Nematic liquid crystals flowing down an incline Namrata Patel - PowerPoint PPT Presentation

Nematic liquid crystals flowing down an incline Namrata Patel

Lubrication theory ๏ Thin film of a viscous fluid ๏ lubrication approximation ๏  = H / L where H  L ๏ velocity gradients in the x,y-directions negligible compared to velocity gradient in z-direction ๏ , can ignore terms  2 Re  1 due to inertia ๏ Lubrication theory simplifies evolution equation

Evolution equation h t + ∇ ·[ h 3 ( C ∇∇ 2 h − B ∇ h ) + N ( m 2 − hm ′ m ) ∇ h ] + U ( h 3 ) x = 0 h 3/2 m = β 3/2 + h 3/2 lubrication approximation adopted ๏ prewetting of substrate with precursory layer of ๏ thickness b  h weak-anchoring model ๏ h: fluid thickness C: inverse capillary number B: Bond number, N: inverse Ericksen number

Linear stability analysis(LSA) of flat film ๏ Assume h is independent of y h t + ∂ x [ h 3 ( Ch xxx − Bh x ) + N ( m 2 − hm ′ m ) h x ] + U ( h 3 ) x = 0 ๏ Perturb profile by a small amplitude h ( x , t ) = h o +  h 1 ( x , t ) + O (  2 ) 3 h 1 xxxx − ( B − NM ( h o ) ) h o h 1 t + Ch o 3 h 1 xx + 3 Uh o 2 h 1 x = 0 M ( h o ) = ho 3/2 − β 3/2 / 2 ( ho 3/2 + β 3/2 ) 3

LSA of flat film cont. ๏ Obtain dispersion relation by assuming solutions of the form h 1 ∝ e σ t + ikx 3 k 4 + ( B − NM ( h o ) ) h o 3 k 2 − i 3 Uh o σ = − Ch o 2 k ๏ Surface tension responsible for stabilizing system for perturbations of short wavelengths NM ( h o ) − B k m = 2 C σ m = ( NM ( h o ) − B ) 2 3 h o 4 C σ : growth-rate k: wavenumber k m : fastest growing wavenumber

Traveling-wave solutions for 2D equation h t + ∂ x [ h 3 ( Ch xxx − Bh x ) + N ( m 2 − hm ′ m ) h x ] + U ( h 3 ) x = 0 m ( h ) = h 3/2 β 3/2 h → b as x → −∞ h → 1 as x → ∞ h x → 0 as x → −∞ h x → 0 as x → ∞ ๏ simplify analysis by assuming β  h ๏ Neumann BCs used since far behind & far in front of the contact line, fluid thickness approximately constant

Traveling-wave solutions cont. ๏ Look at solution in a moving reference frame ๏ make a change of variables ξ = x − Vt where V is the wave speed ๏ ∴ substituting and integrating h o ( ξ ) = h ( x , t ) 3 h o ξξξ − ( B + N 2 β 3 ) h o − Vh o + Ch o 3 h o ξ + U ( h o 3 ) ξ = d ๏ Applying the BCs d = − b (1 + b ) V = U (1 + b + b 2 )

LSA for 3D evolution equation h t + ∇ ·[ h 3 ( C ∇∇ 2 h − B ∇ h ) + N ( m 2 − hm ′ m ) ∇ h ] + U ( h 3 ) x = 0 ๏ constant flux-driven case ๏ fluid is being injected into the film ๏ infinite volume ๏ initially, flow in transverse direction fairly stable ๏ apply perturbations to leading order equation h ( x , y , t ) = h o ( ξ ) +  φ ( ξ ) e σ t + iky + O (  2 )

LSA for 3D evolution equation cont. [ ( ) ] − σ φ = − + φ − φ + φ + φ + φ 4 3 2 3 3 3 3 V g C k h k ( h ) h ( h 3 h h ) ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ o o o o o o N ( ) ( ) + + φ − φ + φ + φ 2 3 3 2 2 B k h ( h 3 h h ) 3 U ( h ) ξ ξ ξ ξ β o o o o o 3 2 ๏ σ , ɸ depend only on even powers of k ๏ looking at the solution in the limit of a small wavenumber φ = φ o + k 2 φ 1 + O ( k 4 ) σ = σ o + k 2 σ 1 + O ( k 4 ) ๏ we modified the position of the contact line by the perturbation, so BCs need to linearized accordingly giving φ o ( ξ ) = h o ξ ( ξ )

LSA for 3D evolution equation cont. ๏ leading-order equation 3 h o ξξξ − ( B + N − σ o h o ξ = [ − Vh o + Ch o 2 β 3 ) h o 3 ) ξ ] ξξ 3 h o ξ + U ( h o ๏ right hand side is our leading order equation, therefore σ o = 0

LSA for 3D evolution equation cont. ๏ O(k 2 ) equation − σ 1 h o ξ = − V φ 1 ξ + C ( − ( h o 3 φ 1 ξξξ ) ξ + 3( ho 2 h o ξξξ φ 1 ) ξ ) 3 h o ξξ ) ξ − h o 3 h o ξξξ + ( h o + ( B + N 2 β 3 )( h o 3 φ 1 ) ξξ ) + 3 U ( h o ξ φ 1 ) ξ 3 h o ξ − ( h o ๏ We integrate and apply the BCs ๏ Finally, 0 σ ≈ k 2 3 h o ξξξ − ( B + N ∫ 2 β 3 ) h o d ξ 3 h o ξ Ch o 1 − b −∞