪1. A plot of these temperature profiles resembles that of tanh(z/e), where the thinness of the membrane ≈e.
Appendix B Two-dimensional channel of finite aspect ratio
The matrix entries for the homogeneous system DˆU=0 are given by the expressions dij for 1≤i≤5, 1≤j≤5, with the exception of Dˆ61=d61+Rq61, Dˆ62=d62+Rq62 and Dˆ63=d63+Rq63; U=[B1B2B3C1C2C3]. We can express the determinant of (Dˆ) as det(Dˆ)=det(D1)+Rdet(D2), where the entries of D1 and D2 are given by the expressions dij with the exception of the last row of D2 being [q61q62q63000] and l=π, i. e.,
d11=−l(Z0+λ)cosh(l(Z0+λ))+sinh(l(Z0+λ)),d12=(Z0+λ)sinh(l(Z0+λ)),d13=(Z0+λ)2sinh(l(Z0+λ)),d14=l(Z0−λ)cosh(l(Z0−λ))−sinh(l(Z0−λ)),d15=−(Z0−λ)sinh(l(Z0−λ)),d16=−(Z0−λ)2sinh(l(Z0−λ)),d21=−(l)2(Z0+λ)sinh(l(Z0+λ)),d22=(Z0+λ)lcosh(l(Z0+λ))+sinh(l(Z0+λ)),d23=(Z0+λ)2lcosh(l(Z0+λ))+2(Z0+λ)sinh(l(Z0+λ)),d24=l2(Z0−λ)sinh(l(Z0−λ)),d25=−(Z0−λ)lcosh(l(Z0−λ))−sinh(l(Z0−λ)),d26=−(Z0−λ)2lcosh(l(Z0−λ))−2(Z0−λ)sinh(l(Z0−λ)),d31=−(Z0+λ)l3cosh(l(Z0+λ))−(Z0+λ)2sinh(l(Z0+λ)),d32=2lcosh(l(Z0+λ))+l2(Z0+λ)sinh(l(Z0+λ)),d33=(2+l2(Z0+λ)2)sinh(l(Z0+λ))+4l(Z0+λ)cosh(l(Z0+λ)),d34=(Z0−λ)l3cosh(l(Z0−λ))+(Z0−λ)2sinh(l(Z0−λ)),d35=2lcosh(l(Z0−λ))+l2(Z0−λ)sinh(l(Z0−λ)),d36=−(2+l2(Z0−λ)2)sinh(l(Z0−λ))−4(Z0−λ)lcosh(l(Z0−λ)),d41=−2l3cosh(l(Z0+λ))−l4(Z0+λ)sinh(l(Z0+λ)),d42=l3(Z0+λ)cosh(l(Z0+λ))+3l2sinh(l(Z0+λ)),d43=(6l+l3(Z0+λ)2)cosh(l(Z0+λ))+6(Z0+λ)l2sinh(l(Z0+λ)),d44=2l3cosh(l(Z0−λ))+l4(Z0−λ)sinh(l(Z0−λ)),d45=−l3(Z0−λ)cosh(l(Z0−λ))−3l2sinh(l(Z0−λ)),d46=−(6l+l3(Z0−λ)2)cosh(l(Z0−λ))−6(Z0−λ)l2sinh(l(Z0−λ)),d51=−l5(Z0+λ)cosh(l(Z0+λ))−3l4sinh(l(Z0+λ)),d52=4l3cosh(l(Z0+λ))+l4(Z0+λ)sinh(l(Z0+λ)),d53=8l3(Z0+λ)cosh(l(Z0+λ))+12l2sinh(l(Z0+λ))+l4(Z0+λ)sinh(l(Z0+λ)),d54=l5(Z0−λ)cosh(l(Z0−λ))+3l4sinh(l(Z0−λ)),d55=4l3cosh(l(Z0−λ))+l4(Z0−λ)sinh(l(Z0−λ)),d56=−8l3(Z0−λ)cosh(l(Z0−λ))−12l2sinh(l(Z0−λ))−l4(Z0−λ)sinh(l(Z0−λ)),d61=4l4(Z0+λ)cosh(l(Z0+λ))+l6(Z0+λ)sinh(l(Z0+λ)),d62=−(Z0+λ)l5cosh(l(Z0+λ))−5l4sinh(l(Z0+λ)),d63=−(20l3+(Z0+λ)2l5)cosh(l(Z0+λ))−10l4(Z0+λ)sinh(l(Z0+λ)),d64=−4l5cosh(l(Z0−λ))−l6(Z0−λ)sinh(l(Z0−λ)),d65=(Z0−λ)l5cosh(l(Z0−λ))+5l4sinh(l(Z0−λ)),d66=(20l3+(Z0−λ)2l5)cosh(l(Z0−λ))+10l4(Z0−λ)sinh(l(Z0−λ)),q61=−(Z0+λ)l3cosh(l(Z0+λ))+l2sinh(l(Z0+λ)),q62=l2(Z0+λ)sinh(l(Z0+λ)),q63=l2(Z0+λ)2sinh(l(Z0+λ)).
The constants Bis and Cis, i=1,2,3, that appear in eqs. (9)–(10) are obtained by setting B1=1 and solving the resulting non-homogeneous linear system of reduced rank AX=B, where the elements of the matrix A are dij, 1≤i≤6, 2≤j≤6, X=[B2B3C1C2c3]T and B=[−d11−d21−d31−d41−d51−d61]T.
Appendix C Open two-dimensional channel
For the case of an unbounded two-dimensional channel, U=[B1ˆB2ˆB3ˆC1ˆC2ˆC3ˆ] and the entries of the matrices D1 and D2 are given by
d11=eπZ0,d12=Z0eπZ0,d13=Z02eπZ0,d14=e−πZ0,d15=Z0e−πZ0,d16=Z02e−πZ0,d21=leπZ0,d22=lZ0eπZ0+eπZ0,d23=2Z0eπZ0+leπZ0Z02,d24=−le−πZ0,d25=−Z0le−πZ0+e−πZ0,d26=2Z0e−πZ0−le−πZ0Z02,d31=l2eπZ0,d32=2leπZ0+l2Z0eπZ0,d33=4lZ0eπZ0+2eπZ0+l2Z02eπZ0,d34=l2e−πZ0,d35=−2le−πZ0+l2Z0e−πZ0,d36=−4lZ0e−πZ0+2e−πZ0+l2Z02e−πZ0,d41=l3eπZ0,d42=l3Z0eπZ0+3l2eπZ0,d43=6leπZ0+l3Z02eπZ0+6l2Z0eπZ0,d44=−l3e−πZ0,d45=−l3Z0e−πZ0+3l2e−πZ0,d46=−l3Z02e−πZ0−6le−πZ0+6Z0l2e−πZ0,d51=l4eπZ0,d52=l4Z0eπZ0+4l3eπZ0,d53=12l2eπZ0+l4Z02eπZ0+8l3Z0eπZ0,d54=l4e−πZ0,d55=l4Z0e−πZ0−4l3e−πZ0,d56=l4Z02e−πZ0+12l2e−πZ0−8l3Z0e−πZ0,d61=l5eπZ0,d62=l5Z0eπZ0+5l4eπZ0,d63=20l3eπZ0+l5Z02eπZ0+10l4Z0eπZ0,d64=−l5e−πZ0,d65=−l5Z0e−πZ0+l45e−πZ0,d66=−l5(Z02)e−πZ0−20l3e−πZ0+10l4Z0e−πZ0,q61=l2eπZ0,q62=l2Z0eπZ0,q63=l2Z02eπZ0.
Upon setting Bˆ1=1, we obtain the rest of the constants in the definitions of F+ and F− by solving the reduced-rank non-homogeneous system as shown in Appendix A for the finite aspect ratio channel.
Appendix D Weakly non-linear analysis
This is the case of a non-linear and open two-dimensional channel, U=[B1˜B2˜B3˜C1˜C2˜C3˜], and the entries to matrices D1 and D2 are given by
d11=e2πZ0,d12=Z0e2πZ0,d13=Z02e2πZ0,d14=e−2πZ0,d15=Z0e−2πZ0,d16=Z02e−2πZ0,d21=(2π)e2πZ0,d22=(2π)Z0e2πZ0+e2πZ0,d23=2Z0e2πZ0+ae2πZ0Z02,d24=−(2π)e−2πZ0,d25=−Z0(2π)e−2πZ0+e−2πZ0,d26=2Z0e−2πZ0−(2π)e−2πZ0Z02,d31=(2π)2e2πZ0,d32=2(2π)e2πZ0+(2π)2Z0e2πZ0,d33=4(2π)Z0e2πZ0+2e2πZ0+(2π)2Z02e2πZ0,d34=(2π)2e−2πZ0,d35=−2(2π)e−2πZ0+(2π)2Z0e−2πZ0,d36=−4(2π)Z0e−2πZ0+2e−2πZ0+(2π)2Z02e−2πZ0,d41=(2π)3e2πZ0,d42=(2π)3Z0e2πZ0+3(2π)2e2πZ0,d43=6(2π)e2πZ0+(2π)3Z02e2πZ0+6(2π)2Z0e2πZ0,d44=−(2π)3e−2πZ0,d45=−(2π)3Z0e−2πZ0+3(2π)2e−2πZ0,d46=−(2π)3Z02e−2πZ0−6(2π)e−2πZ0+6Z0(2π)2e−2πZ0,d51=(2π)4e2πZ0,d52=(2π)4Z0e2πZ0+4(2π)3e2πZ0,d53=12(2π)2e2πZ0+(2π)4(Z02)e2πZ0+8(2π)3Z0e2πZ0,d54=(2π)4e−2πZ0,d55=(2π)4Z0e−nπZ0−4(2π)3e−2πZ0,d56=(2π)4Z02e−2πZ0+12(2π)2e−2πZ0−8(2π)3Z0e−2πZ0,d61=(2π)5e2πZ0,d62=(2π)5Z0e2πZ0+5(2π)4e2πZ0,d63=20(2π)3e2πZ0+(2π)5Z02e2πZ0+10(2π)4Z0e2πZ0,d64=−(2π)5e−2πZ0,d65=−(2π)5Z0e−2πZ0+(2π)45e−2πZ0,d66=−(2π)5(Z02)e−2πZ0−20(2π)3e−2πZ0+10(2π)4Z0e−2πZ0.q61=an2e2πZ0,q62=an2Z0e2πZ0,q63=an2Z02e2πZ0.
As before we set B˜=1 and solve the linear system A[B˜2B˜3C˜1C˜2C˜3]T=B.
The elements of vector H are denoted as follows:
H1=e−2πZ0(C˜4Z03+C˜5Z04)−e2πZ0(C˜4Z03+C˜5Z04),H2=e−2πZ0(3C˜4Z02+4C˜5Z03−2πC˜4Z03−2πC˜5Z04)−e2πZ0(3B˜4Z02+4B˜5Z03+2πB˜4Z03+2πB˜5Z04),H3=e−2πZ0(6C˜4Z0+12C˜5Z02−4π(3C˜4Z02+4C˜5Z03)+4π2(C˜4Z03+C˜5Z04))−e2πZ0(6B˜4Z0+12B˜5Z02+4π(3B˜4Z02+4B˜5Z03)+4π2(B˜4Z03+B˜5Z04)),H4=e−2πZ0(6C˜4+24C˜5Z0−6π(6C˜4Z0+12C˜5Z02)+12π2(3C˜4Z02+4C˜5Z03)−8π3(C˜4Z03+C˜5Z04))−e2πZ0(6B˜4+24B˜5Z0+6π(6B˜4Z0+12B˜5Z02)+12π2(3B˜4Z02+4B˜5Z03)+8π3(B˜4Z03+B˜5Z04)),H5=e−2πZ0(24C˜5−8π(6C˜4+24C˜5Z0)+24π2(6C˜4Z0+12C˜5Z02)−32π3(3C˜4Z02+4C˜5Z03)+16π4(C˜4Z03+C˜5Z04))−e2πZ0(24B˜5+8π(6B˜4+24B˜5Z0)+24π2(6B˜4Z0+12B˜5Z02)+32π3(3B˜4Z02+4B˜5Z03)+16π4(B˜4Z03+B˜5Z04)),H6=−e−2πZ0(−240πC˜5+40π2(6C˜4+24C˜5Z0)−80π3(6C˜4Z0+12C˜5Z02)+80π4(3C˜4Z02+4C˜5Z03)−32π5(C˜4Z03+C˜5Z04)+an2RcC˜4Z03+an2RcC˜5Z04)+e2πZ0(240πB˜5+40π2(6B˜4+24B˜5Z0)+80π3(6B˜4Z0+12B˜5Z02)+80π4(3B˜4Z02+4B˜5Z03)+32π5(B˜4Z03+B˜5Z04)),B˜4=(3Bˆ3−2Bˆ2π)(16Bˆ3π+e−πZ0PrΦ1−(Z0))1536π,B˜5=−Bˆ3(16Bˆ3π+e−πZ0PrΦ1−(Z0))1536,C˜4=(3Cˆ3+2Cˆ2π)(16Cˆ3π+eπZ0PrΦ1+(Z0))1536π,C˜5=Cˆ3(16Cˆ3π+eπZ0PrΦ1+(Z0))1536.
Upon application of the continuity conditions at the interface, we obtain the system DˆU=H, where U=[B˜1,B˜2,B˜3,C˜1,C˜2,C˜3]T. The elements of the matrix Dˆ are defined in Appendix C. The expressions for Θ(2)− and Θ(2)+ are then given by
Θ(2)−=[8e2iπxe2πz[3B˜5(1+8πz+8π2z2)+2π(2B˜3π+B˜4(3+6πz))]+2e2πzπ2[Bˆ22π+Bˆ2(Bˆ3+2Bˆ3πz)+2Bˆ3(−π+Bˆ3z(1+πz))]]sin(2πx),Θ(2)+=[8e2iπxe−2πz[3C˜5(1−8πz+8π2z2)+2π(2πC˜3+C˜4(−3+6πz))]+2e−2πzπ2[−Cˆ22π+Cˆ2(Cˆ3−2Cˆ3πz)+2Cˆ3(Cˆ1π+Cˆ3z(1−πz))]]sin(2πx).
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