ProblemSet5 - Chapter 5 Navier-Stokes Equations Problem 5.1...

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Unformatted text preview: Chapter 5 Navier-Stokes Equations Problem 5.1. Stokes second problem Consider an infinite flat plate y = 0 subject to oscillations with velocity Uw cos !t in the x-direction. The fluid in the half-space y > 0 is Newtonian, homogeneous, and incompressible. Assume that body forces are negligible, that the pressure is uniform and constant, and that the flow driven by the plate is unidirectional along x. (a) Write down the equation and boundary conditions satisfied by vx (y, t) (x-component of the velocity field). (b) Assume that the solution can be written vx (y, t) = Re[U (y)ei!t ], (5.1) where Re denotes the real part. Find an equation and boundary conditions for U (y), and solve that equation. Infer the solution for vx (y, t). (c) Sketch the velocity profile at t = 0. What is the characteristic thickness moving next to the wall? of the layer of fluid that is Problem 5.2. Startup of shear flow A Newtonian incompressible fluid with constant density ⇢ and kinematic viscosity ⌫ is placed between two parallel infinite flat plates separated by a distance h. Initially, both plates are at rest. At t = 0, the bottom plate (at y = 0) starts to translate with a constant velocity U = U ex . Assume that the flow is unidirectional in the x-direction, that body forces can be neglected, and that the pressure is constant and uniform everywhere in the fluid. (a) Determine the velocity field u = u(y, t)ex between the two plates. (b) Sketch the velocity profile for different values of the parameter ⇡ 2 ⌫t/h2 . What flow do you recover when ⇡ 2 ⌫t/h2 ! 1? Problem 5.3. Circular Couette flow Consider the steady flow of a Newtonian fluid between two infinite concentric cylinders of radii Ri (inner) and Ro (outer) that are rotating around their common axis at angular velocities !i and !o . The density ⇢ and viscosity µ of the fluid are both constant and uniform. In cylindrical coordinates (r, ✓, z), assume that the flow is uniform in the z- and ✓-directions, and that its direction is azimuthal in the (r, ✓) plane: u = u✓ (r)e✓ . (a) Write down the Navier-Stokes equations and boundary conditions for this problem using cylindrical coordinates. Simplify as much as you can using the stated assumptions. (b) Solve for the velocity profile u✓ (r), and determine all the integration constants. Sketch the velocity profile. (c) Determine the pressure p(r) up to an additive integration constant. Problem 5.4. Flow down an inclined plane An incompressible liquid (constant density ⇢ and viscosity µ) is flowing under the influence of gravity g 17 (seeplane figure). Assume the flow is the steady, unidirectional along x, and uniform thickn inalon the down a very long at that an angle ✓ to horizontal, forming asteady, film of constant ! inclined (see figure). Assume that the flow unidirectional atmosphere atmosphere downdown a very long plane inclined at an angle ✓angle toishthe forming a a very long plane inclined at an ✓ horizontal, tonthe fo y p n horizontal, = vthe ,isand that the pressure only depends on y: p = p(y). v =! vv (y)e and the pressure only depends on y: = p(y). ! (see figure). Assume flowxthat steady, unidirectional along x, and uniform in the x and z direc y xthat x ,(y)e v Assume = vthat and that theispressure only depends on y: p unifo =x,p(a ! Assume the is steady, unidirectional along x, and ! (see figure). x (y)e x ,flow (seeliquid figure). that the flow steady, unidirectional along " !y: p = p(y). v = vx (y)ex , and that thev pressure only depends onpressure nonly depends nthe pressure = vx= (y)e , and on yny:=p0on = p(y). vxx(y)e and that only depends y: p = p(y). y the y x ,that v(y) y v(y) a ! n y x n ! n atmosphere y y ! ! liquid ! y =yh= h liquid ! atmosphere atmosphere g liquid g atmosp ! ! ! v(y)v(y) n atmosphere ! ! ! atmosphere atmosphere !liquid liquid! x ! ! xh h liquid ! ! v(y) ! ! ! liquid ! ! 18 5 liquid Navier-Stokes Equations liquid v(y) v(y) g ! x v(y) x " ! v(y) !! " v(y)x v(y) y =y0= 0 h y=h down a very long plane inclined at an angle ✓ to the horizontal,x forming a film of constant thickness h g ! !g x x ! ! = h isatmosphere ! y=h ! (see figure). Assume that they flow steady, unidirectional along x, and uniform in the x and z directions: g atmosphere g g ✓ ! v = vx (y)ex , and that the pressure only depends on y: p = p(y). ! h liquid !! h! ! liquid ! ! ! x ! ! ! !h ! nn ! ! ! ! ! y ! "! " atmosphere v(y) ! atmosphere " v(y) " ! "y!=!0 " y ! ! liquid ! ! x y =!0 (a) Justifying the momentum equation, projected g every step in the derivation, show that liquid !along the g atmosphere atmosphere n y directions, leads to the two equations ! h ! ! n atmosphere atmosphere h atmosphereatmosphere liquid liquid ! ! v(y) y=h v(y) 2 ! liquidy = h !! n liquid !! n liquidliquid g d! vx ✓ ! ! n n ⇢g sin ✓n + µg 2 = 0, n ✓ v(y) v(y) h h dy x v(y) v(y) v(y) x g v(y) g y = h dp y g g g y ⇢g gcos ✓y+= h = 0. h h h ✓ ✓ dy (a) Justifying yevery in the that the momentum equation, projected al h derivation, show h = hystep = h h x show (a) Justifying every step in the derivation, that the momentum equation, projected along y two = hy equations x directions, leads to the = h y= h ✓to the y directions, =y hy (b) Knowing that the ✓leads pressure the interfacey between the liquid and the atmosphere is given b ✓attwo ✓ equations ✓ x y = 0 x pxa , solve ✓ constantxatmospheric pressure for p(y). d2 2 vx x ⇢g sin ✓ + µ 0, in the derivation, show that y (a) Justifying every d vx2 =step x⇢gviscous dytensor) y (a) Justifying step inisthe derivation, show that the sin ✓ +stress µevery =to0,the (c) The viscous traction t =yn·⌧y (where ⌧ is the zero at the interface betwee y 2 y directions, leads (a) Justifying every step in the derivation, show that the momentum equation, projected xtwo andequations dyalong (a) Justifying every step in the derivation, show that the momentum equation, projec dp yyevery directions, leads to thethe two equations (a) Justifying every step in the derivation, show that the momentum equation, liquid(a)and the atmosphere. Using this fact, derive a boundary condition for the velocity v (y) at y (a) Justifying every step in the derivation, show that the momentum equatio x (a) Justifying step in the derivation, show that the momentum Justifying every step in the derivation, show that the ✓momentum ⇢g cos +dp = 0.equation, projected along the y directions, leads to theWhat two equations y directions, leads to the twothe equations is the boundary condition at y = 0? y directions, leads to two equations y directions, leads to the two equations dy d y directions, leads to the two equations (a) Justifying every step in the derivation, show that the mom ⇢g cos ✓ + = 0. y directions, leads to the two equations ⇢g sin ✓ +d2µvx dy 2 2 y directions, leads to the two equations ✓ + µ 2d (d) Solve for the velocity vxd(y). vx Sketch the velocity profile. d vx d2 vd2 vx ⇢gd2sin vx dy ⇢g sin ✓that + µ the 2pressure = 0, at the interfaced2⇢g (5.2) x the atmosphere (b) Knowing between the liquid and ✓ +sin µ sin =sin 0, vxsin ⇢g ⇢g ✓ + µ = 0, 2 ⇢g ✓ + µ = 0, is ✓ + µ = 0, dy 2 dy (e) Calculate the volumetric flow rate Q (per unit length in the z direction), defined as ⇢g sin ✓ + µ = 0, 2 2 2 dy (b) Knowing that the pressure at the interface between the liquid and the atmosphere is ⇢g cos + dp dy dy constant atmospheric pressure pa , solve for p(y). d v✓x+✓gi dy 2 ⇢g cos dp dp ⇢g sin ✓ + µ = 0 dp constant atmospheric pressure p , solve for p(y). Z dp a 2 dy dp ⇢g costraction ✓ + t==0.n·⌧ (where ⌧ his the viscous (5.3) ✓ +cos = 0. dp⇢g cosstress (c) The viscous is= zero at the=dy interfac ⇢gtensor) cos ✓⇢g +cos =+0. ✓0. 0. ⇢g ✓ + dy dy ⇢g cos ✓ + = 0. dy Q =fact, vthe dy. dyvelocity x (y) dyat (c) The traction t =Using n·⌧ (where ⌧ isderive viscous stress is zero the interface liquid andviscous the atmosphere. this a boundary condition for vbx dy dpbetw (b) Knowing that the tensor) pressure theatthe interface 0 ⇢g cos ✓+ = 0 Knowing that the pressure at the interface between liquid atmosphere. Using fact, derive a boundary condition for the velocity v (y) What is and the the boundary condition at this y =(b) 0? x constant atmospheric pressure pa , solve for p(y). dy (b) Knowing that the pressure at the interface between the liquid and constant the atmosphere is at given by the (b) Knowing that the pressure atpressure theatmospheric interface between the liquid and the atmosph (b) Knowing that the at the interface between the liquid and the (b) Knowing that the pressure the interface between the liquid pressure pa , solve for p(y). What is the condition y =pressure 0?thebetween Knowing that the at thethe interface between the liquid and the at Problem 6.5. Wind-driven flow inside aSketch lake Knowing thatboundary the(b) pressure theat interface liquid and the atmosphere is given Solve the velocity vxat(y). velocity profile. constant atmospheric pressure pa ,p(y). solve p(y). constant atmospheric pressure(b) pa(d) , solve forfor p(y). constant atmospheric pressure p(c) solve for viscous traction t for = n·⌧ (where ⌧ is thex-dire visco constant atmospheric pressure p solve p(y). a , The a , for Consider a large lake, over which wind is blowing and exerts a constant shear stress S in the constant atmospheric pressure p , solve for p(y). constant atmospheric pressure p , solve for p(y). a (c) The viscous traction t = at n·⌧ (where ⌧ isbetween the viscous (d)Calculate Solve forthe thevolumetric velocity vxa(y). Sketch the velocity profile. (b) Knowing that the pressure the interface the liquid and the atmosphere. Using this fact, derive azer bl (e) flow rate Q (per unit length in the z direction), defined as (c) The viscous traction tand = (where ⌧ is the viscous stress tensor) is viscous traction t= n·⌧ (where ⌧=is the steady viscous stress tensor) isderive zero atain the in (c) The viscous traction t(see = n·⌧ (where ⌧ is(c) theThe viscous tensor) isis zero at then·⌧ between figure below). The goal ofstress this problem totraction determine the flow established the (c) The viscous tinterface n·⌧ (where ⌧the is field the viscous stress tenso liquid the atmosphere. Using this fact, boun (c) The viscous traction t = n·⌧ (where ⌧ is the viscous stress tensor) is zero a (c) The viscous traction t = n·⌧ (where ⌧ is the viscous stress tensor) is zero at the interface betwe constant atmospheric pressure pdirection), for p(y). What is boundary condition ydefined =boundary 0?condition a , solve (e) Calculate theaand volumetric flow rate Q (per unit the zyfact, liquid and atmosphere. Using this fact, derive aatboundary for liquid the atmosphere. Using this fact, athis boundary condition forasThe the condi veloc liquid and the atmosphere. Using derive a0? liquid and the atmosphere. this fact, derive boundary condition for the vxin (y) at = h. by Using the wind. Assume that the lake has athe constant depth hthe before the wind starts wind Zlength 0 What isavelocity the boundary condition y =blowing. hderive liquid and the atmosphere. Using this fact, derive a boundary condition for th liquid and the atmosphere. Using this fact, derive boundary condition for the velocity v (y) at y x is theiscondition boundary condition at vythe= What the(c) boundary at y= 0?v(where boundary at ySolve = The viscous traction t0?= n·⌧ ⌧ is thethe viscous stress What is the boundary condition at y = 0? What is theWhat (d) velocity velocity p Z0?yfor Q =condition x (y). Sketch hthe is the at boundary at = xvelocity 0?(y) dy.vx (y). What is the boundaryWhat condition y = 0? condition (d) Solve for Sketch the velocity profi 0 liquid and the atmosphere. Using this fact, derive a boundary (d) Solve for the velocity v (y). Sketch the velocity profile. x Q = v (y) dy. (d) Solve for the velocity v (y). Sketch the velocity profile. (d) Solveprofile. for the velocity vx (y).(e)Sketch thexvelocity profile. flow rate Q (per unit leng xthe volumetric (d) Solve for the velocity vx (y). Sketch the velocity Calculate (d) Solve for the velocity vx (y). the velocity profile. (d) Solve forSketch the velocity vCalculate (y).boundary Sketch the velocity profile. (e) flow (per unit length 0 the volumetric What isxlake the condition at ylength =rate 0? Q Problem 6.5. Wind-driven flow inside a (e) Calculate the volumetric flow rate Q (per unit in the zin direction), (e) Calculate the volumetric flow rate Q (per unit length the Zz dir (e) Calculate thethe volumetric flowdefined rate Q as (per unit length in the z direction), defined as (e) Calculate the volumetric flow rate Q (per unit length in z direction), (e) Consider Calculate volumetric flowwhich rate Q (per unit length in z vdirection), defined Z hS inhdet (e) Calculate the volumetric flow ratevelocity Qthe (per unit inthe the zasdirection), large lake, over wind is blowing and exerts a length constant shear stress (d) Solve for the Sketch velocity profile. Problemathe 6.5. Wind-driven flow inside a lake x (y). Z Z Q = v h Z h Z The goal of this problem h the steady flow (see figure abelow). determine field establish Q =S vthe Z his to and Consider largehlake, over which wind is blowing exerts aQflow constant shear stress inin Z=hrate 0x (yx v (y) dy. x Q = v (y) dy. (e) Calculate the volumetric Q (per unit length the x 0 Q = h0 before vx (y) dy. Q= vxThe (y) dy. (5.4) by(see thefigure wind. Assume thatgoal theof lake constant depth the wind starts blowing. below). thishas problem is vtox (y) determine the steady flow Qa = dy. Q 0 field established v0flow dy. x (y) 0 = 0 Problem 6.5. Wind-driven inside a lake Z 0 0the h by the wind. AssumeProblem that the 6.5. lake has Problem a constant6.5. depth h0 before starts blowing. The Wind-driven flowwind inside a lake flow inside a lake ProblemWind-driven 6.5.flow Wind-driven flow inside a lake Consider a lake large lake, over which wind is blowing an Problem 6.5. Wind-driven inside a Q = v (y) dy. Problem 5.5. HydrodynamicProblem slip Consider a large lake, over which wind is blowing and ex x 6.5. Wind-driven flow inside lake Consider aWind-driven large lake, over which wind iswind blowing andthis exerts constant she Problem 6.5. flow inside a lake Consider aalarge lake, over which is blowing andaexerts atocons (see figure below). The goal of problem isstress det 0 Consider a large lake, over which wind is blowing and exerts a constant shear S Experiments in microfluidic devices have shown that the no-slip boundary condition can sometimes be in(see figure below). The goal of this problem is to determ Consider a large lake, over which wind isbelow). blowing and exerts athis constant shear stress Sconstant insteady thethe x-dir (see figure below). The goal ofgoal thisAssume is to determine the flow (see figure The ofproblem problem is to determine ste Consider a large lake, over which wind isto blowing and exerts a aconstant shear by the wind. that the lake has dep (see figure below). The goal of this problem is determine the steady flow field est accurate, especially when the (see channel walls are made hydrophobic surfaces. more accurate boundary by the Assume the lake has a constant depth before wind figure below). The of goal of this isthat toA determine the steady flow field established in th th Problem 6.5. Wind-driven inside ahdepth lake 0the bywind. the problem wind. Assume that the lake has constant h0 the before figure below). goal of this problem isflow toahdetermine steady flow fis by the(see wind. Assume thata The the lake has a constant depth the wind starts blow 0 before condition in this case is the following: by the wind. Assume that the lake has constant depth h before the wind starts blowing. The win 0 a constant largehas lake, over which wind blowing exerts by the wind. AssumeConsider that thealake depth h0isbefore theand wind sta vt = b n · rvt at the wall, (see figure below). The goal of this problem is to determine t (5.5) by the wind. Assume that the lake has a constant depth h0 bef where vt is the tangential component of the velocity vector, n is a unit normal vector pointing into the fluid, and b is a given constant. (a) What are the dimensions of b? (b) Consider the pressure-driven flow of an incompressible homogeneous Newtonian fluid in a cylindrical microchannel of radius a (cylindrical Poiseuille flow). Solve for the velocity vx (r) in cylindrical coordinates, using the boundary condition (5.5). < 0 a uniform3µ and removing it6 through the other, cr when ↵the 1plates <0 Navier-Stokes Equations of the plates, and y-direction aligned ⇢gH 3 to the 2p Q = normal ⇣ ⌘ 3 (x, y, z) with originwhere on the centerplane of ↵ : the entrance Q = ⇢gH (6.27) (1 ↵) t 8 = ⌧0 /⇢gH. say component u↵ constant, U3µ . Show that y is : (1 <↵)02 1 + when ↵of < the 1velocity when ↵ 1y-direction plates, and aligned normal to the plate 3µ 2 ⇣ ⌘ 3 ⇢gH ↵ Q= Problem 6.11. Flow in a (6.27) channe where 2 the component u⌧y0 /⇢gH. is constant, say(pU0 . porous Show that : (1 ↵)2 1 +velocity when ↵ < ↵1 =An pL )H incompressible Newtonian fluid of1 de 3µ 2 ux (y) = where ↵ = ⌧0 /⇢gH. Problem length 6.11. Flow in a porous with inje L, separation 2Hchannel ⌧µL L, 2and infinite Re (pthe pLof)H 1 ⇢plate y 0 fluid An incompressible Newtonian density over the length of plates. The two where ↵ = ⌧ /⇢gH. 0 Problem 6.11. Flow in a porous channel with injection/suction ux (y) =⌧ L, and infinite width.and length L, separation 2H Th the plates and removing it through the oth µL Re H wheretwo Reparallel = ⇢U H/µ is the19 crossflow Reynolds nu 5 An Navier-Stokes Equations incompressible Newtonian fluid density channel ⇢ and viscosity µ flows beween plates of over the length the plates. The plates (x,flat y,of z) with origin ontwo thefor centerplane Problem 6.11. Flow in of a porous with injection/suction plates. Sketch the axial velocity profile Reare⌧porou 1,ofR length L, separation 2H ⌧ L, and infinite width. The flow is induced by a pressure difference (p p ) the plates and removing it through the other, unifor 0 L of the plates, and y-direction aligned nor An incompressible Newtonian fluid of density ⇢ and viscosity µRe flows beween two parallel flat plates of anumb where = ⇢U H/µ is the crossflow Reynolds (c)over What the value v (a) ofThe the velocity at theporous: wall? Sketch the velocity profile, andy, give an origin interpreta(x, z) with on of the centerplane of)the say entran theislength of the injecting of the same fluid through one velocity component u is constant, U y length L,xplates. separationtwo 2H plates ⌧ L,are and infiniteby width. Themore flow is induced by a pressure difference (p p Problem 6.12. Pipevelocity flow of two immiscible 0 ReL⌧liquids plates. Sketch theplates, axial profile for ofCartesian the and y-direction aligned normal1,toRe the ⇠ tion the and constant b. it through the other, a uniform crossflow is generated. thefor plates removing Use coordinates over the length of the plates. The two plates are porous: by injecting more of the same fluid through one of Consider the incompressible of two velocity component uy isflow constant, sayimmiscible U . Show (p0 thatp (x, y, z) with origin on theremoving centerplane of the entrance to athe plates, crossflow x-direction aligned with the length ux (y) = pressu plates it through the other, uniform generated. Use Cartesian coordinates Problem Pipe flow of two immiscible liquids dricalis6.12. pipe of radius R, driven by a constant Problem 5.6. the Flow in aand porous channel with injection/suction µL 2 of the plates,(x, and aligned normal to the plates. crossflow means that the transverse y, y-direction z) with origin centerplane of the Uniform entrance the(see plates, aligned the length (p pL )H 1 thex-direction incompressible flow of two immiscible 0 figure below). 1ofuwith (with viscosity µ1 ) hom occ An incompressible Newtonian fluidonofthe density ⇢ and viscosity µ flowstoConsider beween two parallel flatLiquid plates x (y) = of the plates, and y-direction aligned normal the plates. cr...
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