Approximate solutions for the couette viscometry equation. Couette flow by virendra kumar phd pursuing iit delhi 2. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on. Our concern is the motion of an incompressible fluid of density p and kinematic viscosity v which is contained in the gap between two concentric cylinders which. The entire relation or the poiseuilles law formula is given by. Rotating cylinders, annuli, and spheres flow over rotating cylinders is important in a wide number of applications from shafts and axles to spinning projectiles. The dimensionless control parameters are the reynolds numbers of the inner and outer cylinders, the ratio of the cylinder radii, and the aspect ratio. Of course in this case there is no rigid plate at the top and the water surface becomes soon wavy. Finite difference analysis of plane poiseuille and couette. Consider flow between two parallel plates separated by a distance 2 h with a uniform heat flux imposed on both plates. In this study, we apply the energy gradient theory to analyze the taylorcouette flow between concentric rotating cylinders, and aim to demonstrate that the mechanism of instability in taylorcouette flow can be explained via the energy gradient concept. Exact solutions of navierstokes equations example 1.
In this study, we apply the energy gradient theory to analyze the taylor couette flow between concentric rotating cylinders, and demonstrate that the mechanism of instability in. Couette and planar poiseuille flow couette and planar poiseuille. A turbulentlaminar banded pattern in plane couette flow is studied. In this paper we investigate the problem of modulated taylorcouette flow. The well known analytical solution to the problem of incompressible cou. Poiseuille flow university of california, san diego. Pressure and body forces balance each other and at steady state the equation of. If it is not attached to the moving bounding surface for example, a taylor couette flow with a stationary inner cylinder and a rotating outer cylinder, then how is the presence of the moving. Taylorcouette flow, the flow between two coaxial co or counterrotating cylinders, is one of the paradigmatic systems in the physics of fluids.
Couette flow is a laminar circular flow occurring between a rotating inner cylinder and a static one, and the extension via increased speed of rotation to centrifugallydriven instabilities leads to laminar taylor vortex flow, tending to turbulent flow as speed increases. Finally, it is shown that the hagenpoiseuille equation, as well as the expression describing couette flow between parallel plates, can be derived from the equations presented in this work and may thus be viewed as special cases of darcys law. It is distinguished from draginduced flow such as couette flow. The derivation of the flow curve from the torque measurements t. The pressure p is a lagrange multiplier to satisfy the incompressibility condition 3. The momentum equations 1 and 2 describe the time evolution of the velocity. We wish to determine the steady flow pattern set up within the fluid.
Instead of the pressure gradient driving the ow, it is driven by the motion of one of the boundaries and that motion is parallel to the direction of the channel dp dx is in fact absent from this problem. Poiseuille formula derivation hagen poiseuille equation. Some of the fundamental solutions for fully developed viscous. Couette flows 77 stability of couette flows all of the solution previously mentioned are exact steadyflow solutions of the navierstokes equations.
Solving the equations how the fluid moves is determined by the initial and boundary conditions. The flow is driven by virtue of viscous drag force acting on the fluid and the applied pressure gradient parallel to the plates. Highreynolds number taylorcouette turbulence annual. Newtonian fluid flow, considering the effect of viscous dissipation 9,10. Velocity field for taylorcouette flow with an axial flow. Startup and cessation newtonian poiseuille and couette flows.
The flow is two dimensional in is to be driven exclusively by the shea that the velocity profile is formed across the flow. Introduction flow between parallel plates consider steady, twodimensional, viscous flow between two parallel plates that are situated a perpendicular distance apart. Linear and weakly nonlinear analyses of cylindrical. You multiply subtotal by m which is not correct, as you just have to multiply 2upi with subtotal.
This is the generic shear flow that is used to illustrate newtons law of viscosity. For flow between concentric rotating cylinders, the flow instability may be induced by rotation of the inner cylinder or the outer cylinder. The problem of flow development from an initially flat velocity profile in the plane poiseuille and couette flow geometry is investigated for a viscous fluid. In this paper we investigate the problem of modulated taylor couette flow. A compact and fast matlab code solving the incompressible. If it is not attached to the moving bounding surface for example, a taylorcouette flow with a stationary inner cylinder and a rotating outer cylinder, then how is the presence of the moving.
Let be a longitudinal coordinate measuring distance along the plates, and let be a transverse coordinate such that the plates are located at and. Experimental and numerical study of taylor couette flow haoyu wang iowa state university follow this and additional works at. While the fluid mechanics of the original flow are unsteady when, the new flow, called taylorcouette flow, with the taylor vortices present, is actually steady until the flow reaches a large reynolds number, at which point the flow transitions to unsteady wavy vortex flow, presumably indicating the presence of nonaxisymmetric instabilities. Numerical results for wavyvortex flow with one travelling wave by philip s. Note that the average shear driven fluid velocity v xequals 50% of the top surface speed u.
Analytic and numerical solutions of couette flow problem. The difference is that in couette flow one of the plates. Extending previous linear stability analyses of the instabilities developing in permeable taylorcouettepoiseuille flows where axial and radial throughflows are superimposed on the usual taylorcouette flow, we further examine the linear behaviour and expand the analysis to consider the weakly nonlinear behaviour of convectivetype instabilities by means of the derivation of the fifth. Contribute to ctjacobscouetteflow development by creating an account on github. Experimental and numerical study of taylorcouette flow. Instead of the pressure gradient driving the ow, it is driven by the motion of one of the boundaries and that motion is parallel to the direction of the channel dp.
The mass flow rates and mean velocities are the superposition of the pressure flow poiseuille flow and the shear flow couette flow components. One key response of the system is the torque required to retain constant angular velocities. Poiseuille flow is pressureinduced flow channel flow in a long duct, usually a pipe. It is in this context that direct numerical simulations. In nonideal fluid dynamics, the hagenpoiseuille equation, also known as the hagenpoiseuille law, poiseuille law or poiseuille equation, is a physical law that gives the pressure drop in an incompressible and newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a. A derivation of the navierstokes equations can be found in 2. Combined couette poiseuille flow is the steady flow between two flat plates, a fixed distance apart, in which the plates move relative to each other and. In laminar flow regime, the velocity profile is linear. Part of themechanical engineering commons this dissertation is brought to you for free and open access by the iowa state university capstones, theses and dissertations at iowa state university. The fluid is driven between the plates by an applied pressure gradient in the xdirection.
This video is a sequel to torque on rotating cylinder. Couette flow couette flow is steady viscous flow between parallel plates, where top plate is moving parallel to bottom plate noslip boundary conditions at plates x x x x u y u u i u i 0 u u at y h u at y and y u x. Generalized couettepoiseuille flow with boundary mass transfer. Exact solutions to the navierstokes equations i example 1. Startup and cessation newtonian poiseuille and couette. P, the length of the narrow tube l of radius r and the viscosity of the fluid. Aug 04, 2017 this video is a sequel to torque on rotating cylinder. Is there any real life application of taylor couette flow.
The fluid pressure does not vary across the film thickness and fluid inertia effects are. Analytical solution with the effect of viscous dissipa tion was derived for couette poiseuille flow of nonlinear viscoelastic fluids and with the simplified phanthien tanner fluid between parallel plates, with stationary plate. Fluid dynamics derivation of the taylorcouette flow. We consider two plates separated by a distance d from. The couette flow is characterized by a constant shear stress distribution. Heat transfer with viscous dissipation in couettepoiseuille. Write the exact equations for a fluid flow problem incorporating applicable simplifications. It is convenient to adopt cylindrical coordinates,, whose symmetry axis coincides with the common axis of the two shells. The poiseuilles law states that the flow of liquid depends on following factors like the pressure gradient. For discrete problem formulation, implicit cranknicolson method was used. Department of mechanical engineering, northwestern university, 2145 sheridan road, evanston, illinois 60208. Mean flow of turbulentlaminar patterns in plane couette flow. The flow of a fluid between concentric rotating cylinders, or taylor couette flow, is known to exhibit a variety of types of behavior, the most celebrated being taylor vortices taylor 1923. The configuration often takes the form of two parallel plates or the gap between two concentric cylinders.
Shenoy department of chemical engineering, indian institute of technology, powai, bombay 400 076, india the steady laminar axial flow of a powerlaw nonnewtonian fluid in the annular space between. Marcus division of applied sciences and department of astronomy, harvard university received 26 july 1983 and in revised form 23 march 1984 we use a numerical method that was described in part 1 marcus 1984a to solve. The momentum equation for any stress tensor in a couette flow is. May 27, 2015 combined couette poiseuille flow is the steady flow between two flat plates, a fixed distance apart, in which the plates move relative to each other and there is a pressure gradient parallel to. Couettetaylor flow raz kupferman mathematics department, lawrence berkeley national laboratory, 1 cyclotron road, 50a2152, berkeley, california 94720. Write the exact equations for a fluid flow problem incorporating applicable simplifications topicsoutline. In this study, we apply the energy gradient theory to analyze the taylorcouette flow between concentric rotating cylinders, and demonstrate that the mechanism of instability in. Nonetheless the wind exerts a shearing action on the surface that sets the underneath water in motion. In this video i will present you a simple derivation of the velocity distribution profile of the taylor couette flow at laminar speeds. The lecture presents the derivation of the reynolds equation of classical lubrication theory. In this video i will present you a simple derivation of the velocity distribution profile of the taylorcouette flow at laminar speeds. The problem has been studied by a large numberof authors. Later on, in the study of turbulence in fluid film bearings, the mean flow velocities will be. The navierstokes equations academic resource center.
Specifically, it is assumed that there is laminar flow of an incompressible newtonian fluid of viscosity. Analytical solutions for regularized moment equations peyman taheri,1,a manuel torrilhon,2,b and henning struchtrup1,c 1department of mechanical engineering, university of victoria, p. Fluid mechanics, sg2214, ht2009 september 15, 2009 exercise 5. In this study, we apply the energy gradient theory to analyze the taylor couette flow between concentric rotating cylinders, and aim to demonstrate that the mechanism of instability in taylor couette flow can be explained via the energy gradient concept. Couette flow 1 is a viscous flow between two parallel plates distance.
They are called laminar flows and have a smoothstreamline character. Also, instead of using a for statement to determine subtotal, i would suggest to apply the symsum function. Fluid dynamics derivation of the taylorcouette flow youtube. Instability of taylorcouette flow between concentric. Finally, the system of equation tridiagonal is solved with both. We also show that plane couette flow is just the limiting case of taylorcouette flow when the curvature of the walls tends to zero. List and explain the assumptions behind the classical equations of fluid dynamics 3. In fluid dynamics, couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. Winddriven flow in a body of water is a situation where the couette flow can be a useful approximation. Incidentally, this type of flow is generally known as taylor couette flow, after maurice couette and geoffrey taylor 18861975. Analytical solution with the effect of viscous dissipa tion was derived for couettepoiseuille flow of nonlinear viscoelastic fluids and with the simplified phanthien tanner fluid between parallel plates, with stationary plate.
Introduction in fluid dynamics, couette flow is the laminar flow of a viscous fluid in the space between two parallel plates, one of which is moving relative to the other. Couette flow the flows when the fluid between two parallel surfaces are induced to flow by the motion of one surface relative to the other is called couette flow. A simple shear flow is the steady flow between two parallel plates moving at different velocities and called a couette flow fig. Couette flow couette ow is similar to channel ow and has the same geometry but with an important modi cation. Consider a liquid flowing through a thin film region separated by two closely spaced moving surfaces. The flow of a fluid between concentric rotating cylinders, or taylorcouette flow, is known to exhibit a variety of types of behavior, the most celebrated being taylor vortices taylor 1923. The flow is driven by virtue of viscous drag force acting on the fluid, but may additionally be motivated by an applied pressure. The equation of continuity is satisfied by ur frr and w1. Also considered here is the flow in an annulus formed between two concentric cylinders where one or both of the cylindrical surfaces is or are rotating.
1318 462 229 1521 31 1361 1105 647 792 1337 825 1054 136 39 1440 688 970 600 1092 840 1257 877 720 70 793 319 1218 1473 1501 171 1 234 1156 1354 447 21 1330 1499 387 470 680 1024 1188 1276 1272 1420