Double-diffusive two-fluid flow in a slippery channel: A linear stability analysis
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...To discuss the linear stability of the system, Equations (1) and (2) are to be linearized by supposing infinitesimal disturbances in the dependent variable....
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...and with c j M j M = 2 = 0, , 1 1 ( − 1), j ⎨⎩ ≤ ≤ Equations (16) and (17) form the following equation: E X cE X = , 1 2 (20)...
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...Substituting Equation (4) into Equations (1) and (2) and equating the coefficients of leading order (ε̄), we obtain ϕ u z ϕ u z ϕ p x Re ϕ ϕˆ + ( ) ˆ − ′ ( ) ˆ = − ˆ + 1 ( ˆ + ˆ ),tz xz x zxx zzzb b ∂ ∂ (5) ϕ u z ϕ p z A ψ ψ Re ϕ ϕˆ + ( ) ˆ = − ˆ + ( ˆ + ˆ ) + 1 ( ˆ + ˆ ),tx xx zz xx xxx xzzb 2 ∂ ∂ (6) ψ u z ψ ϕ Rm ψ ψˆ + ( ) ˆ = ˆ + 1 ( ˆ + ˆ ),tx xx xx xxx xzzb (7) ψ u z ψ ϕ u z ψ Rm ψ ψˆ + ( ) ˆ = ˆ − ( ) ˆ + 1 ( ˆ + ˆ ).tz zx zx x zxx zzzb b (8) Assuming the solutions in the separable form ϕ x z t ψ x z t ϕ z ψ z e{ ˆ ( , , ), ˆ ( , , )} = ( ( ), ( )) ,iα x ct( − ) (9) where c c ic= +r i is the disturbance wave speed, cr is the phase velocity, ci is the disturbance growth rate, and α is the streamwise wave number, which is real and positive....
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...Also, when A= 0 and Rm = 0, Equations (10) and (11) reduce to Orr–Sommerfeld equation for the clear fluid case....
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...The stream functions ϕ z( ) and ψ z( ) can be approximated in terms of the polynomials T z( )j as follows: ϕ z T z ϕ ψ z T z ψ( ) = ( ) , ( ) = ( ) . j M j j j M j j =0 =0 ∑ ∑ (15) Using Equation (15), Equations (10) and (11) can be written as follows: u c B ϕ α ϕ D u ϕ A B ψ α ψ C ϕ α ϕ α B ϕ ( − ) ( − ) − − ( − ) = + − 2 , k M jk k j j k M jk k j iαRe k M jk k j k M jk k b =0 2 2 b 2 =0 2 1 =0 4 2 =0 ⎛ ⎝ ⎜⎜ ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎞ ⎠ ⎟⎟ ∑ ∑ ∑ ∑ (16) αRm B ψ α ψ ϕ u c ψ1 ( − ) = − ( − ) , k N jk k j j j =0 2 b∑ (17) where j M= 1(1) − 1, ϕ ϕ ψ ψ= = 0, = = 0,M M0 0 A ϕ j M= 0, = 0 andk M jk k=0∑ with ( )A c j k z x j k M N j k N j k = (−1) , /2 1 − 1 = − 1, (2 + 1)/6 = = 0, −(2 + 1)/6 = = 1, jk j k j j j + 2 2 2 ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ ≠ ≤ ≤ (18) B A A C B B= · and = · ,jk jm mk jk jm mk (19) and with c j M j M= 2 = 0, , 1 1 ( − 1),j ⎧⎨⎩ ≤ ≤ Equations (16) and (17) form the following equation: E X cE X= ,1 2 (20) where E1 and E2 are the matrices of order 2(M+ 1) with complex entries, c is the eigenvalue, and X is the discrete eigenfunction....
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