Steady laminar flow

The classical solution of the problem of steady laminar flow in straight ducts is based on a number of assumptions on flow conditions, Hetsroni et al. (2005)... 

This system produces a steady laminar flow with a flat velocity profile at the burner exit for mean flow velocities up to 5m/s. Velocity fluctuations at the burner outlet are reduced to low levels as v /v< 0.01 on the central axis for free jet injection conditions. The burner is fed with a mixture of methane and air. Experiments-described in what follows are carried out at fixed equivalence ratios. Flow perturbations are produced by the loudspeaker driven by an amplifier, which is fed by a sinusoidal signal s)mthesizer. Velocity perturbations measured by laser doppler velocimetry (LDV) on the burner symmetry axis above the nozzle exit plane are also purely sinusoidal and their spectral... 

Example 5-9 Flow Down an Inclined Plane. Consider the steady laminar flow of a thin layer or film of liquid down a flat plate that is inclined at an angle 6 to the vertical, as illustrated in Fig. 5-10. The width of the plate is W (normal to the plane of the figure). Flow is only in the v direction (parallel to... 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

Thus the kinetic energy per unit mass of a Newtonian fluid in steady laminar flow through a pipe of circular cross section is u2. In terms of head this is u2/g. Therefore for laminar flow, a = i in equation 1.14. 

The Hagen-Poiseuille equation for steady laminar flow of Newtonian fluids in pipes and tubes can be written as... 

Consider the steady, laminar flow of an incompressible fluid in a long and wide closed conduit channel subject to a linear pressure gradient, (a) Derive the equation for velocity profile, (b) Derive the equation for discharge per unit width and cross-sectional mean velocity, and compare this with the maximum velocity in the channel, (c) Derive the equation for wall shear stress on both walls and compare them. Explain the sign convention for shear stress on each wall. 

H. Steady Laminar Flow along a Plate with Homogeneous Reaction... 

Let us consider a semiinfinite fluid in steady laminar flow along a plate. Initially, the plate and fluid are at the temperature T, but for times t > 0 the temperature of the wall becomes equal to Tw. In this case, the temperature field satisfies the equation... 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

The equations will here be presented in terms of a cartesian coordinate system, the velocity vector V having components u, v, and w in the three coordinate directions x, y, and z, respectively. For the moment, the discussion will also be restricted to steady laminar flow in which the flow variables u, v, w, p, T do not vary with time. 

This is the continuity equation for turbulent flow when the mean motion is two-dimensional. It will be noted that this equation has exactly the same form as the continuity equation for two-dimensional steady laminar flow with the mean values of the velocity components substituted in place of the steady values that apply in laminar flow. This result can, in fact, be deduced by intuitive reasoning and simply states that if an elemental control volume through which the fluid flows is considered, then over a sufficiently long period of time, the fluctuating components contribute nothing to the mass transfer through this control volume. 

Write out the continuity, Navier-Stokes, and energy equations in cylindrical coordinates for steady, laminar flow with constant fluid properties. The dissipation term in the energy equation can be ignored. Using this set of equations, investigate the parameters that determine the conditions under which similar" velocity and temperature fields will exist when the flow over a series of axisymmetrie bodies of the same geometrical shape but with different physical sizes is considered. 

Attention will initially be restricted to two-dimensional steady laminar flow. The Boussinesq assumptions will again be used and, consistent with this assumption, dissipation effects will be neglected. The governing equations, expressed in Cartesian coordinates, are then [1] ... 

In this theory, equilibrium flow is obtained using thin shear layer (TSL) approximation of the governing Navier- Stokes equation. However, to investigate the stability of the fluid dynamical system the disturbance equations are obtained from the full time dependent Navier- Stokes equations, with the equilibrium condition defined by the steady laminar flow. We obtain these in Cartesian coordinate system given by. 

The exponent in Eqs. (3.2-4,5) is an asymptote given by boundary layer theory for steady laminar flows and for steadily driven turbulent flows. This dependence is consistent with the cited data for values above 0.6 of the Prandtl number Pr = Cpp/k and of the Schmidt number Sc = a/pT ab-... 

The shell balance method will be used to examine steady laminar flow of a fluid in a pipe. For the geometrical system illustrated in Figure 3B-1 and for steady laminar fully developed flow of a fluid, a shell momentum balance can be conducted (Bird et al., 1960 Geankoplis, 1983) using the cylindrical coordinates, r, 6, andz. The momentum balance is conducted on a control volume shell at a radius r with dimensions Ar and Az. 

Reconsider steady laminar flow of a fluid m a circular tube of radius R. The fluid properties p, k, and Cp are constant, and the work done by viscous forces is negligible. The fluid flows along the.r-axis with velocity n. Tlie flow is fully developed so that i< is independent of, v and thus u = n(r). Noting that energy is transferred by mass in the A-direction, and by conduction in the r direction (heat conduction in the. v-direction.is assumed to be negligible), the steady-flow energy balance for a cylindrical shell element of thickness dr and length d. can be expressed as (Fig. 8-21)... 

Some simple heal transfer equipments consist of two concentric tubes, and are properly called double-tube heat exchangers (Fig. 8-27). In such devices, one fluid flows through the tube while the other flows through the aunular space. The governing differential equations for both flow.s are identical. I herefore, steady laminar flow through an annulus can he studied analytically by using suitable boundary conditions. 

The function of the orientational distribution of nwlecules p (, d) is found by solvu the equation of rotatory diffusion in a steady laminar flow >... 

Toh et al. [45] investigated numerically three-dimensional fluid flow and heat transfer phenomena inside heated microchannels. The steady, laminar flow and heat transfer equations were solved using a finite-volume method. The numerical procedure was validated by comparing the predicted local thermal resistances with available experimental data. The friction factor was also predicted in this study. It was found that the heat input lowers the frictional losses, particularly at lower Reynolds numbers. Also, at lower Reynolds numbers the temperature of the water increases, leading to a decrease in the viscosity and hence smaller frictional losses. 

The velocity profile across the tube lumen with pulsatile flow is not of the same parabolic form as that found in a steady laminar flow. The velocity profiles oscillate sinusoidally as discussed in detail by Hale et al. [44]. For example. Figure 8.26 shows the velocity profiles, at intervals of 15°, resulting from a simple sinusoidal pressure gradient (cos[mf]) during the half cycle (0°-180°) as for a simple harmonic motion, the second half is the same. 

A fluid s motion is a function of the properties of the fluid, the medium through which it is flowing, and the external forces imposed on it. For onedimensional steady laminar flow of a single fluid through a homogeneous porous medium, the relationship between the flow rate and the applied external forces is provided by Darcy s law ... 

For steady laminar flow, the relaxation time (C) has the following relationship with shear rate (y), shear stress X12, and the first normal stress difference ... 

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word hydrodynamic is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition. 

Viscosity can also be determined by measuring the total pressure drop (AO = AP + pgAz) and flow rate (Q) in steady laminar flow through a uniform circular tube of length L and diameter D (this is called Poiseuille flow). The shear stress at the tube wall (xj is determined from the measured pressure drop ... 

In this work, heat and fluid flow in some common micro geometries is analyzed analytically. At first, forced convection is examined for three different geometries microtube, microchannel between two parallel plates and microannulus between two concentric cylinders. Constant wall heat flux boundary condition is assumed. Then mixed convection in a vertical parallel-plate microchannel with symmetric wall heat fluxes is investigated. Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. In this study, the usual continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. 

The MRI-based viscosity measurement relies on local calculations of the velocity gradient based on using MRI velocity profiles to provide a wide range of shear viscosity-shear rate data. Two observations are used to characterize fully developed and steady laminar flow in a MRI-based viscometer the velocity profile and the pressure drop per unit tube length measurements. Viscosity measurements were acquired on strawberry milk samples using H P LG tubing for the sample channel. 

M. Tachibana, and Y. lemoto, Steady Laminar Flow in the Inlet Region of Rectangular Ducts, BulLJSME, (24/193) 1151-1158,1981. 

H. C. Topakoglu, and O. A. Arnas, Convective Heat Transfer for Steady Laminar Flow between Two Confocal Elliptical Pipes with Longitudinal Uniform Wall Temperature Gradient, Int. J. Heat Mass Transfer, (17) 1487-1498,1974. 

The NMRI technique uses an induction coil surrounding the sample to image nuclear spin density that results from the nuclear spin system rearrangement. An initial magnetic pulsed field orients the nuclear spin system and then it relaxes back toward a random state. Because of the relaxation time and that tomographic reconstruction is needed to extract the 3-D details, there are time limitations (Altobelli et al., 1992) (currently of the order of 10-ms, at best). Consequently, the technique has been used mostly for steady or quasi-steady laminar flows because of the rather low data acquisition rate. However, modifications to allow turbulent and unsteady flows to be investigated have been reported and new... 

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