Convective derivatives and Reynolds transport in curvilinear time-dependent coordinates

Abstract

A fully covariant formulation of kinematics and dynamics of fluid flows and heat transfer is developed in time-dependent curvilinear coordinate systems. These moving and deformable reference frames have the same properties of a fluid motion and they can be successfully applied to a variety of problems ranging from numerical analysis to theoretical physics. The classical Reynolds transport theorem, the Euler formula and the acceleration addition theorem are extended to these general types of coordinates through a generalization of the convected differentiation concept originally introduced by Oldroyd (1950 Proc. R. Soc. A 200 523–41). A rigorous formulation of dynamical equations and conservation laws in curvilinear time-dependent coordinates could be the key for the construction of variational principles based on the method of constrained variations.

Rheological evaluation of colloidal dispersions using the smoothed profile method: formulation and applications

“…When the coordinate system used to solve the Navier-Stokes equation is timedependent, which is the case we are interested in here, the proper form for the equations of motion is given by Luo & Bewley (2004), Venturi (2009) δu µ δt = ρ −1 ∇ ν σ νµ + φf µ p , (A 4) where δ/δt denotes the intrinsic time derivative, defined for a contravariant quantity as…”

Conjugate flow action functionals

“…We remark both the conjugate flow perturbation as well as the perturbation induced in the field equations can be represented relative to arbitrary curvilinear coordinates 1,48,51 . The choice of a coordinate system advected by the conjugate flow, however, is convenient since the flow map x µ (σ ν ; u j ) then appears explicitly in the equations of motion, and it can be selected in order to satisfy existence conditions of an action functional (see section IV).…”

“…(A14) and (A15)). This is why we have included the second-order derivative of the conjugate flow in the second-order vector field equation (48). A substitution of (56)- (57) into (49) yields the following operator G x (see Eq.…”

“…The curvilinear coordinate system advected by such flow is known as Lagrangian system and it can be determined by integrating out the definition of the velocity field 48 . In this sense, physical fluid flow of classical mechanics can be considered as a very particular type of conjugate flow.…”

A fully symmetric nonlinear biorthogonal decomposition theory for random fields

“…As an example, consider the following inner product (Einstein's summation convention on repeated indices is assumed) (19) inducing a norm in the space of vector fields represented in a curvilinear coordinate system [54] with metric g ij (x). If we assume that g ij is a possibly nonlinear functional of a field u(x), then (20) defines a local inner product, which is nondegenerate only if g ij (x; u) is nonsingular.…”