ID 54

Adjoint-based flow and electrostatic control of an isolated liquid drop

 

Abstract:

A continuous adjoint theory is developed to control the deformation of a clean, neutrally buoyant axisymmetric droplet under two distinct flow conditions. In the first case, an initially spherical drop is suspended in a low-Reynolds-number uniaxial extensional flow. The applied flow strength expressed in terms of the non-dimensional capillary number Ca is taken to be the control. The theory is demonstrated for dynamic viscosity ratios of λ = 0.1, 1 and 10. The second study concerns the adjoint-based control of a weakly conducting dielectric drop suspended in another weakly conducting liquid subjected to a uniform electric field. Here, the electric capillary number Ca_E is the control and the theory is validated for λ = 1 and various resistivities R and permittivities Q. In both cases, the control generates a distinct steady-state distorted droplet. This information is utilized in the inverse problem to compute the adjoint velocities and the corresponding control gradient at the drop interface. The resultant gradient is subsequently used to update Ca or Ca_E. The algorithm is repeated until the desired shape is realized. Both the forward and adjoint velocities as well as the forward electric field are obtained by solving a set of boundary integral equations. In both studies, the cost functional is successfully minimized. The adjoint gradient is shown to be in good agreement with the corresponding finite-difference approximation of it.