Couette Flow
Couette flow is a type of viscous fluid flow between two parallel plates, where one plate is stationary and the other is in constant motion. This flow is characterized by the shearing motion of the fluid, with the velocity of the fluid increasing linearly from the stationary plate to the moving plate. Couette flow is an important concept in fluid dynamics and has applications in various engineering and scientific fields.
7 Key excerpts on "Couette Flow"
eBook - ePub- H. W. Liepmann, A. Roshko(Authors)
- 2013(Publication Date)
- Dover Publications(Publisher)
...The simplicity of the problem will enable us to exhibit the effects of compressibility upon shear flow, without the additional difficulties entailed in the boundary-layer problem. It will make the following study of boundary-layer flow much easier. 13.2 Couette Flow The flow is two-dimensional, between two infinite, flat plates, distance d apart (Fig. 13.1). The coordinates are x, in the direction of flow, and y, normal to the plates. The space between the plates is filled with a gas. The upper plate slides in the x -direction with a constant velocity U. The problem is to compute the flow of the gas. In perfect fluid theory the sliding motion of the upper wall would have no effect whatsoever on the gas, since the only boundary condition available applies to the velocity component normal to the surface. In real fluid theory it is necessary to add another boundary condition, for the velocity component parallel to the wall. This is the so-called no-slip condition: the fluid at a solid boundary has the same velocity as the boundary. In this problem, then, the gas next to the upper wall is moving with the wall at velocity U, while the gas at the lower wall is at rest. The no-slip condition permits the wall to transmit a shear force τ to the fluid. Conditions are the same at every section x, and so the shear τ can depend only on y. For the same reason, there are no accelerations and no pressure gradients in the x -direction. Therefore, the equilibrium of forces on a fluid element shows that (Fig. 13.2) Thus τ is constant throughout the flow, and must be equal to the shear stress, τ w, on the walls. F IG. 13.1 Couette Flow. F IG. 13.2 Force balance in Couette Flow. Similarly, a momentum balance for the y -direction shows that dp/dy = 0, since v = 0. Thus the pressure is uniform throughout. The shear stress is related to the velocity field u (y) by Newton’s form of the friction law This equation defines the viscosity μ...
eBook - ePubTransport Phenomena
An Introduction to Advanced Topics
- Larry A. Glasgow(Author)
- 2010(Publication Date)
- Wiley(Publisher)
...There is an important difference between this class of flows and the Poiseuille flows we examined previously. Consider a steady Couette Flow between parallel planar surfaces—one plane is stationary and the other moves with constant velocity in the z -direction: (3.42) Note that the velocity distribution is independent of viscosity. A closely related problem, and one that is considerably more practical, is the Couette Flow between concentric cylinders. The general arrangement is shown in Figure 3.7. Figure 3.7 The standard Couette Flow geometry for concentric cylinders. In this scenario, one (or both) cylinder(s) rotates and the flow occurs in the θ- (tangential) direction. Flows of this type were extensively studied by Rayleigh, Couette, Mallock, and others in the late nineteenth century; work continued throughout the twentieth century, and indeed there is still an active research interest in the case in which the flow is dominated by the rotation of the inner cylinder. This particular flow continues to attract attention because the transition process is evolutionary, that is, as the rate of rotation of the inner cylinder is increased, a sequence of stable secondary flows develops in which the annular gap is filled with Taylor vortices rotating in opposite directions. We will examine this phenomenon in greater detail in Chapter 5. For present purposes, we will write down the governing equation for the Couette Flow between concentric cylinders: (3.43) For the steady flow case, (3.44) If the outer cylinder is rotating at a constant angular velocity ω and the inner cylinder is at rest, then (3.45) The shear stress for this flow is given by. Consider the case in which the radii R 1 and R 2 are 2 and 8 cm (a very wide annular gap), respectively, and the outer cylinder rotates at 30 rad/s...
- Nicholas P. Cheremisinoff, Paul N. Cheremisinoff(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
...As a matter of useful terminology, flows solely due to an imposed pressure gradient are referred to as Poiseuille flows (or pressure flows), whereas flows solely due to a bounding surface in motion are referred to as Couette Flows (or drag flows). Flows with an imposed pressure gradient as well as moving bounding surface(s) are termed generalized Couette Flows (or combined pressure and drag flows). A. Axial Generalized Couette Flow The axial generalized Couette Flow problem in an annulus for a power-law fluid has been investigated by Lin and Hsu [ 1 ]. However, their solution is not complete because the possibility of the pressure gradient opposing the drag flow is not considered. Furthermore, their volumetric flow-rate expressions are in the form of definite integrals necessitating numerical quadrature. Malik and Shenoy [ 2 ] have presented the complete solution and evaluated the flow integrals analytically. However, their derivation involves two separate cases, one in which there is a maximum or minimum in the velocity profile within the annular region and another in which there is not. In what follows, a single exact analytical expression for the volumetric flow rate is derived that holds for both cases and has a simple algebraic form useful in performing quick practical calculations. Malik and Shenoy [ 2 ] considered a power-law fluid between two long coaxial cylinders of length L (Fig. 1), with the outer cylinder (r = R) being stationary and the inner cylinder (r = kR) moving at a constant axial velocity V in the positive z -direction. Furthermore, there was a constant pressure gradient (Δ P/L) in the z -direction. For this one-dimensional flow problem, the equation of motion in cylindrical coordinates simplifies to d (r τ r z) d r = (Δ P L) r (1) If ξ denotes r/R, then Eq...
- Amithirigala Widhanelage Jayawardena(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
...Flow separation can take place when the pressure gradient is positive. When d p d x < 0 o r C > 0, using a nominal value of +1 for C, the velocity becomes u u 0 = 2 Y − Y 2. In this case, the pressure decreases with distance resulting in an increase in velocity. The average velocity is given as V = ∫ [ u 0 y h + C u 0 (y h − y 2 h 2) d (y h) ] = (1 2 + C 6) u 0 (7.84) For simple Couette Flow, Eq. 7.84 simplifies to V = u 0 2. When C = −3, V = 0. Flow rate Q = h V = (1 2 + C 6) u 0 h (7.85) Maximum velocity u max = u 0 4 ((1 + C) 2 C) for C ≥ 1 (7.86) Minimum velocity u min = u 0 4 ((1 + C) 2 C) for C ≤ − 1 (7.87) Shear stress τ = μ d u d y = μ u[--=PLGO. -SEPARATOR=--]0 h + μ u 0 C h (1 − 2 y h) (7.88) 7.5 Boundary-layer separation In the discussion so far, the pressure gradient has been assumed to be zero i.e. (∂ p ∂ x = 0). But there are instances where the pressure gradient is not zero, such as for example, when a fluid is flowing over a curved surface where the radius of curvature is much greater than the boundary-layer thickness. The flow in converging sections gives negative pressure gradients while diverging flow gives positive pressure gradients. A positive pressure gradient is called an adverse pressure gradient and occurs in the flow around curved boundaries and towards a stagnation point. Increase in pressure is of course accompanied by a reduction of velocity, which at some point along the length may decrease to zero. When this happens, the flow separates from the boundary at a point of inflexion of the velocity profile and is called the boundary-layer separation. The separation is followed by a region called the wake where eddies dissipate energy. When the pressure gradient is negative, it is called a favourable pressure gradient. When it is so, there is a resultant force in the direction of flow due to pressure. This pressure acts against the boundary shear, thereby reducing the retarding action of the shear...
eBook - ePub- A. D. Johnson(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...14.1 Frictionless flow in a pipe. A pressure drop is required to overcome friction in the flow. The friction is created due to the viscous nature of the fluid together with the nature of the flow. These will be discussed later in this chapter. For the present, it can be stated that fluid flow in pipes or ducts can be either laminar or turbulent. In laminar flow the friction is entirely due to the viscous nature of the fluid, defined by the property ‘viscosity’, which is a measure of the fluid’s resistance to flow. 14.3 Viscosity One characteristic of the behaviour of real flows with friction is that immediately adjacent to a solid surface the velocity of the flow is zero with respect to that surface. This is because the layer of molecules of the fluid in contact with the surface will adhere to that surface. Moving out from the surface each subsequent layer of fluid will gain increasing velocity as it ‘shears’ past the lower layers. Consider the behaviour of a fluid between two parallel solid surfaces, as shown in Figure 14.2. If one surface is stationary, the layer of fluid in contact with it will be stationary. If the other surface is moving with velocity v, the fluid in contact with that surface will also have velocity v. Across the fluid the velocity will change from 0 to v and, if the gap width y is relatively small, the velocity profile will be as shown in Figure 14.2. However, to ensure that the moving surface achieves a constant velocity v, a force F must be applied such that F ∝ A v y where A is the area in contact with the fluid of the moving surface. This can be re-expressed as F A ∝ v y But (force/area) is a stress and in this case (F/A) is a shear stress, τ, so that τ = constant (v y) (14.1) Fig...
eBook - ePub- Zeki Berk(Author)
- 2008(Publication Date)
- Academic Press(Publisher)
...Elements of Fluid Dynamics 2.2.1. Viscosity Consider a mass of fluid confined between two flat plates (Figure 2.1). The lower plate is held stationary. The upper plate moves in the x direction at a constant velocity v x. Assume that the liquid layer in immediate contact with each plate moves at the velocity of that plate (no slippage). What has been described is clearly a ‘shearing’ action on the fluid. Figure 2.1. Definition of viscosity Newton's law states that the shearing force F x required to maintain the upper plate in movement is proportional to the area of the plate A and to the velocity gradient −(dv x /dz). It is assumed that there is no movement other than in the x direction. (2.1) The proportionality factor μ is called viscosity. Viscosity is a property of the fluid and represents the resistance of the fluid to shearing action. Its units in the SI system are Pascal.second (Pa.s). The traditional c.g.s unit is the poise, after the French physicist Poiseuille (1799–1869). The conversion factor is: The viscosity of liquids is strongly temperature-dependent and almost pressure-independent. The viscosity of gases increases with pressure and decreases slightly with increasing temperature. Viscosity data for various materials of interest in food process engineering are given in the Appendix. The shearing force per unit area is the shear stress, shown by the symbol τ. The velocity gradient −(dv x /dz) is called the shear rate, represented by the symbol γ. Eq. (2.1) can now be written as: (2.2) For many fluids, the viscosity is independent of the shear rate and Eq. (2.2) is linear. Such fluids are qualified as ‘Newtonian’. Gases, water, milk and dilute solutions of low molecular weight solutes are practically Newtonian. Other fluids, such as solutions of polymers and concentrated suspensions are non-Newtonian. Their viscosity depends on the shear rate. For non-Newtonian liquids Eq. (2.2) is not linear...
eBook - ePub- Rose G Davies(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
...3 Viscous Flow and Boundary Layer In Chapter 2, gas was treated as ideal gas. An ideal gas is the gas, within which there is sufficient space between the gas molecules, so that the interaction between molecules can be ignored. In reality, the interaction between the molecules of any fluid not only exists, but can be very significant as well. So the real fluids, either liquids, or gases, are viscous fluids. The dynamics of viscous flow is very different from the ideal fluid flow. Viscous fluid flow can produce more drag, in particular, on the surfaces and at the places, where there is a sudden change on the surface of an object and within the viscous flow. For example, on the surface of an aerofoil and the fuselage of aircraft in flight; inside the pipes of hydraulic fluid and lubricate oil; and around the undercarriage when it is down. This chapter will explain the property of viscous fluid, the characteristics of viscous flow, and the types of drag caused by viscous fluid flow. Viscosity Consider a fluid particle in a 2-D fluid flow field, shown in Figure 3.1 : u is the velocity of the fluid along its streamline, y -direction is perpendicular to the direction of u. The fluid is assumed as a continuum, a “Newtonian fluid”. The force due to the interaction between the fluid particles is shown as τ, the shear stress. τ is proportional to the velocity gradient (change) in y -direction. In Figure 3.1, the direction of the fluid velocity is x, and the direction perpendicular to the velocity is y, then the shear stress can be written as: FIGURE 3.1 A particle of viscous fluid. τ = μ ∂ u ∂ y ; SI unit : [ N m − 2 ] (3.1) where the proportionality μ is called the dynamic viscosity of the fluid ; u is the speed of fluid particle; and y is the direction, which is perpendicular to u, as suggested in the reference (Shandong Engineering College, 1979). In fact, τ is the viscous friction on a unit area. Viscosity is a property of a fluid...
