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Does a droplet roll or slide on an inclined surface ? Ronojoy Adhikari The Institute of Mathematical Sciences Sumesh Thampi and Rama Govindarajan Jawaharlal Nehru Center for Advanced Scientific Research

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An ancient observation ...

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An ancient observation ... http:/www./religiousleftlaw.com/2012/02/the-three-yogas-of-the-bhagavad-gītā-an-introduction.html

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An ancient observation ... brahmany adhaya karmani sangam tyaktva karoti yah lipyate na sa papena padma-patram ivambhasa ---Bhagavadgita ~ 200BCE http:/www./religiousleftlaw.com/2012/02/the-three-yogas-of-the-bhagavad-gītā-an-introduction.html

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An ancient observation ... brahmany adhaya karmani sangam tyaktva karoti yah lipyate na sa papena padma-patram ivambhasa ---Bhagavadgita ~ 200BCE http:/www./religiousleftlaw.com/2012/02/the-three-yogas-of-the-bhagavad-gītā-an-introduction.html One who performs his duty without attachment, surrendering the results unto the Supreme God, Is not affected by sinful action, as the lotus leaf is untouched by water.

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An ancient observation ... brahmany adhaya karmani sangam tyaktva karoti yah lipyate na sa papena padma-patram ivambhasa ---Bhagavadgita ~ 200BCE http:/www./religiousleftlaw.com/2012/02/the-three-yogas-of-the-bhagavad-gītā-an-introduction.html One who performs his duty without attachment, surrendering the results unto the Supreme God, Is not affected by sinful action, as the lotus leaf is untouched by water. प"प#िमवा(भसा

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प"प#िमवा(भसा

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An ancient observation ... brahmany adhaya karmani sangam tyaktva karoti yah lipyate na sa papena padma-patram ivambhasa ---Bhagavadgita ~ 200BCE http:/www./religiousleftlaw.com/2012/02/the-three-yogas-of-the-bhagavad-gītā-an-introduction.html One who performs his duty without attachment, surrendering the results unto the Supreme God, Is not affected by sinful action, as the lotus leaf is untouched by water. प"प#िमवा(भसा

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...and a modern question. ? rolling sliding

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Why is this important ?

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Why is this important ? self-cleaning surfaces want minimum amount of maintenance avoid economic costs of cleaning

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Why is this important ? self-cleaning surfaces want minimum amount of maintenance avoid economic costs of cleaning

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Why is this important ? self-cleaning surfaces want minimum amount of maintenance avoid economic costs of cleaning

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Why is this important ? adhering surfaces want minimum amount of pesticide avoid contamination of soil and water self-cleaning surfaces want minimum amount of maintenance avoid economic costs of cleaning

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Why is this important ? adhering surfaces want minimum amount of pesticide avoid contamination of soil and water self-cleaning surfaces want minimum amount of maintenance avoid economic costs of cleaning

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Why is this important ? adhering surfaces want minimum amount of pesticide avoid contamination of soil and water self-cleaning surfaces want minimum amount of maintenance avoid economic costs of cleaning

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The physics of drops and surfaces sessile droplet solid liquid vapour pendant droplet s o l i d liquid vapour

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Statics and energy minimization

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Statics and energy minimization lv sv ls ✓ Alv Als liquid-vapour interface area liquid-solid interface area surface tension

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Statics and energy minimization lv sv ls ✓ Alv Als liquid-vapour interface area liquid-solid interface area surface tension E = lv Alv ( sv sl)Asl “Hamiltonian’’ equilibrium : minimize surface area at constant volume

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Statics and energy minimization lv sv ls ✓ Alv Als liquid-vapour interface area liquid-solid interface area surface tension E = lv Alv ( sv sl)Asl “Hamiltonian’’ equilibrium : minimize surface area at constant volume E = 0 with V constant gives drop shape and contact angle ✓ “constrained variational problem’’

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The variational solution lv sv ls ✓ Alv Als liquid-vapour interface area liquid-solid interface area surface tension cos ✓ = sv sl lv hemispherical cap Young-Laplace Law + +

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Wetting water droplet in oil on brass water droplet in oil on glass good wetting poor wetting images from wikipedia

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Where is gravity ?

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Where is gravity ? E = lv Alv ( sv sl)Asl + Z V ⇥gz dV

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Where is gravity ? E = lv Alv ( sv sl)Asl + Z V ⇥gz dV gravity surface tension = R ⇥gz dV lv Alv

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Where is gravity ? E = lv Alv ( sv sl)Asl + Z V ⇥gz dV gravity surface tension = R ⇥gz dV lv Alv Bo = ⇥gL2 lv Bond Number

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Where is gravity ? E = lv Alv ( sv sl)Asl + Z V ⇥gz dV gravity surface tension = R ⇥gz dV lv Alv Bo = ⇥gL2 lv Bond Number small Bond number - gravity not important - small droplets large Bond number - gravity important - large droplets

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Where is gravity ? E = lv Alv ( sv sl)Asl + Z V ⇥gz dV gravity surface tension = R ⇥gz dV lv Alv Bo = ⇥gL2 lv Bond Number small Bond number - gravity not important - small droplets large Bond number - gravity important - large droplets Too much of Bond - cannot solve constrained variational problem analytically, need numerical solutions for shape and contact angle.

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Shape and contact angle changes with Bo movie from mit opencourseware project

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Shape and contact angle changes with Bo movie from mit opencourseware project

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Drop dynamics

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Drop dynamics cartoon

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Drop dynamics cartoon reality leading contact angle trailing contact angle

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Drop dynamics cartoon reality leading contact angle trailing contact angle contact angles and shape no longer determined by energy minimization.

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Drop dynamics cartoon reality leading contact angle trailing contact angle contact angles and shape no longer determined by energy minimization. now, a full fluid dynamical problem has to be solved.

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Drop dynamics cartoon reality leading contact angle trailing contact angle contact angles and shape no longer determined by energy minimization. now, a full fluid dynamical problem has to be solved. this is a “free boundary problem”, where the surface where boundary conditions are imposed is part of the problem!

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Drop dynamics cartoon reality leading contact angle trailing contact angle contact angles and shape no longer determined by energy minimization. now, a full fluid dynamical problem has to be solved. this is a “free boundary problem”, where the surface where boundary conditions are imposed is part of the problem! hard! hard! hard!

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Landau-Ginzburg theory = ⇢l ⇢v ⇢l + ⇢v F = Z A 2 2 + B 4 4 + K 2 (r )2 mathematical interface physical interface = 2 3 p 2KA3/B2 = p 2K/A } “phi4 soliton”

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Dynamics with Landau-Ginzburg theory @t + · j = 0 @t(⇢v) + · = 0 j = ⇥v + Mr F ⇥ = ⇤vv + rp + ⇥rv + r F ⌅ conservation laws constitutive equations

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What’s new ? previous work our work approximate solutions exact numerical solution small Bond number arbitrary Bond number small OR large contact angle arbitrary contact angle

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Droplet motion under gravity

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Droplet motion under gravity

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What is rolling for a drop ?

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A scalar from shape q = 4 area (perimeter)2

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A rolling law for drops

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प"प#िमवा(भसा

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Thank you for your attention!