Non-Newtonian Fluids and Finite Elements

Non-Newtonian Fluids and Finite Elements Janice Giudice Oxford University Computing Laboratory Keble College

Talk Outline Motivating Industrial Process Multiple Extrusion of Pastes Governing Equations The Governing Equations Numerical Approximation Current Methods Finite Element Approximation Some Results Adaptivity A posteriori Error Analysis Conclusion and Further Work

Extrusion of Paste A soft, smooth, thick mixture of materials being forced through a cylindrical chamber out of an exit nozzle. The material is tightly confined.

Multiple Extrusion of Pastes Pastes with differing rheological properties layered undergoing the same extrusion process with interfacial stability.

Multiple Extrusion of Pastes Pastes with differing rheological properties layered undergoing the same extrusion process with interfacial stability. Complex Geometries: multiple nested chambers, the design of the extrusion chamber may be complex with internal moving parts and varying nozzle geometry.

Non-Newtonian Fluids Nonlinear stress tensor T = pi + k (x, e(u) ) e(u) e(u) = 1 ( 2 u + ( u) ) is the rate of strain tensor. k (x, e(u) ) is the apparent viscosity, is the Frobenius norm.

Non-Newtonian Fluids Nonlinear stress tensor T = pi + k (x, e(u) ) e(u) e(u) = 1 ( 2 u + ( u) ) is the rate of strain tensor. k (x, e(u) ) is the apparent viscosity, is the Frobenius norm. Conservation of momentum and mass «u ρ + (u ) u t = p + (k (x, e(u) ) e(u)) + f u = 0

Simplifying the System «u ρ + (u ) u t = p + (k (x, e(u) ) e(u)) + f u = 0 Assuming that the flow is tightly confined and slow, we can drop the non-linear term and neglect inertial effects. Also, we assume steady flow. u + (u ) u = 0 t

Choice of Constitutive Relation Possible choices are: Stokes flow: k( e(u) ) 1 Power-law model: k( e(u) ) = µ e(u) r 2, 1 < r < Carreau model: k( e(u) ) = µ + (µ 0 µ )(1 + λ e(u) 2 ) (θ 2)/2, µ 0 > µ 0, λ > 0, θ (0, )

The Case for Numerical Simulation Mathematically non-trivial problems can be contemplated

The Case for Numerical Simulation Mathematically non-trivial problems can be contemplated Experiments involving complex industrial fluids and processes can be expensive and time consuming

What exactly is Numerical Approximation? A two-stage process Discretize the PDEs on some grid using a convergent representation of the solution

What exactly is Numerical Approximation? A two-stage process Discretize the PDEs on some grid using a convergent representation of the solution Solve the discrete problem accurately and efficiently Au = f

Current Numerical Approximation Methods Finite Difference Methods Derivatives approximated locally using Taylor series expansions Finite Element Methods A versatile method based on the variational formulation Finite Volume Methods A method suited to conservation equations Spectral Methods Approximations to derivatives are global.

Outline of The Finite Element Method Consider a much simplified problem; 2 u = f in Ω u = 0 on Ω

Outline of The Finite Element Method Consider a much simplified problem; 2 u = f in Ω u = 0 on Ω Multiply by test function v and integrate over Ω ( 2 u ) v dω = fv dω v V, Ω Ω u v dω = fv dω v V, Ω Ω a (u, v) = (f, v) v V.

Outline of the Finite Element Method The problem is now to find u in V such that holds for every v in V. a (u, v) = (f, v)

Outline of the Finite Element Method The problem is now to find u in V such that holds for every v in V. a (u, v) = (f, v) Let V h be a finite-dimensional subspace of V, then we seek u h in V h such that a (u h, v h ) = (f, v h ) holds for every v h in V h.

Outline of the Finite Element Method We seek u h in V h such that a (u h, v h ) = (f, v h ), v h V h. Let {φ 1,..., φ n } be a basis of V h, then u h = n U i φ i i=1

Outline of the Finite Element Method We seek u h in V h such that a (u h, v h ) = (f, v h ), v h V h. Let {φ 1,..., φ n } be a basis of V h, then u h = n U i φ i i=1 Then the problem can be expressed as ( n ) a U i φ i, φ j = (f, φ j ) j = 1,..., n. i=1 This is system of n equations in n unknowns; Au = f

Variational Formulation of Governing PDEs Find u V = [W 1,r 0 (Ω)]d and p Q = L r 0 (Ω) = L r (Ω)/R a(u, v) + b(p, v) = (f, v) v V b(q, u) = 0 q Q.

Variational Formulation of Governing PDEs Find u V = [W 1,r 0 (Ω)]d and p Q = L r 0 (Ω) = L r (Ω)/R a(u, v) + b(p, v) = (f, v) v V b(q, u) = 0 q Q. Here Z a(u, v) = k(x, e(u) )e(u) : e(v) dω Ω Z b(q, v) = ( v)q dω. Ω

Variational Formulation of Governing PDEs Find u V = [W 1,r 0 (Ω)]d and p Q = L r 0 (Ω) = L r (Ω)/R Here a(u, v) + b(p, v) = (f, v) v V Z a(u, v) = b(q, u) = 0 q Q. Ω k(x, e(u) )e(u) : e(v) dω Z b(q, v) = ( v)q dω. Inf-sup condition [Amrouche & Girault (1990)]: c 0 > 0 s.t. Ω inf sup b(q, v) c 0 q Q. q Q v V q Q v V

Finite Element Approximation Let V h V and Q h Q be finite-dimensional spaces consisting of p.w. polynomial functions, defined on a triangulation T h = {T } of the computational domain Ω.

Finite Element Approximation Let V h V and Q h Q be finite-dimensional spaces consisting of p.w. polynomial functions, defined on a triangulation T h = {T } of the computational domain Ω. Find u h V h and p h Q h such that a(u h, v h ) + b(p h, v h ) = (f, v h ) v h V h b(q h, u h ) = 0 q h Q h.

Some Preliminary Numerical Results r = 2 r = 1 + 1 2 r = 1 + 1 4 r = 1 + 1 8 r = 1 + 1 16

What are we doing? Numerical simulation of incompressible, viscous extrusion flows for shear-thinning power-law fluids.

What are we doing? Numerical simulation of incompressible, viscous extrusion flows for shear-thinning power-law fluids. Accurate capturing of the thin boundary layers in the flow.

What are we doing? Numerical simulation of incompressible, viscous extrusion flows for shear-thinning power-law fluids. Accurate capturing of the thin boundary layers in the flow. The accurate prediction of the free surface between two pastes with different rheological properties flowing in channels or extruders.

Adaptivity in a Channel 48 simplices;113 nodes 124 simplices;281 nodes 332 simplices;725 nodes 1064 simplices;2241 nodes

A Posteriori Error Analysis How to quantify the size of the error u u h, in terms of a computable bound? p p h

A posteriori error bound Theorem. Let (u, p) V Q denote the solution to b.v.p., and let (u h, p h ) V h Q h denote its finite element approximation. Then, there is a positive constant C = C(K 1, K 2, c 0, c 0, r, f V ) s.t. u u h R V + p p h ˆR Q C S 1 R V + S2 ˆR Q, where R = max{r, 2}, ˆR = max{r, 2}, 1/R + 1/R = 1, 1/ˆR + 1/ˆR = 1, and S 1 and S 2 residual functionals which are computably bounded. [Barrett, Robson, Süli (2004)]

Some Numerical Results r = 2 U V P

Some Numerical Results r = 1.3 U V P

Some Numerical Results r = 3.3 U V P

Some Numerical Results r = 2 r = 1.3 r = 3.3

Conclusions, ongoing and future research We developed the a posteriori error analysis of finite element approximations to a class on non-newtonian flows. Ongoing research: implementation into an adaptive finite element method in 2D. Future work: application to multiple fluids, time-dependent problems in time-dependent geometries.