Fluid flow in curved geometries

Presentation LayoutAnalysis of fluid behaviourGoverning equationsManifold and Metric∇ operationsStretching a curved surfacePeristaltic flow in a curved channelSummary

Analysis of fluid behaviour Analysis of any problem in fluid mechanicsnecessarily includes statement of the basic lawsgoverning the fluid motion. The basic laws, whichapplicable to any fluid, are:Conservation of massNewton’s second law of motionThe principle of angular momentumThe first law of thermodynamicsThe second law of thermodynamics

Analysis of fluid behaviour NOT all basic laws are required to solve any oneproblem. On the other hand, in many problems itis necessary to bring into the analysis additionalrelations that describe the behavior of physicalproperties of fluids under given conditions. Many apparently simple problems in fluidmechanics that cannot be solved analytically.In such cases we must resort to morecomplicated numerical solutions and/orresults of experimental tests.

Manifold and metricWhat is a manifold?A manifold is a geometric ‘thing’ which hasopen charts, subsets where a flat set ofcoordinates is given. In general, however,they can be built by patching together on atlasof open charts.We need to review open charts

Open charts:Manifold and metricThe same point is contained in more than one openchart. Its description in both charts is related by acoordinate transformation

Manifold and metricDifferentiable structure

Manifold and metricDifferentiable structure

ManifoldManifold and metricParallel transportA vector field is parallel transported along a curve,when it mantains a constant angle with the tangentvector to the curve

Manifold and metricThe metric: a rule to calculate the lenghtof curves!!Connection and covariant derivativeA connection is a map∇:TM× TM→TMFrom the product of thetangent bundle with itself tothe tangent bundle

In a basis...Covariant derivativeThis defines the covariant derivative of a (controvariant) vector fieldFluid Mechanics Group (FMG)14

Covariant derivativeCovariant derivative of a general tensorVector (first order tensor, r + s = 1)ii ∂Ai n ∂AiA, m= + ΓnmA ; Ami ,m= − Γm∂ξ∂ξ• Second-order tensor (r + s = 2)nimAnijij ∂Ti nj jT,m= + ΓnmT+ Γ Tmnm∂ξ∂TijnnTij,m= − ΓimTnj− ΓjmTmin∂ξii∂T•j i n nT• j ,m= + ΓnmTj jmTm•− Γ∂ξini• netc.r = 2, s = 0r = 0, s = 2r = 1, s = 1

For the velocity field∇ operationsV =i jV ( u )Gradient of the velocity field iseiLiik= ∇V= V ; j = g gjmm,[( )]mmm llg V k + Γ g VlklDivergence of the velocity field isi∇ ⋅ V = V ; i = ,( )iii mmg Vii + Γmig VmWhere ; represent the covariant derivative

Divergence of a tensor∇ ⋅ T =( ) [( )]kk ii jji mm jj j ii mmV⋅ ∇ T = g g g V g g T , k + Γ g g T + Γ g g Tiijjgiik( )ii jjg g T⎡⎢⎢⎣+ Γjmj(V⋅∇) of a vectorijgiiijg,jmmmk+ ΓTim( )( )ii kk∇ V = g g V g VV⋅ ,kk(V⋅∇) of a tensor∇ operationsiimjgmmmjgjjmkTmj[ ]k mmi + Γ g Vkmi⎤⎥⎥⎦mim

Stretching a curved surfaceGeometry of the problemFig. 1 (a)Fig. 1 (b)r, vs, uRO(a) a flat stretching sheet, (b) a curved stretching sheet.

Stretching a curved surfaceMathematical FormulationCurvilinear coordinates ( r + R,s)metriccontinuity equationg ab⎛1⎜⎝0=2∂∂r0⎞( r + R) ⎟⎠∂u∂s{( r + R)v} + R = 0,components of equation of motion2u 1 ∂p=r + R ρ ∂r,v∂u∂r+Rur + R∂u∂s+uvr + R1= −ρ rR+ R∂p∂s⎛+ ν⎜⎝2∂ u2∂r+r1+ R∂u∂r−( ru+ R)2⎞⎟,⎠

Stretching a curved surfaceThe boundary conditions applicable to the presentflow areuas,u → 0,∂u∂rUsing the similarity transformationu = asf ′The problem takes the form=Rr + Rv=0atr=0,→ 0 as r → ∞.2 2( η ),v = − aνf ( η) , p = ρas P(η),η = r.2kf ′′P = f ′′′ + −η + k η + kf ′( η + k)kη + k2∂Pf ′=∂ηη + kkη + kk( η + k)′ 2− f + ff + ff2 2′′aν′,ff =′ →0,0,ff′ = 1 at η = 0,′′ → 0 as η → ∞,

Stretching a curved surfaceEliminating pressuref iv2 f ′′′+η + k−f ′′ f ′ kk ′ 2k+ − ( f f′′′ − ff ′′′ ) − ( f − ff ′′)− ff ′ = 0.2 323( η + k)( η + k)η + k( η + k)( η + k)The skin friction coefficientf ′(0)1Re 1 / 2xC f= f ′′(0)+ = f ′′(0)k+ kIt is important to point out that the Crane’sproblems can be recovered by taking k → ∞ . Thenumerical solution of the problem is developed by ashooting method using a Runge-Kutta algorithm.

Stretching a curved surface1.00.81.21.0k = 1000 ; k = 20 ; k = 10 ; k = 50.60.8f 'HhL0.4k = 1000 ; k = 20 ; k = 10 ; k = 5f HhL0.60.40.20.20.00 1 2 3 4 5 6h0.00 1 2 3 4 5 6hr-component of velocity, s-component of velocity

Stretching a curved surfacePHhL0.020.00-0.02-0.04-0.06-0.08-0.10k = 1000 ; k = 20 ; k = 10 ; k = 50 1 2 3 4 5 6Dimensionless pressurehk1/2−C f Re x5 0.7576310 0.8734920 0.9356130 0.9568640 0.9675950 0.97405100 0.98704200 0.993561000 0.99880Skin friction coefficient

Peristaltic flow in a curved channelGeometry of the problem⎡2π( ) = + ( − )⎤H X, t a bsin ⎢ X ct , Upper wall⎣ λ ⎥⎦⎡2π( ) ( )⎤− H X, t =−a−bsin ⎢ X −ct. Lower wall⎣ λ ⎥⎦

Peristaltic flow in a curved channelMathematical FormulationGoverning equations in fixed frame:∂∂R{( ) }R R V R∂U+ * + * =∂X*2V V R U V U p 1V+ V + − =− + R + R* * *∂ ∂ ∂ ∂ *ν ⎡ ∂ ⎧ ∂ ⎫⎢ ⎨( ) ⎬∂t ∂ R R + R ∂ X R + R ∂ R ⎢⎣R + R ∂R⎩ ∂R⎭*22 *⎤⎛ R ⎞ ∂ V V 2R ∂U+ ⎥⎜ ,* ⎟ − +2 2 2R R * *⎝ + ⎠ ∂ X ( R R) ( R R)∂X⎥+ + ⎥⎦* *∂U ∂U R U ∂U UV R ∂p 1* UV ν ⎡ ∂ ⎧ ∂ ⎫+ + − =− +* * *⎢* ⎨( R+R ) ⎬∂t ∂ R R+ R ∂ X R+ R R+ R ∂ X ⎢⎣R+ R ∂R⎩ ∂R⎭*22 *⎤⎛ R ⎞ ∂ U U 2R ∂V+ ⎥⎜ .* ⎟ − +2 2 2R R * *⎝ + ⎠ ∂ X ( R R ) ( R R )∂X⎥+ + ⎥⎦0,

Peristaltic flow in a curved channelGoverning equations in wave frame:∂∂r{( ) }r R v R( ) ( )*R u c u c∂u+ * + * =2∂v ∂v + ∂v + ∂p ⎡ 1 ∂ ⎧* ∂v⎫− c + v + − =− + ν* *⎢* ⎨( R + r)⎬∂x ∂ r R + r ∂ x R + r ∂ r ⎢⎣R + r ∂r ⎩ ∂r⎭*22 *⎤⎛ R ⎞ ∂ v v 2R ∂u+ ⎥⎜ ,* ⎟ − +2 2 2R r * *⎝ + ⎠ ∂ x ( R r) ( R r)∂x⎥+ + ⎥⎦( ) ( )**∂u ∂u R u+ c ∂u u+ c v R ∂p ⎡ 1 ∂ ⎧* ∂u⎫− c + v + − =− + ν* * *⎢* ⎨( r+R ) ⎬∂x ∂ R R+ R ∂ x R+ R R+ R ∂ x ⎢⎣r+ R ∂r ⎩ ∂r⎭*22 *⎤⎛ R ⎞ ∂ u u 2R ∂v+ ⎥⎜ .* ⎟ − +2 2 2v R * *⎝ + ⎠ ∂ x ( r R ) ( r R )∂x⎥+ + ⎥⎦∂x0,

Peristaltic flow in a curved channelNon-dimensional variables:2πx r u v ρcax= , η = , u = , v= , Re = ,λ a c c µ2 *2πa H RP = p, h= , k = ,λµ c a aStream function: ∂ψk ∂ψ2u =− , v=δ where δ =∂ η η+ k ∂xUnder the long wavelength approximation we have∂ P = 0,∂η2∂P1 ∂ ⎧ ∂ψ⎫ 1 ⎛ ∂ψ⎞− − ⎨( η + k )1 0.2 ⎬− ⎜ − ⎟=∂x k ∂η ⎩ ∂ η ⎭ k( η+ k)⎝ ∂η⎠π a.λ

Peristaltic flow in a curved channelEliminating pressure⎧ ⎫ ⎧ ⎛ ⎞⎪⎫⎨ + ⎬+ ⎨ 1− ⎟⎬=0.∂ ⎩ ∂ ⎭ ∂ ⎪⎩( + k)⎝ ∂ ⎠⎪⎭2 21 ∂ ∂ ψ ∂ ⎪ 1 ∂ψ2( η k )2⎜k η η η η ηq ∂ψψ =− , = 1 at η = h= 1+φsin x,2 ∂ηq ∂ψψ = , = 1 at η =− h=−1−φsin x,2 ∂ηSolution of the problemC( ) ( ( ) ) ( ) 2 ( ) 2C ⎧1 ⎪η+ k ⎫⎪η+kψ = η+ k + ⎨ ln η+ k − 1 ⎬+ C + C ln ( η+ k)+ C2 ⎪⎩2 ⎪⎭42 3 4C ⎡η+ k η+k C3⎤u = + ( η k) { ( η k)} C22⎢ + + + − + +2 2 η + k⎥⎣⎦11 ln 1 .( )− 8hk 2h + q=− 4hk + h −k ln k− h + ln k+ h −2 h −k ln k− h ln k+h2 2( ) ( ) ( ) ( )( ) ( ) ( )1 2 2 2 2 2 2 22

Peristaltic flow in a curved channeldp=CC( )2 2( ) ( ) ( ) ( )2 2( h q) hk ( k h) ( k h) ( k h) ( k h)− 2 2 + 2 + − ln − − + ln +=− 4h k + ( h −k ) ln( k− h) + ln( k+ h)−2 h −k ln k− h ln k+h2 22 ⎛k−h⎞( h − k ) ( 2h+ q)ln⎜ ⎟k+h=⎝ ⎠2 2 2 2 2 22 2− 4h k + h −k ln k− h + ln k+ h −2 h −k ln k− h ln k+h22 2 2 222 22( ) ( ) ( ) ( )( ) ( ) ( )3 2 2 2 2 2hk ( h hk h q k q) ( h k )3 2 22 ⎛k−h⎞2 2 − 2 + + + − ln⎜⎟×⎝k+h⎠( −2h− q+ ( 2( h+ k)+ q) ln ( k− h) + ( 2h− 2k+ q) ln ( k+h))C4 =−2 2 2 22 2 22 222( − 4h k + ( h −k ) ( ln ( k− h) + ln ( k+ h)) −2( h −k ) ln ( k− h) ln ( k+h))The axial pressure gradient turned out to be( + )8h 2h q2 2( ) ( ) ( ) ( )( ) ( ) ( )dx 2 2 2 222 224 h k h k ln k h ln k h 2 h k ln k h ln k h− + − − + + − − − +The dimensionless pressure rise over onewavelength is defined by 2∆ Pλπ= ∫0dpdxdx.

Peristaltic flow in a curved channel

Peristaltic flow in a curved channelk = 3.5k = 5k =10k →∞

Summary The ∇ operations for curvilinearcoordinates have been discussed. Flow of a viscous fluid due toStretching a curved surface isanalyzed.Peristaltic flow in a curved channel isinvestigated.

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