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ev.f90
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!3d two fluid newtonian pipe flow solver!
MODULE ev
USE prms
USE data
USE cheb
IMPLICIT none
INTEGER NOUT
PARAMETER (NOUT=6)
INTEGER NB, NMAX, lt
PARAMETER (NB=4000,NMAX=4000)
INTEGER LDA, LDB, LDVR, LWORK
PARAMETER (LDA=NMAX,LDB=NMAX,LDVR=NMAX,LWORK=NMAX+NMAX*NB)
COMPLEX *16 ii
PARAMETER (ii=(0,1.0))
!FLOW PARAMETERS
Real AA1,AA2
integer iter,iterMax,itermin
CONTAINS
SUBROUTINE evsolve
INTEGER INFO,LWKOPT, N, U, W
COMPLEX *16 A(LDA,NMAX), ALPHA(NMAX), B(LDB,NMAX),BETA(NMAX), DUMMY(1,1), VR(LDVR,NMAX),WORK(LWORK)
DOUBLE PRECISION RWORK(8*NMAX), SMALL, DLAMCH
integer :: i,j
character(30) :: fn
Real, allocatable, dimension(:,:,:) :: xDddr,x2Dddr,xDddt,x2Dddt,x3Dddr,x3Dddt,x4Dddr,x4Dddt
!stop if the number of collocation points in each domain is not the same
If (Ndt .ne. Ndr) Stop
allocate(xDddr(0:Ndr,0:Ndr,0:3))
allocate(x2Dddr(0:Ndr,0:Ndr,0:3))
allocate(x3Dddr(0:Ndr,0:Ndr,0:3))
allocate(x4Dddr(0:Ndr,0:Ndr,0:3))
allocate(xDddt(0:Ndr,0:Ndr,0:3))
allocate(x2Dddt(0:Ndr,0:Ndr,0:3))
allocate(x3Dddt(0:Ndr,0:Ndr,0:3))
allocate(x4Dddt(0:Ndr,0:Ndr,0:3))
lt=1
N=6*(Ndr+1)
iterMin=1
iterMax=100
!fix the sign error for odd derivatives
Dddr(:,:,1) = -1*Dddr(:,:,1)
Dddr(:,:,3) = -1*Dddr(:,:,3)
Dddt(:,:,1) = -1*Dddt(:,:,1)
Dddt(:,:,3) = -1*Dddt(:,:,3)
!make the xDddr matrix that premultiplies Dddr(:) by XXdr
Do i=0,3
xDddr(:,:,i) = Matmul(XXdr(:,:),Dddr(:,:,i))
xDddt(:,:,i) = Matmul(XXdt(:,:),Dddt(:,:,i))
End Do
!make the x2Dddr matrix that premultiplies Dddr(:) by x^2 vector matrix
Do i=0,3
x2Dddr(:,:,i) = Matmul(XX2dr(:,:),Dddr(:,:,i))
x2Dddt(:,:,i) = Matmul(XX2dt(:,:),Dddt(:,:,i))
End Do
Do i=0,3
x3Dddr(:,:,i) = Matmul(XX3dr(:,:),Dddr(:,:,i))
x3Dddt(:,:,i) = Matmul(XX3dt(:,:),Dddt(:,:,i))
End Do
Do i=0,3
x4Dddr(:,:,i) = Matmul(XX4dr(:,:),Dddr(:,:,i))
x4Dddt(:,:,i) = Matmul(XX4dt(:,:),Dddt(:,:,i))
End Do
!write the xDddr matrix
open(unit=13, file='graphDddr0.txt', ACTION="write", STATUS="replace")
do i=0,Ndr
write(13, '(1600F9.2)')( Real(xDddr(i,j,0)) ,j=0,Ndr)
end do
close(13)
!write the x2Dddr matrix
open(unit=157, file='graphDddt0.txt', ACTION="write", STATUS="replace")
do i=0,Ndr
write(157, '(1600F9.2)')( Real(xDddt(i,j,0)) ,j=0,Ndr)
end do
close(157)
Do iter=itermin,itermax,1
MU1 = MU2 * 8 * Real(iter)/Real(itermax) !joseph
print*, 'Mu1/Mu2 =', Mu1/Mu2
!print*, 'R1/R2 =', aa/HH
AA1 = (kap/4)*( (aa**2)*((1/mu1)-1/mu2) - ((HH**2)*(1/mu2)) )
AA2 = (-kap/4)*(HH**2)/mu2
call buildA(A,xDddr,x2Dddr,xDddt,x2Dddt,x3Dddr,x3Dddt,x4Dddr,x4Dddt)
call buildB(B,xDddr,x2Dddr,xDddt,x2Dddt,x3Dddr,x3Dddt,x4Dddr,x4Dddt)
!lapack gevp solver
IF (N.LE.NMAX) THEN
CALL ZGGEV('No left vectors','Vectors (right)',N,A,LDA,B,LDB,&
& ALPHA,BETA,DUMMY,1,VR,LDVR,WORK,LWORK,RWORK,INFO)
!write eigenvalues to files in directory D
Do J=1,N
write(fn,"('D/eval',I9.9)")iter
!write(fn,"('D/eval',I9.9)")Int(10000.*aa/HH)
open(1,file=fn)
If (Real(beta(j))>0.001) Then
write(1,"(3F30.20)") ALPHA(J)/BETA(J)
End if
End Do
close(1)
!write eigenvectors to files in directory V
Do lt=1,N
Do I=1,N
!If (Real(Beta(lt))>0.00001) Then
write(fn,"('V/x.',I9.9)")lt
open(2,file=fn)
write(2,"(3F30.20)") VR(I,lt)
!End if
End Do
End Do
close(2)
IF (INFO.GT.0) THEN
WRITE (NOUT,99999) 'Failure in ZGGEV. INFO =', INFO
ELSE
SMALL = DLAMCH('Sfmin')
DO 20 J = 1,N
IF ((ABS(ALPHA(J)))*SMALL.GE.ABS(BETA(J))) THEN
!WRITE (NOUT,99998) 'Eigenvalue(', J, ') is numerically infinite or undetermined'
ELSE
!print eigenvalues to the shell
!WRITE (NOUT,99997) 'Eigenvalue(', J, ') = ',ALPHA(J)/BETA(J)
END IF
!print eigenvectors to the shell
!WRITE (NOUT,*)
!WRITE (NOUT,99996) 'Eigenvector(', J, ')',&
!& (VR(I,J),I=1,N)
20 CONTINUE
LWKOPT = WORK(1)
IF (LWORK.LT.LWKOPT) THEN
WRITE (NOUT,99995) 'Optimum workspace required = ',&
& LWKOPT, 'Workspace provided = ', LWORK
END IF
END IF
ELSE
WRITE (NOUT,*) 'NMAX too small'
END IF
99999 FORMAT (1X,A,I4)
99998 FORMAT (1X,A,I3,2A,/1X,2(A,I3,A,'(',1P,E11.4,',',1P,E11.4,')'))
99997 FORMAT (1X,A,I2,A,'(',1P,E11.4,',',1P,E11.4,')')
99996 FORMAT (1X,A,I2,A,/3(1X,'(',1P,E11.4,',',1P,E11.4,')',:))
99995 FORMAT (1X,A,I5,/1X,A,I5)
End Do
END SUBROUTINE evsolve
SUBROUTINE buildA(A,xDddr,x2Dddr,xDddt,x2Dddt,x3Dddr,x3Dddt,x4Dddr,x4Dddt)
integer :: i,j
COMPLEX, allocatable, dimension(:,:) :: A11, A12, A13, A14, A15, A16, A21, A22, A23, A24, A25, A26, A31, A32, A33, A34, A35, A36
COMPLEX, allocatable, dimension(:,:) :: A41, A42, A43, A44, A45, A46, A51, A52, A53, A54, A55, A56, A61, A62, A63, A64, A65, A66
Real, allocatable, dimension(:,:,:) :: xDddr,x2Dddr,xDddt,x2Dddt,x3Dddr,x3Dddt,x4Dddr,x4Dddt
COMPLEX *16 :: A(LDA,NMAX)
!COMPLEX *16, allocatable, dimension(:,:) :: A
!allocate(A(0:LDA,0:NMAX))
allocate(A11(0:Ndr,0:Ndr))
allocate(A12(0:Ndr,0:Ndr))
allocate(A13(0:Ndr,0:Ndr))
allocate(A14(0:Ndr,0:Ndr))
allocate(A15(0:Ndr,0:Ndr))
allocate(A16(0:Ndr,0:Ndr))
allocate(A21(0:Ndr,0:Ndr))
allocate(A22(0:Ndr,0:Ndr))
allocate(A23(0:Ndr,0:Ndr))
allocate(A24(0:Ndr,0:Ndr))
allocate(A25(0:Ndr,0:Ndr))
allocate(A26(0:Ndr,0:Ndr))
allocate(A31(0:Ndr,0:Ndr))
allocate(A32(0:Ndr,0:Ndr))
allocate(A33(0:Ndr,0:Ndr))
allocate(A34(0:Ndr,0:Ndr))
allocate(A35(0:Ndr,0:Ndr))
allocate(A36(0:Ndr,0:Ndr))
allocate(A41(0:Ndr,0:Ndr))
allocate(A42(0:Ndr,0:Ndr))
allocate(A43(0:Ndr,0:Ndr))
allocate(A44(0:Ndr,0:Ndr))
allocate(A45(0:Ndr,0:Ndr))
allocate(A46(0:Ndr,0:Ndr))
allocate(A51(0:Ndr,0:Ndr))
allocate(A52(0:Ndr,0:Ndr))
allocate(A53(0:Ndr,0:Ndr))
allocate(A54(0:Ndr,0:Ndr))
allocate(A55(0:Ndr,0:Ndr))
allocate(A56(0:Ndr,0:Ndr))
allocate(A61(0:Ndr,0:Ndr))
allocate(A62(0:Ndr,0:Ndr))
allocate(A63(0:Ndr,0:Ndr))
allocate(A64(0:Ndr,0:Ndr))
allocate(A65(0:Ndr,0:Ndr))
allocate(A66(0:Ndr,0:Ndr))
!FLUID 1 EQUATIONS
!R-MOMENTUM
!A11
A11(:,:) = MU1 * x2Dddr(:,:,2) + MU1 * xDddr(:,:,1) - MU1*(K**2.)*x2Dddr(:,:,0) &
& - MU1*Dddr(:,:,0) - RHO1*ii*K*(KAP/(4.*MU1))*x4Dddr(:,:,0) - RHO1*ii*K*AA1*x2Dddr(:,:,0)
!BC for singularity condition on Ur at r=0
A11(0,:) = Dddr(0,:,0)
!BC for normal stress condition at r=aa
!!!! check the sign here !!!! this reflects page 15 of notes
A11(Ndr,:) = (-1)*Dddr(Ndr,:,0)*GAM*((K**2.)-1./(aa**2.)) -ii*K*(kap/(4.*MU2))*((HH**2.) - aa**2.) * (-2.*mu1*Dddr(Ndr,:,1))
!A12
A12(:,:) = 0
!BC for singularity on Ur
A12(0,:) = 0
!BC for stress at r=aa
A12(Ndr,:) = 0
!A13
A13(:,:) = (-1)*x2Dddr(:,:,1)
!BC singularity condition on Ur
A13(0,:) = 0
!BC for normal stress at r=aa
A13(Ndr,:) = -ii*K*(kap/(4.*MU2))*((HH**2) - aa**2) * Dddr(Ndr,:,0)
!A14
A14(:,:) = 0
!BC for normal stress at r=aa
A14(Ndr,:) = -ii*K*(kap/(4.*MU2))*((HH**2.) - aa**2.) * (2.*MU2*Dddt(0,:,1))
!A15
A15(:,:) = 0
!A16
A16(:,:) = 0
!BC for normal stress at r=aa
A16(Ndr,:) = -ii*K*(kap/(4.*MU2))*((HH**2.) - aa**2.) * (-1.*Dddt(0,:,0))
!X-MOMENTUM
!A21
A21(:,:) = -RHO1*(KAP/(MU1*2.))*x2Dddr(:,:,0)
!BC for singularity on Ux
A21(0,:) = 0
!tang stress BC at r=a
A21(Ndr,:) = (-1.)*MU1*ii*K*Dddr(Ndr,:,0)
!A22
A22(:,:) = MU1*xDddr(:,:,2) + MU1*xDddr(:,:,1) - K*MU1*xDddr(:,:,0) - &
& RHO1*ii*K*(KAP/(MU1*4.))*x3Dddr(:,:,0) - RHO1*ii*K*AA1*xDddr(:,:,0)
!BC singularity condition on Ux
A22(0,:) = Dddr(0,:,1)
!BC for tang stress at r=a
A22(Ndr,:) = (-1.)*MU1*Dddr(Ndr,:,1)
!A23
A23(:,:) = (-1.)*ii*K*xDddr(:,:,0)
!BC singularity condition on Ux
A23(0,:) = 0
!BC for tang stress at r=a
A23(Ndr,:) = 0
!A24
A24(:,:) = 0
!BC for tang stress at r=a
A24(Ndr,:) = MU2*ii*K*Dddt(0,:,0)
!A25
A25(:,:) = 0
!BC for tang stress at r=a
A25(Ndr,:) = MU2*Dddt(0,:,1)
!A26
A26(:,:) = 0
!CONTINUITY
!A31
A31(:,:) = Dddr(:,:,0) + xDddr(:,:,1)
!BC singularity condition on P
A31(0,:) = 0
!A32
A32(:,:) = ii*K*xDddr(:,:,0)
!BC singularity condition on P
A32(0,:) = 0
!A33
A33(:,:) = 0
!bc on singularity condition on P
A33(0,:) = Dddr(0,:,1)
!A34
A34(:,:) = 0
!A35
A35(:,:) = 0
!A36
A36(:,:) = 0
!FLUID 2 EQUATIONS
!R-MOMENTUM
!A41
A41(:,:) = 0
!bc for continuity at interface
A41(0,:) = Dddr(Ndr,:,0)
!A42
A42(:,:) = 0
!A43
A43(:,:) = 0
!A44
A44(:,:) = MU2*xDddt(:,:,1) + MU2*x2Dddt(:,:,2) - MU2*(K**2.)*x2Dddt(:,:,0) &
& - MU2*Dddt(:,:,0) - RHO2*ii*K*(KAP/(MU2*4.))*x4Dddt(:,:,0) - RHO2*ii*K*AA2*x2Dddt(:,:,0)
!bc for no penetration at r=H
A44(Ndr,:) = Dddt(Ndr,:,0)
!bc for continuity at interface
A44(0,:) = (-1.)*Dddt(0,:,0)
!A45
A45(:,:) = 0
!A46
A46(:,:) = (-1.)*x2Dddt(:,:,1)
!bc for no penetration at r=H
A46(Ndr,:) = 0
!bc for continuity at interface
A46(0,:) = 0
!X-MOMENTUM
!A51
A51(:,:) = 0
!bc for no slip at interface
A51(0,:) = Dddr(Ndr,:,0) * kap*aa*0.5*((1/mu1) - 1/mu2)
!A52
A52(:,:) = 0
!bc for no slip at interface
A52(0,:) = Dddr(Ndr,:,0) * ii*K* kap*0.25*(1/mu2)*((aa**2.) - HH**2.)
!A53
A53(:,:) = 0
!A54
A54(:,:) = -RHO2*(KAP/(2.*MU2))*x2Dddt(:,:,0)
!bc no slip at r=H
A54(Ndr,:) = 0
!bc no slip at interface
A54(0,:) = 0
!A55
A55(:,:) = MU2*xDddt(:,:,2) + MU2*Dddt(:,:,1) - (K**2.)*MU2*xDddt(:,:,0) - RHO2*ii*K*(KAP/(MU2*4.))*x3Dddt(:,:,0) &
& - RHO2*ii*K*AA2*xDddt(:,:,0)
!bc no slip at r=H
A55(Ndr,:) = Dddt(Ndr,:,0)
!bc no slip at interface
A55(0,:) = (-1.)*Dddt(0,:,0) * ii*K* kap*0.25*(1/mu2)*((aa**2) - HH**2)
!A56
A56(:,:) = -1.*ii*K*xDddt(:,:,0)
!bc for no slip at r=H
A56(Ndr,:) = 0
!bc no slip at interface
A56(0,:) = 0
!CONTINUITY
!A61
A61(:,:) = 0
!A62
A62(:,:) = 0
!A63
A63(:,:) = 0
!A64
A64(:,:) = Dddt(:,:,0) + xDddt(:,:,1)
!bc for wall pressure 2
A64(Ndr,:) = 0
!A65
A65(:,:) = ii*K*xDddt(:,:,0)
!bc for wall pressure 2
A65(Ndr,:) = 0
!A66
A66(:,:) = 0
!bc for wall pressure 2
A66(Ndr,:) = Dddt(Ndr,:,1)
!put together the LHS
!first row of blocks
Do i=0,Ndr
Do j=0,Ndr
A(i+1,j+1) = A11(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1,j+1+ 1*(Ndr+1)) = A12(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1,j+1+ 2*(Ndr+1)) = A13(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1,j+1+ 3*(Ndr+1)) = A14(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1,j+1+ 4*(Ndr+1)) = A15(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1,j+1+ 5*(Ndr+1)) = A16(i,j)
End Do
End Do
!second row of blocks
Do i=0,Ndr
Do j=0,Ndr
A(i+1+Ndr+1,j+1) = A21(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+Ndr+1,j+1+ 1*(Ndr+1)) = A22(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+Ndr+1,j+1+ 2*(Ndr+1)) = A23(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+Ndr+1,j+1+ 3*(Ndr+1)) = A24(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+Ndr+1,j+1+ 4*(Ndr+1)) = A25(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+Ndr+1,j+1+ 5*(Ndr+1)) = A26(i,j)
End Do
End Do
!third row of blocks
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 2*(Ndr+1),j+1) = A31(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 2*(Ndr+1),j+1+ 1*(Ndr+1)) = A32(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 2*(Ndr+1),j+1+ 2*(Ndr+1)) = A33(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 2*(Ndr+1),j+1+ 3*(Ndr+1)) = A34(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 2*(Ndr+1),j+1+ 4*(Ndr+1)) = A35(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 2*(Ndr+1),j+1+ 5*(Ndr+1)) = A36(i,j)
End Do
End Do
!fourth row of blocks
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 3*(Ndr+1),j+1) = A41(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 3*(Ndr+1),j+1+ 1*(Ndr+1)) = A42(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 3*(Ndr+1),j+1+ 2*(Ndr+1)) = A43(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 3*(Ndr+1),j+1+ 3*(Ndr+1)) = A44(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 3*(Ndr+1),j+1+ 4*(Ndr+1)) = A45(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 3*(Ndr+1),j+1+ 5*(Ndr+1)) = A46(i,j)
End Do
End Do
!fifth row of blocks
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 4*(Ndr+1),j+1) = A51(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 4*(Ndr+1),j+1+ 1*(Ndr+1)) = A52(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 4*(Ndr+1),j+1+ 2*(Ndr+1)) = A53(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 4*(Ndr+1),j+1+ 3*(Ndr+1)) = A54(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 4*(Ndr+1),j+1+ 4*(Ndr+1)) = A55(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 4*(Ndr+1),j+1+ 5*(Ndr+1)) = A56(i,j)
End Do
End Do
!sixth row of blocks
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 5*(Ndr+1),j+1) = A61(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 5*(Ndr+1),j+1+ 1*(Ndr+1)) = A62(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 5*(Ndr+1),j+1+ 2*(Ndr+1)) = A63(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 5*(Ndr+1),j+1+ 3*(Ndr+1)) = A64(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 5*(Ndr+1),j+1+ 4*(Ndr+1)) = A65(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
A(i+1+ 5*(Ndr+1),j+1+ 5*(Ndr+1)) = A66(i,j)
End Do
End Do
call matrixAPrint(A,A11,A12,A13,A21,A22,A23,A31,A32,A33)
END SUBROUTINE buildA
SUBROUTINE buildB(B,xDddr,x2Dddr,xDddt,x2Dddt,x3Dddr,x3Dddt,x4Dddr,x4Dddt)
integer :: i,j
COMPLEX, allocatable, dimension(:,:) :: B11, B12, B13, B14, B15, B16, B21, B22, B23, B24, B25, B26, B31, B32, B33, B34, B35, B36
COMPLEX, allocatable, dimension(:,:) :: B41, B42, B43, B44, B45, B46, B51, B52, B53, B54, B55, B56, B61, B62, B63, B64, B65, B66
Real, allocatable, dimension(:,:,:) :: xDddr,x2Dddr,xDddt,x2Dddt,x3Dddr,x3Dddt,x4Dddr,x4Dddt
COMPLEX *16 :: B(LDB,NMAX)
allocate(B11(0:Ndr,0:Ndr))
allocate(B12(0:Ndr,0:Ndr))
allocate(B13(0:Ndr,0:Ndr))
allocate(B14(0:Ndr,0:Ndr))
allocate(B15(0:Ndr,0:Ndr))
allocate(B16(0:Ndr,0:Ndr))
allocate(B21(0:Ndr,0:Ndr))
allocate(B22(0:Ndr,0:Ndr))
allocate(B23(0:Ndr,0:Ndr))
allocate(B24(0:Ndr,0:Ndr))
allocate(B25(0:Ndr,0:Ndr))
allocate(B26(0:Ndr,0:Ndr))
allocate(B31(0:Ndr,0:Ndr))
allocate(B32(0:Ndr,0:Ndr))
allocate(B33(0:Ndr,0:Ndr))
allocate(B34(0:Ndr,0:Ndr))
allocate(B35(0:Ndr,0:Ndr))
allocate(B36(0:Ndr,0:Ndr))
allocate(B41(0:Ndr,0:Ndr))
allocate(B42(0:Ndr,0:Ndr))
allocate(B43(0:Ndr,0:Ndr))
allocate(B44(0:Ndr,0:Ndr))
allocate(B45(0:Ndr,0:Ndr))
allocate(B46(0:Ndr,0:Ndr))
allocate(B51(0:Ndr,0:Ndr))
allocate(B52(0:Ndr,0:Ndr))
allocate(B53(0:Ndr,0:Ndr))
allocate(B54(0:Ndr,0:Ndr))
allocate(B55(0:Ndr,0:Ndr))
allocate(B56(0:Ndr,0:Ndr))
allocate(B61(0:Ndr,0:Ndr))
allocate(B62(0:Ndr,0:Ndr))
allocate(B63(0:Ndr,0:Ndr))
allocate(B64(0:Ndr,0:Ndr))
allocate(B65(0:Ndr,0:Ndr))
allocate(B66(0:Ndr,0:Ndr))
!FLUID 1 EQUATIONS
!R-MOMENTUM
!B11
B11(:,:) = RHO1*x2Dddr(:,:,0)
!singularity bc on Ur
B11(0,:) = 0
!normal stress at r=a
B11(Ndr,:) = -2.*MU1*Dddr(Ndr,:,1)
!B12
B12(:,:) = 0
!B13
B13(:,:) = 0
!bc for normal stress at r=a
B13(Ndr,:) = 1.*Dddr(Ndr,:,0)
!B14
B14(:,:) = 0
!bc for normal stress at r=a
B14(Ndr,:) = 2.*MU2*Dddt(0,:,1)
!B15
B15(:,:) = 0
!B16
B16(:,:) = 0
!bc for normal stress at r=a
B16(Ndr,:) = (-1.)*Dddt(0,:,0)
!X-MOMENTUM
!B21
B21(:,:) = 0
!B22
B22(:,:) = RHO1*xDddr(:,:,0)
!singularity BC on Ux
B22(0,:) = 0
!tang stress at r=a
B22(Ndr,:) = 0
!B23
B23(:,:) = 0
!B24
B24(:,:) = 0
!B25
B25(:,:) = 0
!B26
B26(:,:) = 0
!CONTINUITY
!B31
B31(:,:) = 0
!B32
B32(:,:) = 0
!B33
B33(:,:) = 0
!B34
B34(:,:) = 0
!B35
B35(:,:) = 0
!B36
B36(:,:) = 0
!FLUID 2 EQUATIONS
!R-MOMENTUM
!B41
B41(:,:) = 0
!B42
B42(:,:) = 0
!B43
B43(:,:) = 0
!B44
B44(:,:) = RHO2*x2Dddt(:,:,0)
!bc for no penetration at r=H
B44(Ndr,:) = 0
!bc for continuity at interface
B44(0,:) = 0
!B45
B45(:,:) = 0
!B46
B46(:,:) = 0
!X-MOMENTUM
!B51
B51(:,:) = 0
!B52
B52(:,:) = 0
!bc no slip at interface
B52(0,:) = (-1.) * Dddr(Ndr,:,0)
!B53
B53(:,:) = 0
!B54
B54(:,:) = 0
!B55
B55(:,:) = RHO2*xDddt(:,:,0)
!bc for no slip at r=H
B55(Ndr,:) = 0
!bc no slip at interface
B55(0,:) = Dddt(0,:,0)
!B56
B56(:,:) = 0
!CONTINUITY
!B61
B61(:,:) = 0
!B62
B62(:,:) = 0
!B63
B63(:,:) = 0
!B64
B64(:,:) = 0
!B65
B65(:,:) = 0
!B66
B66(:,:) = 0
!put together the RHS
!first row of blocks
Do i=0,Ndr
Do j=0,Ndr
B(i+1,j+1) = B11(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1,j+1+ 1*(Ndr+1)) = B12(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1,j+1+ 2*(Ndr+1)) = B13(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1,j+1+ 3*(Ndr+1)) = B14(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1,j+1+ 4*(Ndr+1)) = B15(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1,j+1+ 5*(Ndr+1)) = B16(i,j)
End Do
End Do
!second row of blocks
Do i=0,Ndr
Do j=0,Ndr
B(i+1+Ndr+1,j+1) = B21(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1+Ndr+1,j+1+ 1*(Ndr+1)) = B22(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1+Ndr+1,j+1+ 2*(Ndr+1)) = B23(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1+Ndr+1,j+1+ 3*(Ndr+1)) = B24(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1+Ndr+1,j+1+ 4*(Ndr+1)) = B25(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1+Ndr+1,j+1+ 5*(Ndr+1)) = B26(i,j)
End Do
End Do
!third row of blocks
Do i=0,Ndr
Do j=0,Ndr
B(i+1+ 2*(Ndr+1),j+1) = B31(i,j)
End Do
End Do
Do i=0,Ndr
Do j=0,Ndr
B(i+1+ 2*(Ndr+1),j+1+ 1*(Ndr+1)) = B32(i,j)