USC Summer School on Mathematical Fluids An Introduction to Instabilities in Interfacial Fluid Dynamics

This is a short course on instability in interfacial fluid mechanics taught by Professor Ian Tice. This course is one of the three series of courses offered in the USC Summer School on Mathematical Fluids organized by Professor Juhi Jang in University of Southern California during May 22 and May 26, 2017. 1 A crash course on basic fluid mechanics 1.1 Equations of motion for a fluid For a fluid evolving due to purely mechanical effects, i.e., neglecting thermodynamics, we can specify the state of the fluid with the following for t ≥ 0. 1.1.1 Notations Let Ω(t) ⊂ R be an open set in which the fluid resides. We refer this set as fluid domain. We have following functions describing features of fluid: ρ(·, t) : Ω(t) → (0,∞) denotes the density of fluid, u(·, t) : Ω(t) → R denotes the velocity of fluid, p(·, t) : Ω(t) → R denotes the pressure of fluid, S(·, t) : Ω(t) → Sym3 = {M ∈ R3×3 |M = M } denotes the stress tensor, and f(·, t) : Ω(t)→ R denotes the external force on fluid. In these notes, we use Einstein’s summation convention. The k-th partial derivative of φ will be denoted by φ,k = ∂φ ∂xk . Repeated Latin indices i, j, k, etc., are summed from 1 to 3, and repeated Greek indices α, β, γ, etc., are summed from 1 to 2. For example, φ,iθ,i = ∑3 i=1 ∂φ ∂xi ∂θ ∂xi , and φ,αAαβθ,β = ∑2 α=1 ∑2 β=1 ∂φ ∂xα Aαβ ∂θ ∂xβ . 1.1.2 The Equations In these notations, the equations of motion are ∂tρ+ div(ρu) = 0 in Ω(t), ∂t(ρu) + div(ρu⊗ u) + divS = f in Ω(t). (1.1) Here, (A⊗B)ij = AiBj , (divM)i = Mij,j . By direct computation ∂t(ρu)i + [div(ρu⊗ u)]i = ∂tρui + ρ∂tui + ρ,juiuj + ρui,juj + ρuiuj,j = ui(∂tρ+ ρ,juj + ρuj,j) + ρ(∂tui + ui,juj) = ρ(∂tu+ u · ∇u)i. It follows that the system (1.1) is equivalent to ∂tρ+ div(ρu) = 0 in Ω(t), ρ(∂tu+ u · ∇u) + divS = f in Ω(t), (1.2) where (u · ∇u)i = ujui,j . The interests mainly lie on two types of fluid. Type 1: Incompressible. For the incompressible fluid, we assume (i) div u = 0 in Ω(t), and (ii) the stress tensor is S = pI − μDu where I is the 3 × 3 identity matrix, Du = Du + (Du) is the symmetric gradient, and μ ≥ 0 is the (shear) viscosity.