Velocity profile, in laminar flow

Velocity Profiles In laminar flow, the solution of the Navier-Stokes equation, corresponding to the Hagen-PoiseuiUe equation, gives the velocity i as a Innction of radial position / in a circular pipe of radius R in terms of the average velocity V = Q/A. The parabolic profile, with centerline velocity t ce the average velocity, is shown in Fig. 6-10. 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

The shearing characteristics of non-Newtonian fluids are shown in Fig. 3.24 of Volume 1. This type of fluid remains rigid when the shear stress is less than the yield stress Ry and flows like a Newtonian fluid when the shear stress exceeds Ry. Examples of Bingham plastics are many fine suspensions and pastes including sewage sludge and toothpaste. The velocity profile in laminar flow is shown in Fig. 3c. 

Seeley, L.E., Hummel, R.L., and Smith, J.W., Experimental velocity profiles in laminar flow around spheres at intermediate Reynolds numbers, J. Fluid Mechs., 6S, 591-608 (1975). 

Fig. 3.5. The velocity profile in laminar flow through a tube...

SHELL MOMENTUM BALANCE AND VELOCITY PROFILE IN LAMINAR FLOW ... 

Figure 2.9-2. Velocity and momentum flux profiles for laminar flow in a pipe. Sec. 2.9 Shell Momentum Balance and Velocity Profile in Laminar Flow...

Sec. 2.9 Shell Momentum Balance and Velocity Profile in Laminar Flow... 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

The entry length Leto achieve a near steady-state velocity profile in laminar flow where inertia is important is given approximately by... 

Figure 6 The developing velocity profile in laminar flow.

It is shown in Example 1.9 that the velocity profile for laminar flow of a Newtonian fluid in a pipe of circular section is parabolic and can be expressed in terms of the volumetric average velocity u as ... 

The relative shapes for the velocity profiles in laminar and turbulent flow are indicated in Fig. 5-1. The laminar profile is approximately parabolic, while the turbulent profile has a portion near the wall which is very nearly linear. This linear portion is said to be due to a laminar sublayer which hugs the surface very closely. Outside this sublayer the velocity profile is relatively flat in comparison with the laminar profile. 

Flfl. 6-3 Influence of heating on velocity profile in laminar tube flow. 

Typical average velocity profiles in laminar and turbulent flow are also given in Fig. 6-12. Note that the velocity profile in turbulent flow is much fuller than that in laminar flow, with a sharp drop near the surface. The turbulent boundary ... 

An interesting point is the dependence of the turbulent Nu numbers on the boundary condition. For laminar flow it was shown that the dependence of Nu on the boundary condition rapidly fades away when the relative pitch is increased The difference in Nu between the two limiting boundary conditions is less than 30% for relative pitches larger than 1.1. Because of the flatter velocity profile in turbulent flow, the dependence on the boundary conditions is weak in turbulent flow, except for small relative pitches. An estimate of the maximum influence of the boundary condition on the turbulent Nu number can be obtained from the respective values for laminar flow. 

Determine the axial velocity profile for laminar flow in a square duct. Also determine the pressure drop per unit length for water in a square duct with H = pm (each side is 2 pm)... 

FIGURE 1.4 Velocity profile for laminar flow in a tube. 

As discussdd in Sec. 6.3, the velocity profile for laminar flow in a tube is parabolic. For turbulent flow it is much closer to plug flow, i.e., to a uniform velocity over the entire pipe cross section. Furthermore, as seen from Fig. 11.7, as the Reynolds number is increased, the velocity profile approaches closer and closer to plug flow. At the wall the turbulent eddies disappear so the shear stress at the wall for both laminar and turbulent flow of newtonian fluids is given byj dVJdy. Although it is ve difficult experimentally to... 

Figure 3.24 Development of the boundary layer and velocity profile for laminar flow in the entrance region of a pipe...

For Re below a critical value of about 2300, we have laminar flow with a parabolic velocity profile. The maximum velocity in the center of the tube is twice as much as the mean value Umean defined by Eq. (3.2.61). The radial velocity profile for laminar flow in a round tube is given by ... 

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