VISCOUS FLUID FLOWS Mechanical Engineering

NEER ENGI STRUCTURE PRESERVING FORMULATION OF HIGH VISCOUS FLUID FLOWS Mechanical Engineering Technical Report ME-TR-9 grad curl div constitutive div curl grad

DATA SHEET Titel: Structure preserving formulation of high viscous fluid flows. Subtitle: Mechanical Engineering Series title and no.: Technical Report ME-TR-9 Author: Kennet Olesen Department of Engineering Mechanical Engineering, Aarhus University Internet version: The report is available in electronic format (pdf) at the Department of Engineering website http://www.eng.au.dk. Publisher: Aarhus University URL: http://www.eng.au.dk Year of publication: 2014 Pages: 31 Editing completed: November 2014 Abstract: This report contains the progress status of the PhD project titled Structure preserving formulation of high viscous fluid flows. It describes the first ideas to a numerical scheme, which conserves mass and momentum in a discrete sense. It is based on spectral expansion polynomials, and it is thought to be applicable to arbitrary constitutive fluid models. Keywords: Computer aided engineering, Continuum mechanics, Numerical Modelling, Fluid mechanics, Non- Newtonian fluids, Spectral methods, Structure preserving methods Referee/Supervisor: Bo Gervang (main supervisor), Henrik Myhre Jensen and Marc Gerritsma Please cite as: Kennet Olesen, 2014. Structure preserving formulation of high viscous fluid flows. Department of Engineering, Aarhus University. Denmark. 31 pp. - Technical report ME-TR-9 Cover image: Kennet Olesen ISSN: 2245-4594 Reproduction permitted provided the source is explicitly acknowledged

STRUCTURE PRESERVING FORMULATION OF HIGH VISCOUS FLUID FLOWS Kennet Olesen, Aarhus University Abstract This report contains the progress status of the PhD project titled Structure preserving formulation of high viscous fluid flows. It describes the first ideas to a numerical scheme, which conserves mass and momentum in a discrete sense. It is based on spectral expansion polynomials, and it is thought to be applicable to arbitrary constitutive fluid models.

t ts Pr tr t 2 t s r t s r t r2 ts t t s t t 1 r r t s r r t t r2 t r2 r 1 t t s s t r r r t t 1 r t r 2 ss r2

tr t r t s s r t rt t2 s s rt rs t t q t s r t s r t t t st t t r t t r s t r st t t t st t t r t s 2 r t s P r t s t r r str t r r s r r t t t s q t r r tr r2 r ss r ss t r r s r t r s s t s t q t s r s t r ss r s s str ss s t st t t r t r t s s s r t q t r s r t t st t t q t s s t s t s2st s t t2 ts t r ss r t s s str ss ts r s2st s st ts 2 2 2 t t2 ts t r ss r r s r t s ts s t t t s2st ss r 2 s s t 2 t t r t s 2 t 1 r ss r t str ss s s t r t s t s t r tr r2 st t t s r r t r t s t r t t r r t t s r s t st s s t t s t s r s 2 t s t t s r t r s r t s s t t t t r2 s t r 2s tr rr r s t t r t r r t t s 1t s 2 r 2 r t tr2 s t s r t 2 t s r rt s t s t 2 t t r2 t r t t r r t t t r s s r r ss r rt s str t r s s rst r 2 s 1 1 s r t 2 s st r 2 t ss t s r 1 1t t t r2 t r r t s 1 r 1 s t s t s r r r t t r str t r r s r r s 1 r rt s s 2 s t r r s t

2 t r 2 s t t t s t r 2s s r t t s tt r 2s r s s r 2 t t s r t t ss s r t (ρu) t + (ρuu) = Π+f ext, ρ t + (ρu) = 0 s st t t q t s r s t r t t str ss s r t 2 s t st 2 s r t s ρ s t s t2 t u s t t2 t r Π s t t t str ss t s r f ext s r t r s r t 1t r 2 r s r t2 2 f ext = ρg r g s t r t t r t t r r 2 t t t ss r tr r2 1 s r t r s r q t t t t r tr t s t t str ss t s r Π s t t 1 r t r s t r tr r2 t s t rts Π = pi +τ. t r 2 r ss r p s r t t t s t2 t r t r t t r q t st t t s s s str ss t s r τ t t r s r t t t r t r t t s 3 r t s st t r2 r s t r s t s s r q r t s r t s r t t r r 2 st t t r t s r t t t r t r t t t r r 2 t r r s t t 2 ss t t t s2st s s t r s s st t r t 1 t2 t t r st t s q t s r ss t2 ss t tr t t s t rr rs s s t q t st t r t s t s t2 s st t s r s t ( ) (u) ρ +u u = Π+f ext, t u = 0. s t q t s r r ss s t r s t 2 t t t s r t s r t s r t r2 t s t t t s t s s str ss str r t r t r t tt r q t t s r s s t2 t t r st r t r2 ts r s t t s r r s ss s r r t t t s s str ss t r t r t s r t s t s t s r t s r t s s r2 t s t 2 t t t r t r st t s 2

τ = 2µ γ. γ γ = 1 2 ( u+( u) T). µ η( γ) = m γ n 1, n m η η η 0 η = (1+(λ γ) a ) n 1 a. γ η

r2 ts r n s t t r t s t 1 t r η 0 s t s s t2 t t s r r t s η s t s s t2 t t s r r t s a s t r t r t t t tr s t r λ t r s t s r r t r t tr s t st rts t s s r t t t r s r t s s t 0 < n < 1 s r t s r t n > 1 r t rr s 0 < n < 1 s r t s s t η 0 > η r s r t η 0 < η t s r r r r r s r t t r s t 2 rr s s t t t t t r r r ts t t s t r t 2 r 1 r t t r2 ts s s r t 2 s t s s ts t s t t t r 2 s ss t r r s t r r r t s r st 2 s s st ts r rt t str ss s t r r r t s s s s ts t ss r t s s r s s r ss r t s s r st q s st rr t t 2 ss t t t tr r s t 2 r s t t s s r t t t s t s st 1 r t t s s t 1 r s t rs s t s str t r r sq 3 t s s t t r r r s s s t r s t s s t s s s s r st 1 ts r t r s t st str ss s r r s t q str r s s t r 2 t r r t st str ss s t s s st s t t t s s s t2 t s s s t2 s r r tt t r r t s t t 1 3 r t t tt t s st t r2 t t s t2 V s r s s str ss s r τ yx.visc = µ du dy = µ γ yx(t), r u s t 1 t t t2 t r st s r str ss s 2 s r s s ts τ yx.elas = G dd dy = Gγ yx(t ref,t), 10 8 10 6 Power law Newtonian λ = 0 λ = 0.01 λ = 0.1 λ = 1 λ = 10 λ = 100 10 8 10 6 Power law Newtonian λ = 0 λ = 0.01 λ = 0.1 λ = 1 λ = 10 λ = 100 viscosity 10 4 viscosity 10 4 10 2 10 2 10 0 10 4 10 2 10 0 10 2 10 4 strain-rate 10 0 10 4 10 2 10 0 10 2 10 4 strain-rate r r s t r t rr s r s r t P r n = 0.1 m = 10000 rr s n = 0.1 η 0 = 10000 η = 10 a = 2 r r s t r t rr s r s r t P r n = 2 m = 10000 rr s n = 0.1 η 0 = 10 η = 10000 a = 2

G D γyx (tref, t) t tref ˆt γij (tref, t) = γ ij (t ) dt tref τyx + µ τyx = µγ yx (t ), G t τyx t τyx t = Gγ yx τyx (t) = G t 0 τyx t τyx γ yx (t ) dt = Gγyx (tref, t) tref f = GDspring, Dspring f = µ dddash, dt Ddash Dtotal = Dspring + Ddash. u=v Initial state Dtotal y Final state x u=0 f

t t s t t 1 t t r t s rt s f + µ df G dt = µdd total, dt t r r t s t 1 r 1 t t s s λ = µ G s r s t st ts r r s t t r t r 1 t t 2 s s ts r r s t t r r r r r t s r tt t s t t G µ t λ t s2st 2s t 2 st ts G µ t λ 0 t s2st r s st st t st st t s 2 r r t r s r s t s2st s t t r q t s t t s t 2 rt s r s r 2 rt ss t s r 2 1 r s 2 r t s r q t t s str ss str r t s t t s r q t t s τ +λ τ t = η 0 γ. r s s t r rs t s r s t t r 2 e λ t λ t r t r st t s t t t t r st t 2 r q r t t t str ss s t t t = t t r r t 1 s t τ(t) = ˆt [ η0 e (t t ) λ λ ] γ(t ) dt, r t t r t sq r r ts r t r r tt t γ(t ) s t str r t t st t s t s s t t t t 1 t t r r s 3 r t t t t str ss s s r s r t r r tt t r s s str rs 2 τ(t) = ˆt [ η0 2e (t t ) λ λ ] γ(t,t ) dt, t t s t t 1 s r 2 st t t t t r q r ts t t r st r 2 t s t t t s r r t s t r t s t s r r ss t r t r t s2st rt s r r t t t 1 s t t s s t s t r t 1 r t r t r s s 2 t t t st s t r tt s r 2 s s s 1 t r t r t t t st t r t2 Ω 2 ss t t t s s r 2 t 1 2 t t str r t t str ss r t t r r rt s s t r r r t s r t r r r s r t r rr s t t t r s t t t t 123 r t s2st t t r r rt s r t Ω s t s s t s s t 1 s st t t t t s s t r s t s str r t s st r t t r s t tr s r t t s r q t t t t s r q t r s t t t t s str t s r t t s r rs s t tr s r r t 2 r st t r2 r r r t r t t 2 2 t rt r t t τ t q t s r r t q t

2 y y x y 0 r r t s x 0 r str t t t r t 1 x t r 1 s t t t s s t t s str t s r s t s r t r 2 r t t r t r t r t s 2 s str s r r r 2 2 t r 2 t s s t r t s t r s t t r st t s t t 2 t s r A r rs A 1 1 sts s t q s t s A = R U = V R, r U =(A T A) 1 2 V =(A A T ) 1 2 A R = (A T A) 1 2 = A U 1. R s r r t t t s r t s t r t r tr r2 t r t t s t ts t s s t t U V t t str t r t A t 2 r r t t 2 r str t t s rs r s t 2 t t r t str t t s r r t t t s r s s r r tr r2 t r u s tr s r t v 2 t t s r A t s t s r s s t r t t rt R str t rt U t U tr s r s u t w R r t t s w t r v w v q t str t r t U r t t rs r u t w s t r t s t r t r t str t t s r s tt r r t t s t s t U V r t tr s r t s r r t r r t U r 2 ts t rs ξ i t s r t t t t r r t V 2 ts t rs ζ i U V r t str t r tr r2 t r t 2 t s λ U s s t rr s s t t s t r r rst t str t s r 2 U t t r t t s 2 R s V s t s s t t s r 2 rst 2 t r t t 2 R t str t t 2 t s V s r 2 r r t s r r t st t t r tr r2 ts r s r t t st s t r t t t t r 2 ts 1 r t t t t t t s 2 dr = dr r r.

r r t s t s r r r s t r t r t t s r F(t,t ) = r r. t r t 2 dr = dr r r, r r r s t rs r t r t t s r F 1 (t,t ) = r r. r t s r t t r 2 r s t t t t r 2 r t t s r s t s 2 s t s C = F F T C 1 = (F 1 ) T F 1, r C s t 2 str t s r C 1 s t r str t s r t t s str t s r s t t r str rs t r 1 s r s t str r st t s t t rr t t t t t s str t s r γ s r t t r str t s r C 1 τ(t) = ˆt [ η0 2e (t t ) λ λ ] C 1 (t,t ) dt, s s t q t s t s t st t2 s s r rr t 2 t t r s r r ts r t r s 2 r t t t r s t t t 2 t r t 2 rts s t r t r r t t s rt 2 s ] τ λ + [ dτ dt ( u)t τ τ u = η 0 λ I, r u s t t2 t r I s t t t2 t s r r t s t ss t s r t r s s str ss s r t r ss r r t s ts r s t r s s str ss r 3 r s r s τ s t r r 2 t s tr st t s 2 str ss t s r t ς = τ + η 0 λ I, u U R U R A w v i i R V r P r s t r t t r t str t t s rs

2 P At time t' dr' Q z r' r P dr At time t Q y x r 2 r r t t q t s [ ] dτ τ +λ dt ( u)t τ τ u = τ +λ τ= η 0 γ. r t 1 r ss t r ts s t r t r t t str ss t s r r t r t str ss t s r t s t 2 τ q t s t r t 1 t r t s r t s s t r s t r t s q t t 2 2 t s st t r r 2 s s t r t st s t 2 s t ˆx 1ˆx 2ˆx 3 r t s2st tt t t r s t t 2 t t t = t t t r r s t t st t r2 xyz r t s2st t t s t r ts P Q S t s t r t r r t s t t s s ê 1 ê 2 ê 3 r t t r r 1 r ss s ts r t s 2 r = ˆx 1 ê 1 + ˆx 2 ê 2 + ˆx 3 ê 3, dr = r ˆx 1dˆx1 + r ˆx 2dˆx2 + r ˆx 3dˆx3. s s t r r t t r t s2st s 2 g (i) = r ˆx i s t s s s t rs t 1 r ss t s s str ss t s r r t t t t rt s r t s2st rr s t Material grid at time t' dˆτ dt = dτ dt [ τ u+( u) T τ ]. S P ^x 2 Q y, ^x 2 Material grid at time t Q P x^ 1 S x, ^x 1 r t r r s r t r t st s t

r r t s s t s s str ss t ˆx 1ˆx 2ˆx 3 r t s2st r t t t r s t t t t s s t t t s t r t str ss t s r τ t t rr s t t s t r t r 2 t r t r r t rr s r r s r t r s r 1t s s t t 1 s t s rt 2 t ❼ r t r 2s r r 2 τ +λ 1 τ= η0 ( γ +λ 2 γ ) r 2s t s t str r t t r r t s t t st t λ 2 s t r t r t t s r s t s r t str ss ❼ t t3 r τ + η( γ) G 0 τ= η( γ) γ t t3 r t s s r t t t t 2 r t s s t2 t r G 0 s st t s r t r ❼ r 2 st t 1 τ +λ 1 τ + 2 (λ 1 µ 1 )( γ τ +τ γ)+ 1 2 µ 0(tr(τ)) γ + 1 2 ν 1(τ : γ)i [ = η 0 γ +λ 2 γ +(λ2 µ 2 )( γ γ)+ 1 ] 2 ν 2( γ : γ)i s r s 2 r 2 t r t s 2 r2 t st ts λ 1 λ 2 µ 0 µ 1 µ 2 ν 1 ν 2 r s r s t r st t t s ❼ s s τ +λ τ + αλ τ τ = η 0 γ η 0 s s r r t s r str ss t r s r r ts

r r t t r2 s t r s r s t ts s r r r s tr s s tr t t r t tr t s 2 t s t r ss str t r r s r t Pr rt r P t r q t s r t t r t r r 1 s s t s s 2 t s ts t s t r s2st q t s t s s s r2 2s r r t q r q t s r t t r tr r2 rt s t t t r s t r t r t t r st r rt r r t s t r t r r t t ss r t r t r t q t s r tr t s r t t r r 2 r s 1t r r s r r 2 t t t t t s 2 s s r t t t s st rt s t rt t2 s t t r q t s r t t rt r s r 2 t s t t t s t r r t t r q t s s r tr r2 t st r t t s rr s s t r t 2 r q t t s t r s r 33 r ss t st t s r s r t t2 r ss r t t t r s s r t t s t t r t t r t st s s r 1 r s tt rs3 r s t t t r2 s t t r s tr t s t r s q t t t r t r t s t s 2 r s r ø q st r s s tr t s s s r r r r t s r s t t s r t tr t t r2 tt s r P rt r t q t P L(u(x)) = f(x), r u s t f s t t t t s t r t s s2 s 2 t s t t r x P tt t s r t r 2 t 2 t t t t rt t r v(x) t r t r t Ω ˆ ˆ (L(u(x)),v(x)) = (f(x),v(x)). Ω Ω

t r2 t r r t r L t q t r r tt t r r u(x) v(x) r r s t s s r s 1 s u(x) = a i φ i (x) i=0 v(x) = b i ψ i (x), i=0 r a i b i r ts φ i ψ i r s s t s r 1 t s r tr 2 tr t t s t t s s N u(x) u N (x) = a i φ i (x) i=0 N v(x) v N (x) = b i ψ i (x), i=0 r t s t s t t t t r tr t r t tt r r 1 t r 2 r t s r t t r r s r r t s tr t s t s t r 1 t t 1 t t t s s s t t s t s s r s r s t s t t r s r t r 1 t s s tr t s r 2 t s t r s t t r s s t r t s t ts t t t r s t 1 s t r r 2 r t s r t t b i r t rs s t s r r r t r 3 r t 2 t r r t s s r t ts a i s s r r 2 s r t 1 s 2 s ψ i (x) r s t r st ❼ t t s 2 s tt ψ i (x) = δ(x x j ) r δ(x x j ) s t r t t q s t t t t x j t t 3 r r2 r s ❼ st sq r s t s s t (R,R) r R = f(x) L(u(x)) s t r s r 1 t s t s 3 r s s 2 s tt ψ i (x) = R a i ❼ r t s ts t 1 s 2 r t t st t t q t t 1 s 2 r t ψ i (x) = φ i (x) ❼ P tr r t s s 2 t r 1 s 2 t t 1 s 2 r t ψ i (x) φ i (x) t s t r r s t t st r s s t t t t t r t t t r s t t r r s t t r t t t t a i q s t s r t t u N (x) t t t t t t r t s s t st r r tt rs3 st sq r s t s t r rt2 t t t r ts t s t t t t s s2 tr tr 1 r t t r t r t r t s t2 s t s2 tr s s r t t t s t s t q t s2st s s t r 1 t r s s N t t 1 s 2 s r r r t t 2 t t st t s r t t t r s s t 2 r r r t r t t t t t s t s r r s ts s t s r t t s r rt t 2 r r 2 t t r s t s s r s t r t 2 r s s s t s2st q t s t t 2 s s t s r s t s t

r r t t r2 P 2 s r s t rt t r [a,b] ˆb a w(x)p m (x)p n (x)dx = δ mn c n, r w(x) s t t δ mn s t r r t c n s st t rt 2 s s s s s t 2 2 s t s r r r 3 r t r r r r s ss s 2 s t t s r rt2 t 2 r t2 2 t s t t t t r r t q t t st 2 s 2 s r t 2s t r 2 s t r t r t r t t r [ 1,1] t t 2s 2 s w(x) = 1 1 x 2 c n = r t r 2 s w(x) = 1 c n = 2 2n+1 { π r π 2 t r s 1 r t s r r s r r r s t s r t s t r t r tt t r q t s2st (v N,u N ) = (v N,f), Ma = b, r t ss tr 1 s 2 M pq = (φ p,φ q ) t r t s s 2 b p = (φ p,f) s t s 2 t 2 t t rs M r r t 1 s 2 s r r t t r [ 1, 1] ❼ t 1 s s 2 r s s t r r x s s t t t s t r r P t s s ts r r r X P 1 X P s X 2 = {1,x,x 2 } t X 3 = {1,x,x 2,x 3 } = X 2 {x 3 } ss tr 1 s 2 M pq = ˆ1 1 x p x q dx = { 2 p+q+1 r q 0 r q. ❼ 1 s s 1 2 r 2 s s sts P 2 s r r P 1 r r r (X P 1 X P ) s s t s 2 φ p (x q ) = δ pq s r s s t t t t t t r st t t t t t t t q t s r s 1 t 2 t t ts t 1 s t ts r r s t r 1 t s t r s 1 t r t ss tr 1 s t s t s r t r t t r q s s r t s tr 1 ❼ 1 s s r 2 s s s 1 r ss r t ss tr 1 M pq = ˆ1 1 L p (x)l q (x)dx = 2 2p+1 δ pq, s s s tr 1 s r2 s2 t rt s r t tr 1 s s t t r r t r t r t r t t tr 1 s r s s t t t r t 1 s 2 s t t r s r t L2 r κ 2 t r r r s2 tr tr 1 t L2 r s t r t t t 1 s r t r 2 1 s t r r P t s s κ 2 = 2 2 = 2P +1 2 r s s t t r t s t rs ss tr 1 s 2P+1

r 1 r t t r t t tt r t 1 s 2 t r t r rs r t r r 1 s r s t 2 t r t t t r s t s t t 2 r r r r s r r t t t r 2 1 s s t t r r s r t 2 s t 2 r r s r s s s r r r κ 2 10 P r t r 2 s rst st rts t s tr t r P 5 2 s rt 1 s s t ss t s r r r 1 t s t st t s2st s r t r s rt 2 s s t t t r r r r t s t t r rs t ts r t s s rr r t s r r t r t 2 s t t t t t r rr r t r 2 s s nl n (x) = (2n 1)xL n 1 (x) (n 1)L n 2 (x), L 0 (x) = 1, L 1 (x) = x (1 x 2 )L n(x) = nxl n (x)+nl n 1 (x) = (n+1)xl n (x) (n+1)l n+1 (x). r rr r t 2s 2 s s T n (x) = 2xT n 1 (x) T n 2 (x), T 0 (x) = 1, T 1 (x) = x (1 x 2 )T n(x) = nxt n (x) nt n+1 (x). r 1 r t s r r r r t s tr t s r t s s t t s t st s s t s r s 1 s 2 s t st s s d 2 u dx 2 +2du +10u = f(x) r x = [ 1,1] dx u( 1)+ du dx = 3 x= 1 u(1)+ du dx = 1. x=1 r f(x) = sin(x) t s t s t t r r r t s r r r t t2 r t s r t t t r t s s r r t t r t s t r t s r r r t t r t s 1 t

r r t t r2 4 3 2 10 5 10 0 Linear interpolation, Varying elements Quadratic interpolation, Varying elements Cubic interpolation, Varying elements Varing interpolation, One element Varing interpolation, Two element u 1 0 1 uex u 10 5 10 10 2 1 0.5 0 0.5 1 x 10 15 10 0 10 1 10 2 10 3 DOFs r P t t st s t f(x) = sin(x) r r s r r t s r r t t f(x) = sin(x) r r t s t t s tr t s s t r r r s t t s f(x) s s t t t s st t 0 r 1 1 f(x) = H(x) = 2 r 1 1 r 1 r s t s s r t s r2 s r t t s t s r t t r r t r t s t r2 t r t t s r 2 t r r t s s r t s s s t t r2 t r t t ts st 1 t r s s t t t t t t s t t2 t r t s t t r t ts 5 4 3 10 5 10 0 Linear interpolation, Varying elements Quadratic interpolation, Varying elements Cubic interpolation, Varying elements Varing interpolation, One element Varing interpolation, Two element u 2 1 uex u 10 5 0 1 2 1 0.5 0 0.5 1 x 10 10 10 15 10 0 10 1 10 2 10 3 DOFs r P t t st s r r s r r t s r r t

t t s s s t s r s t t r str t r r s r t s t r t r2 s t r t r r t rr t s st t str t r r s r r t s r s t q r q t s r t2 2 r 2 s r r tr r2 t t t 1t r t r tr t s s 2s q t t2 r s r 2 ss t r t q t s r t 2 2 t r s t r t r t r t t 1t r tr t s s r s t t t t tt t s 3 t r t s r s t s t t tr ss t s r st t q r r q t s s st s t s 2 s t 2 ts s 2 str t r r s r t s t t t t r s s r t s s t t t q t s r s t s t r s t 2 t r s r t s t r t s r t t st t t q t s r r r t s r s s r t s P r t t t s r 1t s 2 1 t tt r r r t t s r r t s s r t s t r P r t P r t s ts t t s q t r (p)+µ u = f, t t rt t2 ω = r (u) s s rt r (p)+µ r (ω) = f, s t ss t 2s q t t2 t tr t t t s t 1 t r t s 1 sts t r tr r2 tr ts t r r s 2 t t t r t r s ˆb a r (f) ds = f(b) f(a), t t s t r r (F) n ds = F ds, S t r t r (F) dω = F n ds. Ω Ω S r f s s r F s t r s s s S s s r s t t t r t r n Ω s s s s r s t s s r t 1 s r r t t 2s q t t2 st t t t r t t s r s 2s q t t2 s ss t t tr t 2s q t t2 ss t t

t t s u topological - intrinsically discrete inner oriented grad curl div 3 v constitutive outer oriented div curl grad topological - intrinsically discrete r s r t 1 r tr ss t s t t2 rt t2 s r r st r ss t s r t 1 r r t t r r t r t r t t t t s r t 1 t r r t r t r t t s s r r t s s s r t t t st t t q t s r t t t t r t t s P r t s sts t r q t s 2 ω = r (u) q t t r ss t t2 q t (u) = 0. r r s t ss t t s t tr ts s t t 1 t s r t r t s st s t rt t t t t2 q t t t s s r s t ss 1 r t s r s t s t s r ss t s t2 1 r s r 2 ū = u n ds r 2 t t 2 s r t t2 Ω 1 ss t t s r s s t t t t2 q t t s r 2 s s r 1 s t Ω N S (u) dω = ū i, r ū i s t t2 1 r s r t N S s t r s r s t s s t s s t t t t2 q t 2s s t s t s s s 2 s t t t2 1 ss t t t r s t s s s s q t ω = r (u) s s t s 1 t s r t 2 r s s t t s r t t2 1 ts ū v r s rt t2 s t r t t t s t t t2 1 s r r t t r ss t s r s t r t t s r t s st ss t t t t t s r s s r t rt t2 t r t r t s t s s s s ω = ω ds t t r t t s t r t t2 1 S s r 2 s t rt t2 t r s r ts s Ω S i=1 N s ū = u n ds = r (ω) n ds = ω i, r ω i s t rt t2 t r s r t s r N s s t r s t s r t s t ss t 2 2 t t r s t s 2 t t 1 t s r t r t s t s t s q t s t r t t s t s t s r t s s s rt t r 1 t s s rt t s t t q t s t t s t s tt r s 3 s t s t s r t t i=1

s s r 2 t t s q t t r t r r (p)+ (τ) = 0, t r ss t t2 q t t s s s t t r t q t s r t t t r s t s s r 1 t r s s s s s r t s st t t q t r s s t t s s s t2 t q t s r s r t r st t t q t s r s t t τ s s r r t s r t r t s st t t 1 r 2 s r s t r t t s r s t r s 2 2s t 2s q t t s r s r t 1 t r t r t q t s r r 2 t t r r tr r2 r t t s q t t rt t r s t t r s r τ p r 1t r tr t s t s r t r t r f s t 2 r t t s s s t t 3 r t s r t s s t r s s r s t s s str ss t r t r t ss t s r s r s r r r s t 1 t s r r ss r s r r r t t s r t s t s s tr t s 3 t r r 2 t t r t s r t s s t s t t t t s 3 s t s st t r ss r s r r t t ss t t t t s tr s r s s rr s s t t s t r r t r ts s rs s r t ss s r 2 t t ss r tr r2 t 2 ss r ss t2 t 2 tr t s t ss 1 r ts r s s s t tr t r r t t t t2 r t t t s s t 2 t t str r t s st t t s str r t s t r t s t t t r tr r2 ts r s str r t s tr 2 ss t t s t ts t2 s t r r t r 2 ss t t t ts t str r t r r r t tr t r r t t r t t2 s t r ss t t t s s t t r r t r t t t2 q t t r 2 ts t rs r s s t t t s r t s r 1 r r str t r s s t t t t t2 ts r tr 2 s r t t t t 2 s t t2 1 s r t s r s t s t s r t q r r ss r s s r s s s s r t r t 1 r t r t r t 2 r t r t r t t s r t s r s r 2 r r r r s t s ss tt r ts r t s t r t str t r r s str t s ss r ts r t r s t r t str t r s s r r P r t r q t s r s r r s t t t r t r st t t r t s r s s r r s t t t 2s q t t s r t t t r t r t t t t t r s ss t t t 1 t r 2 t t 1 t2 ts t r s ss t t t 2 t r 2 t t 2 t2 ts t t s s t st t t r t s s s t s st t t t 2s r r str ss s s t s s s t str r t s s t r t t t2 t str ss s r r r t t t2 t sq r tr 1 s2st t q t s r t t r t r t t t r t t2 q t s s t t s s s t r ss r t t

& & + $ $ $ +, $ + $ +, ++, + $ ① + & / & & 0 & + + + & & + + + & &② & & & / & +& & & 0 & + + & & + + + & + + / + + + & &/ & / / & / / & ++ & + / & / + / & & / &② & +& + & + + / + & & + &② / + / & & / + & + + & ② / / + / + + + / + / & + +& & / & & & / & + & + + / + + & + + / & ② & + + / & & & &② & +&/ ++ &② + & + & + & &② + + /& ② ① / 0 / & / & & + / & & +&/ / & + 0 &② ++ & & & / & +&/ / & & + / & & + & / & & &② & &② + & 0 +&/ / & & + + & & + / & + & + & & / & +&/ / & + ② / / &② + & &② / &/ /② & + + / & & + + + ② & / & & + / / 0 / & & & && / & / & & & & + + & & / + & + & + + + & & &② / 0 / & + / 0 / & + + & ②+ + & + / & & + &② / ++ / / & / & +&/ ++ + & + / / + & &② / 0 / & & + / & / & & + & + + & / & / ③③ + ② & & & / ++ / / ① & + +& + / & & &② / ① & + ② S & / & + + & & / + &/ &+ & / ++ / / ① & + + PN 2 & &② / ① & + + PN +&/ & / / + / & & + / 0 / & & & +&/ ++ + / + / + & &② & +&/ ++ &② + / 0 / // &+ S + & & &② / 0 / &+ / & &② S/ ++ / &/ ++ S / & + +& & &② / ++ / / 0 / & +& + ② S & / & &② +&/ ++ / 0 / & + & & & & / ① & + & +&/ ++ +! $! & + + ③ + & &② + + + / 0 & /& & + + + & +&/ ++ + + + / & & &② + + + & // & +& & + & / & // & & & ++ & &②

1e 05 0 1e 05 0.01 0 0 y y 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1e 05 0.0001 1e 05 0.001 0.0001 0.001 0 1e 05 0.001 0.01 0.001 0.05 0.07 0.07 0.03 Streamlines, N p = 6 0 0.09 0.05 0.03 1e 05 1 0 1 0.5 0 0.5 1 x 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 0.07 0.05 0.07 0.05 1e 05 1e 05 0.01 0.001 0 0 0.03 0.01 0.001 0 Streamlines, N p = 24 0.09 0.03 0.01 1e 05 0.0001 1 1e 05 1 0.5 0 0.5 1 x 0 0.001 0.01 1e 05 0.0001 1e 05 0.001 0.001 0.001 0 0 1e 05 y Inifinite norm of flux balance 1 0.8 0.6 1e 05 0.4 0.2 0 1e 05 0.2 0.4 1e 05 0.6 0.8 0.0001 1e 05 0.0001 0.01 0.01 1e 05 0.001 0.001 0.07 0 0.03 0.05 0.07 0.05 0.09 0.03 1e 05 1 1 0.5 0 0 0.5 1 x 1e 05 13 x 10 7 mass x momentum 6 y momentum 5 4 3 2 1 Streamlines, N p = 12 1e 05 1e 05 0 0 2 4 6 8 10 12 N p 0.01 1e 05 1e 05 1e 05 0.001 1e 05 1e 05 0.0001 0.0001 r tr ts t r t 2 r r t st str t s s t ss t s r t t ss t s r s t 2 t t t t2 r s t s r r s r t str ss t 2 1tr str ss r2 t s t s r t t t str t r s r s r s st r s r t s t t r t r s rt t s t r t s r t r ts t st t s t t s ss t r r ss r rt r s s rt t t r t t s r rt

t t s P t s r r rr ts str t s r s r s s r t ss t t s t r s 2 s t s s t s r r t r t ss t r 1 t 2 s r t L r t s t t r t2 r tr t s r t s st t t q t s 2 r2 st t r s 2s s s r t s 1 r ss t r t s t s t s r t r s r s st t t q t s r r s t r s r t s r t r ss s t t r s s r t s r t r r t rs st t t q t s t r t rs r t r 3 t t r r st 2 r r st s r t t s r t s 1 t 2 r r s t t t s r t t r r 1 t rs t st t t q t s t s r t t 1t t s s t t r s r t s s s s r t r t t s r r t s t s 2 s r t ss 1 t 2 t t s r t t t2 s s st r s r t t s 2 r t s st r t t s s r s t t s t r t2 ts r r 2 t t r2 t t t t2 ts r s st ts s s t s s s q s r t t q t r t s r s t t r str ss ts r tr t r t 2 r t s r str ss t t r t r s t t t s tr s s t s t tr t t r2 t s tr s ts t s s tr s t r t s t r t2 s s t r r r s ss t r t t s r t t Ω t s t3 r2 Ω t t r ss t s t r t t s s r 2 t t q t σ = r (p)+ τ = 0 Ω, t t r t s r t ss r r ss u = 0 Ω, s t 2 s st t t s t str r t t t str ss t s r t s s s t2 s s τ = 2µ D. t s r s rs t2 rt t r s r s r r r s rs t2 rt t r s r s r s r rr ts t rs t2 2 t2 r s r 2 r t t r s rr ts t t

r p s t r ss r u s t t2 τ s t s r r s s str ss t s r µ s t st t s s t2 t t D s t s2 tr rt t str r t t s r D = 1 2 ( ) r T u+ r u. s s t 2 t r2 t s u = 0 Ω. r r r t s tr t t s r t s str t r t rr ts s t rt t2 t2 r ss r P r t t s r P r t s 2 r st t s s t2 ts s t s r s r t r t s r st s s t t2 r ss r str ss r t s r st t s s t2 ts s r t s r q r s r r s s r s t t r s r t t r s r t s tr t t rr ts P s 1 t2 r ss r str ss r t s r r r t s r s r r s t s t 2 t L( v,q,σ) = 1 (σ : σ) b(σ, v)+(q, v), 4µ r q L 2 (Ω) v H(div;Ω) σ [H(div;Ω)] 2 2 s s t s s2 tr 2 t s rs t sq r t r ts ts r s sq r t r t t t s r σ 2 ( σξξ σ σ = σ σ ηη ), t t r v = (u,v) T t t r r b(σ, v)) s 2 ˆ ( b(σ, v)) := σ ( ξξ u ξ u+σ η + v ) σ ) ηη ξ η v dω. Ω st t r2 t ( u, p, τ) t r s t s t s 2 t r t r t ( v,p) b(τ, v) = 0 v H(div;Ω) ( u,q) = 0 q L 2 (Ω) b(σ, u) + 1 2µ (τ : σ) = 0 σ [H(div;Ω)]2 2 s s s2st st t t s s2 tr 2 str 1 r t s rr ts P s r s ss t r s t s s t s t s 2 r r t s t s t s r t s tt s t t s s s s Q L 2 (Ω) V H(div;Ω) T [H(div;Ω)] 2 2 s r t r t st t t r s ( u h,p h,τ h ) V Q T s t t ( v h,p h ) b(τ h, v h ) = 0 v h V ( u h,q h ) = 0 q h Q b(σ h, u h ) + 1 2µ (τh : σ h ) = 0 σ h T s s r t s s t s t s 2 t t2 t s t s r q s t t Z V V t s s r t t r s v h s t s 2 v h = 0 t ZV Q s t 2 ss s r s ZV s t r 2 r 1 t ss s r t ZV s t s s r s r ss r s t s r s t t s s s V Q s t s 2 ZV Q st str t r t rr ts s t s r

t t s r 2 t s t t t s t2 s s V T t s s r t s Z T V s st t r s v h r t s2 tr rt t t2 r t s s Z T = { v h V v h + T v h = 0} { v h V b(σ, v h ) = 0, σ [H(div;Ω)] 2 2 s } t t t Z T s t s s 2 t s tr s t s r t t s t t2 r q r s t t ZT T Z T s t r s t s 2 s r t t 2 t s r s t2 s ZT s t s t 2 r 1 t s r t t s r s r t r t t r2 t s r t t2 s t s r t t t s 2 t s Z T = tr t s s t s t s r s s s t s t ξ i i = 0,...,N t ss tt r ts 2 r N ξ i i = 1,...,N t ss r ts t t t ξ i 1 < ξ i < ξ i r i = 1,...,N rt r r tr t 1t ss r ts ( ξ 0 = 1, ξ i, ξ N+1 = 1) t ss r ts s t ts 1 1 r 2 s ss t t t ts t 2 h i (ξ) t r 2 s ss t t t ts t 2 h i (ξ) t r 2 s ss t t t ts r rr t s h e (ξ) r t r 2 s str t t 2 t s 2 rr ts i 1 e i (ξ) = dh k (ξ), i = 1,...,N k=0 i 1 ẽ i = d h e (ξ), i = 1,...,N +1. e i (ξ) s 2 r N 1 ẽ i (ξ) s 2 r N t t s s s t s 1 r ss t t2 s tr t s k=0 N N N N u h (ξ,η) = u i,j h i (ξ) h j (η)+ v i,j hi (ξ)h j (η). i=0 j=1 i=1 j=0 t t t t ξ t t t2 s 2 r N t ξ r t 2 r N 1 t η r t u P N,N 1 r 2 t η t t t2 v P N 1,N t s t t u P N 1,N 1 t r r t r ss r 1 s N N p h (ξ,η) = p i,j e i (ξ)e j (η). i=1 j=1 s t t2 r ss r 1 s r t s t t ZV Q t r r 1 t ss s r t t t s r s r ss r s 2 s r t r ss r s t 2s s r st t s t t t r ss r s t r t st t r t s 2 s 1 1tr str ss ts s τ h ξξ(ξ,η) = ˆ N+1 Ω i=0 j=1 p h (ξ,η)dξdη = 0 N i=1 j=1 N N τ ξξi,j he i (ξ)e j (η), τηη(ξ,η) h = N p i,j = 0. N+1 i=1 j=0 τ ηηi,j e i (ξ) h e j(η), N N τξη(ξ,η) h = τηξ(ξ,η) h = τ ξηi,j h i (ξ)h j (η). i=0 j=0

t t s 1 s s t t τξξ h N+1 ξ = i=1 j=1 i=1 j=1 N (τ ξξi,j τ ξξi 1,j )ẽ i (ξ)e j (η) P N,N 1, τηη h N η = N+1 (τ ηηi,j τ ηηi,j 1 )e i (ξ)ẽ j (η) P N 1,N. t r s 2 r τξξ h/ ξ s t s s s t ξ t uh τ ηη / η s t s 2 s s t η t u h r s u s t 3 r t t r t r2 t str τξξ h 2 s tt τ h ξξ ξ = 0 r ξ = ±1. r 2 s τ h ηη η = 0 r η = ±1. t t s t str ts s r t t τξξ h / ξ 1 r ss s t s r t u t2 s rs r2 u 1 r ss t r s t s τξξ h r t s s s r τξξ h / ξ v r t r t t s r str ss t s r t 3 t s r r t s u/ η v/ ξ s r 2 N u η = i=0 j=0 v N ξ = i=0 j=0 N (u i,j+1 u i,j )h i (ξ)ẽ j (η), N (v i+1,j v i,j )ẽ i (ξ)h j (η), t t t tr 1 s ts r t t t2 1 s s t ts r s t s rt t r s r t t t2 ts t r2 r r r t u i,0 r u i,n+1 r t 2 t t t t2 r r r t v 0,j v N+1,j r t s s 2 t r s r t t t2 t t t r t r2 r s t 2 t t t u/ η v/ ξ r t s 2 s s t s r str ss r r s t t

s t t t s r r ss r rt r ts t rr t st t s t s P r t r t2 t t s r s t s t r rst t t r2 r 2 r r t s t t t s s t t r q r ts rs s str t r r s r r t t t s r s s st t t r s r ss t r r tr r2 st t t s 2 s t t r2 r 2 s r s t t s r rt r t s s s st t rt t t r r t s s t t t r r t s r r t s s s r r t r r t s s t t s r t t r 2 r r t s ss t s t t s t t t t r t ts s t r r t s t r s s r t s r r ss r rt r r s t t s ss ss t t r r t s r s r t s t rst t ss t t r P r t s s r s r t s s s r r rr ts t 1t s 2 r s r t tr2 r t 2 s tt t s r rt t r t2 r t s r r s r s ss t t r s t t r s t t s t s s s t r t2 t st s r s t2 s t s r s r t tr t t s t s t t2 t t t2 str ss s 2 s r r t r 1 t s s r t str ss rt t 2 t str ss s s t s r t t t2 s r s s t s r s t t t s t r t t r str t r r s r t s st r t s t s s t s t r q r ts s rr t 2 r

t r r s s t s t t s r t r t t P r t t s t s t r s t s 2 t r t rt 1 r t rt t rt r r t s s t s rr s s t t r r t t 2 1 t t r t r t r s t r2 t t t r st s t t r t s st t 2 t t st t t s s r s r s t r t 1 s r t s t t t t r2 s t ss t s st s r t r s r st t t s t s r t s s t s r ss t t 1 t 1 r ss s r t str ss s t r 1 r ss s r q r t r t t s rr t s t s r s s r2 t s t t s t r t r s t 2 r t rs r 2 t s t r t s t s r tt r s r s t s r t t t t r t r t tr t r r t s s r2 r s t ts r st ss r s t st s s r t r r r t s 2 t t2 t r r t rr t tt r s r s t r r t r r t s r t r t t s 1 r t q Pr ss r r r r s s t s r tr t t r t r t q s 2 r rr t 2 t r t s rs t r s rt t 1 r t r s t r r ts t r r ts s s r 1 r 1 s s q s t r 2 2 s r r s r s s r t r s s tr t t s s s t s s q s r t t r r t 2 r 1 t r t r t s s s tr s s r r s s st t q s r r s t q s t s s t r rt s t t r st t t r t s r rt s s t tr s r r s t 2 2 r r t st t t tt s r t r tt t r tt tt q

t r r r s r 1 r s st r r t t s s t s s

r 2 t t t t r t rs r t r 2r r str rt ss r 2 s P 2 r q s s r 2 t 2 s r 33 r t 1 st q ss r 1 t s t r s r s r r t rs rt s P s tt rt ts t s t t 2s s r 4 t t 2 s rr ts r t s r s tr t t s tr r r t s r rt r t q t s t r t s t t s r rr ts r P s t 2 t s tr r 1 t s r t t2 r ss r str ss r t t st s r tt rs3 t r 2s s tr t s r2 t s str s s s s é rr ts r r r tr t s t 1 t s r t r rt s r t t P 2s s r s r r r tr t t s r r 2 t 1 r rs t2 Pr ss r s r t st r r 2s t t r r ss s r t t P 2s s r t s r rr ts r 1 t s tr t t r st s t s r r s t r t t P 2s s r t s r s t tr t t s r t 3 t tr2 P 2s s P t s s r t 2 P t r t 2 tr t t s r t r ss r st s q t s t t t rt s r 2s t t s s rr s t rst 2 r 1 st t 1 r rs t2 Pr ss

r 2 r 2 s t r t r q t s st t Pr s t 2 t2 P rt r r r t s r t 3 t t t t t r s r tr r2 q r t r s P t s s r t ø q st r t tr t t s r t st 2 r s r ss r t s q t s P t s s r ss s tts st t t 2 s s s s é rr ts r tr r t r s t s r t t r t s t t r t r s t t st t P 1t s r 2 s r t r st r t r s s s t

ss r2 t t P P P s r t t t 2 s t r t t t t t t ss r ss tt r P rt r t q t Pr rt r tr t t rt t2 t2 Pr ss r

Olesen, Kennet. Structure preserving formulation of high viscous fluid flows, 2014 Department of Engineering Aarhus University Inge Lehmanns Gade 10 8000 Aarhus Denmark