Theory of capillary-gravity wave scattering by a fixed, semi-immersed cylindrical barrier with contact line dissipation Published online by Cambridge University Press 06/25/2025 10609235 10.1017/jfm.2025.10158 Journal of Fluid Mechanics 0022-1120 1013 Guoqin LiuLikun Zhang The scattering of surface waves by structures intersecting liquid surfaces is fundamental in fluid mechanics, with prior studies exploring gravity, capillary and capillary–gravity wave interactions. This paper develops a semi-analytical framework for capillary–gravity wave scattering by a fixed, horizontally placed, semi-immersed cylindrical barrier. Assuming linearised potential flow, the problem is formulated with differential equations, conformal mapping and Fourier transforms, resulting in a compound integral equation framework solved numerically via the Nyström method. An effective-slip dynamic contact line model accounting for viscous dissipation links contact line velocity to deviations from equilibrium contact angles, with fixed and free contact lines of no dissipation as limiting cases. The framework computes transmission and reflection coefficients as functions of the Bond number, slip coefficient and barrier radius, validating energy conservation and confirming a$90^\circ$phase difference between transmission and reflection in specific limits. A closed-form solution for scattering by an infinitesimal barrier, derived using Fourier transforms, reveals spatial symmetry in the diffracted field, reduced transmission transitioning from gravity to capillary waves and peak contact line dissipation when the slip coefficient matches the capillary wave phase speed. This dissipation, linked to impedance matching at the contact lines, persists across a range of barrier sizes. These results advance theoretical insights into surface-tension-dominated fluid mechanics, offering a robust theoretical framework for analysing wave scattering and comparison with future experimental and numerical studies.

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. 1c ) p 0 + p = P 0 + σ ∇ • n , at |x | > r , y = η , ( 2. 1d ) w h er e v = v ( x , y , t ) is t h e v el o cit y fi el d, p = p ( x , y , t ) is t h e p ert ur b e d pr ess ur e of t h e w a v e fi el d, η = η (x , t ) is t h e s urf a c e el e v ati o n wit h r es p e ct t o its e q uili bri u m p ositi o n a n d Ω = { (x , y ) ∈ R 2 | y < 0 a n d x 2 + y 2 > r 2 } d e fi n es t h e li q ui d d o m ai n. H er e, n = ∇ ( y -η ) /|∇ ( y -η )| is t h e n or m al v e ct or of t h e fr e e s urf a c e p oi nti n g o ut w ar d of t h e li q ui d d o m ai n a n d ∇ • n r e pr es e nts t h e c ur v at ur e of t h e fr e e s urf a c e. I ntr o d u ci n g t h e v el o cit y p ot e nti al wit h v = ∇ φ a n d c o nsi d eri n g a li n e aris e d pr o bl e m, (2. 1 ) si m pli fi es t o ∇ At t h e fr e e s urf a c e of t h e li q ui d of i n fi nit e d e pt h, t h e i n ci d e nt s urf a c e w a v e of a n g ul ar fr e q u e n c y ω a n d w a v e n u m b er k h as t h e f or m of φ I (x , y , t ) = φ A e k y e i(k x -ω t ) ,

( 2. 5a )

) , ( 2. 5b ) w h er e a m plit u d e φ A a n d η A h as t h e li n e ar r el ati o n φ A = -i ω k η A , ( 2. 5c ) s atisf yi n g t h e ki n e m ati c b o u n d ar y c o n diti o n ( 2. 2 c ), a n d ω a n d k h a v e t h e dis p ersi o n r el ati o n ω 2 = σ ρ k 3 + g k = σ ρ k 3 (1 + B ), ( 2. 5d ) s atisf yi n g t h e d y n a mi c b o u n d ar y c o n diti o ns ( 2. 2 b ) a n d (2. 2 d ), w h er e B = ρ g σ k 2 ( 2. 6) is t h e B o n d n u m b er, c h ar a ct erisi n g t h e r el ati v e eff e ct b et w e e n gr a vit y a n d s urf a c e t e nsi o n. At l ar g e dist a n c es fr o m t h e b arri er, t h e s urf a c e el e v ati o n t a k es t h e f ar-fi el d f or m:

w h er e η R = R η I (-x , t ) = R η A e i(-k x -ω t ) , ( 2. 7c ) η T = T η I (+ x , t ) = T η A e i(k x -ω t ) .

( 2. We a p pl y t h e f oll o wi n g F o uri er tr a nsf or m t o ( 3. 6 ): A n i n v ers e F o uri er tr a nsf or m yi el ds t h e s ol uti o n t o ( 3. 6 ): 1 0 1 3 A 2 7-1 1

We t h e n d eri v e e q u ati o ns f or t h e n e ar fi el ds h e a n d h o :

w h er e I is t h e i d e ntit y o p er at or, w hi c h t a k es a f u n cti o n t o t h e s a m e f u n cti o n. B as e d o n t h e s y m m etr y of (4. 7 ), w e s e e k s ol uti o ns of h e a n d h o i n t h e f or m: h e = 1 2 (T + R )h e ,+ + 1 2 h e ,-, ( 4. 8a )

a n d h χ , ± s atis fi es (4. 1 0 ):

Gi v e n t h at H its elf is t h e li q ui d d o m ai n, n o tr a nsf or m ati o n i nt o a l o w er h alf-s p a c e is r e q uir e d. T h us, t h e c o nf or m al m a p pi n g w = z / r + r / z i n § 3. 2. 1 si m pli fi es t o t h e i d e ntit y w = z . E m pl o yi n g a si mil ar a p pr o a c h as i n § 3. 2. 2 , w e f or m ul at e t h e i nt e gr al e q u ati o n pr o bl e m f or f ( x ) = φ ( x , 0 ):

fr o m t h e mi ni m a of t h e m a g nit u d e of t h e gr a di e nts of t h e dissi p ati o n i n t h e p ar a m et er s p a c e. T h e li n e r e pr es e nts t h e p at h of l e ast d es c e nt f or dissi p ati o n i n t h e p ar a m et er s p a c e. I ntri g ui n gl y, it is o bs er v e d t h at t his dissi p ati o n ri d g e li n e fits wit h t h e r el ati o n ( w hit e d as

2) R e c o g nis e t h at t h e e x pr essi o n i n p ar e nt h es es is t h e i n v ers e F o uri er tr a nsf or m of 1 / |λ |, n a m el y ± i e ir 2

x Ei ir (± x -r )

x + e -ir 2 x Ei ir (± x + r ) x + e ix Ei (i(-x ± r )) + e -ix Ei (i(x ± r )) -2 e ± ir l n x r -r x , ( C 9a )

( D 1 6) R e arr a n gi n g, w e h a v e ∞ 0 ĥ e ,c (k )

N oti c e t h at o n t h e p ositi v e k -a xis, o nl y t h e si n g ul arit y t er ms i n h c c orr es p o n d t o t h e t er ms i n h t h at pr o p a g at e as x → ∞ . I n t his i nst a n c e, t h er e ar e t w o si n g ul arit y t er ms o n t h e p ositi v e k -a xis i n h e ,c : (k -1 ) -foot_9 a n d δ ( k -1 ), w hi c h c orr es p o n d t o si n x a n d c os x , r es p e cti v el y, at x → ∞ i n h e . Si n c e t h e e q u ati o ns f or c al c ul ati n g T a n d R (( 4. 1 4a ) a n d ( 4. 1 4b )) o nl y r e q uir e t h e r ati o h e ,-/ h e ,+ , it is s uf fi ci e nt t o c al c ul at e t his r ati o usi n g t h e c o ef fi ci e nts of t h e (k -1 ) -1 t er ms f or ĥ e ,-,c a n d ĥ e ,+ ,c . It c a n als o b e s h o w n ( al b eit t hr o u g h a n ot h er l e n gt h y d eri v ati o n) t h at t h e c o ef fi ci e nt of δ ( k -1 ) is e q u al t o t h e c o ef fi ci e nt of (k -1 ) -1 ; w e o mit t h at pr o of f or br e vit y. B y s u bstit uti n g t h e e x pr essi o ns f or α , ŵ c a n d ĝ e ,c i nt o (D 1 8 ) a n d e xtr a cti n g t h e c o ef fi ci e nts of t h e (k -1 ) -1 t er ms, w e o bt ai n t h e f oll o wi n g e x pr essi o n f or C e : , ( E 3a )

w h er e α = 2 K (1 + 3 K )( J 1 ( K ) -iλ (1 + K ) -1 / 2 ) , ( E 3c ) β = J 2 ( K ) + i(1 + 3 K )(1 + K ) 1 / 2 / 2 K λ , ( E 3d ) a n d J 1 ( K ) = 2 K π ( 1 + 3 K ) 1 2 l n K + 1 K + 3 K + 2 K 1 / 2 (3 K + 4 ) 1 / 2 t a n -1 3 K + 4 K 1 / 2

, ( E 3e ) J 2 ( K ) = 1 π K -K 2 l n K + 1 K + 3 K 2 + 6 K + 2 K 1 / 2 (3 K + 4 ) 1 / 2 t a n -1 3 K + 4 K 1 0 1 3 A 2 7-4 1

√ B -e -2i ζ e √ B (2 r -| x + x |) + e - √ B |x -x | , ( 3. 2) 1 0 1 3 A 2 7-8 htt p s:// d oi. o r g/ 1 0. 1 0 1 7/jf m. 2 0 2 5. 1 0 1 5 8 P u bli s h e d o nli n e b y C a m b ri d g e U ni v e r sit y P r e s s

0 1 3 A

7-1 0 htt p s:// d oi. o r g/ 1 0. 1 0 1 7/jf m. 2 0 2 5. 1 0 1 5 8 P u bli s h e d o nli n e b y C a m b ri d g e U ni v e r sit y P r e s s

√ B -( √ B ± i)e -2i ζ + (-√ B ± i) e ( √ B ± i)r I 1 , χ + I 2 , χ ,± , ( 4. 1 2) w h er e t h e e x pr essi o ns f or I 1 a n d I 2 ar e gi v e n i n A p p e n di x C . 1 0 1

A 2 7-1 2 htt p s:// d oi. o r g/ 1 0. 1 0 1 7/jf m. 2 0 2 5. 1 0 1 5 8 P u bli s h e d o nli n e b y C a m b ri d g e U ni v e r sit y P r e s s

0 1 3 A

7-1 7 htt p s:// d oi. o r g/ 1 0. 1 0 1 7/jf m. 2 0 2 5. 1 0 1 5 8 P u bli s h e d o nli n e b y C a m b ri d g e U ni v e r sit y P r e s s

0 1 3 A

7-2 7 htt p s:// d oi. o r g/ 1 0. 1 0 1 7/jf m. 2 0 2 5. 1 0 1 5 8 P u bli s h e d o nli n e b y C a m b ri d g e U ni v e r sit y P r e s s

0 1 3 A

7-3 7 htt p s:// d oi. o r g/ 1 0. 1 0 1 7/jf m. 2 0 2 5. 1 0 1 5 8 P u bli s h e d o nli n e b y C a m b ri d g e U ni v e r sit y P r e s s

0 1 3 A

7-3 9 htt p s:// d oi. o r g/ 1 0. 1 0 1 7/jf m. 2 0 2 5. 1 0 1 5 8 P u bli s h e d o nli n e b y C a m b ri d g e U ni v e r sit y P r e s s

0 1 3 A

7-4 2 htt p s:// d oi. o r g/ 1 0. 1 0 1 7/jf m. 2 0 2 5. 1 0 1 5 8 P u bli s h e d o nli n e b y C a m b ri d g e U ni v e r sit y P r e s s