The Viscous Stress

In the decoupled scheme the solution of the constitutive equation is obtained in a separate step from the flow equations. Therefore an iterative cycle is developed in which in each iterative loop the stress fields are computed after the velocity field. The viscous stress R (Equation (3.23)) is calculated by the variational recovery procedure described in Section 1.4. The elastic stress S is then computed using the working equation obtained by application of the Galerkin method to Equation (3.29). The elemental stiffness equation representing the described working equation is shown as Equation (3.32). 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

To understand how the dispersed phase is deformed and how morphology is developed in a two-phase system, it is necessary to refer to studies performed specifically on the behavior of a dispersed phase in a liquid medium (the size of the dispersed phase, deformation rate, the viscosities of the matrix and dispersed phase, and their ratio). Many studies have been performed on both Newtonian and non-Newtonian droplet/medium systems [17-20]. These studies have shown that deformation and breakup of the droplet are functions of the viscosity ratio between the dispersity phase and the liquid medium, and the capillary number, which is defined as the ratio of the viscous stress in the fluid, tending to deform the droplet, to the interfacial stress between the phases, tending to prevent deformation ... 

The elastic stress curve in figure perfectly follows elastic strain [2]. This constant is the elastic modulus of the material. In this idealized example, this would be equal to Young s modulus. Here at this point of maximum stretch, the viscous stress is not a maximum, it is zero. This state is called Newton s law of viscosity, which states that, viscous stress is proportional to strain rate. Rubber has some properties of a liquid. At the point when the elastic band is fully stretched and is about to return, its velocity or strain rate is zero, and therefore its viscous stress is also zero. 

In the context of the preceding model, a drop is said to break when it undergoes infinite extension and surface tension forces are unable to balance the viscous stresses. Consider breakup in flows with D mm constant in time (for example, an axisymmetric extensional flow with the drop axis initially coincident with the maximum direction of stretching). Rearranging Eq. (26) and defining a characteristic length Rip113, we obtain the condition, for a drop in equilibrium,... 

The second term on the right-hand side of Eqs. (145) and (146) contains the viscous-stress models ag and asm. Even for laminar flow, suitable forms for these models are difficult to determine a priori. Typical models used in CFD introduce an effective viscosity pea for each phase, and describe the viscous stresses as follows. 

As the fluid s velocity must be zero at the solid surface, the velocity fluctuations must be zero there. In the region very close to the solid boundary, ie the viscous sublayer, the velocity fluctuations are very small and the shear stress is almost entirely the viscous stress. Similarly, transport of heat and mass is due to molecular processes, the turbulent contribution being negligible. In contrast, in the outer part of the turbulent boundary layer turbulent fluctuations are dominant, as they are in the free stream outside the boundary layer. In the buffer or generation zone, turbulent and molecular processes are of comparable importance. 

Figure 1.26. For clarity, the magnitude of the viscous stress is exaggerated...

Equations 2.3 to 2.6 are true, irrespective of the nature of the fluid. They are also valid for both laminar and turbulent flow. In the latter case, the shear stress is the total shear stress comprising the viscous stress and the Reynolds stress. 

The terms on the left hand side of equation A.22 represent inertial stresses, the first due to acceleration and the others to advection. The first and second terms on the right hand side are the component of the gravitational force and the pressure gradient. The remaining terms represent the viscous stress components acting in the x-direction. 

From dimensional considerations, the drag coefficient is a function of the Reynolds number for the flow relative to the particle, the exponent, nm, and the so-called Bingham number Bi which is proportional to the ratio of the yield stress to the viscous stress attributable to the settling of the sphere. Thus ... 

In contrast, in a model proposed by Voigt and Kelvin (Figure 5.5), in which the spring and dashpot are in parallel, the applied stress is shared, and each element is deformed equally. Thus the total stress S is equal to the sum of the viscous stress ij (dy/dt) plus the elastic stress Gy ... 

Moreover, if one assumes that the (Ur) changes very slowly on the length scale of the porous media, (i.e., ), then the viscous stress term in the Brinkman equation can be neglected and this equation reduces to ... 

Here a is the elastic stress which arises from the change in the (dynamic) free energy in the macroscopic flow, while o(V) and a(S) are the viscous stresses produced by the polymer-solvent friction and the solvent-solvent friction, respectively. In concentrated isotropic polymer solutions, the elastic stress overwhelms the viscous stresses, so the latter are often neglected. However, it should be noticed that the viscous stresses may become significant in more dilute solutions as well as in nematic solutions where the elastic stress diminishes. 

Consider a fluid element of constant mass pAxAyAz moving along with the local fluid velocity v. The x component of momentum of this fluid element is pvxAxAyAz. The momentum of the fluid element as it moves along with the local fluid velocity is a function of both space and time. The total derivative of the momentum of the fluid element with respect to time is then pAxAyAz Dvx/Dt). According to Newton s second law this quantity is to be equated to the forces acting on the element of mass the net force in the x direction due to the difference in pressure on faces a and b, which is [p x)AyAz — p(x + Ax)AyAz], the net force in the x direction due to the difference in the viscous stresses,2 which is... 

When a fluid is in turbulent flow past a rigid surface, fluctuations of velocity in the direction normal to the surface are inhibited, and very close to the surface they may he negligible. Then the Reynolds shear stress is small compared with the viscous stresses, and it has been common to describe the region as a laminar sublayer. In fact, turbulent fluctuations of velocity in planes parallel to the wall are considerable in comparison with the mean velocity. 

The pressure drop across the cyclone is an important parameter in the evaluation of cyclone performance. It is a measure of the amount of work that is required to operate the cyclone at given conditions, which is important for operational and economical reasons. The total pressure drop over a cyclone consists of losses at the inlet, outlet and within the cyclone body. The main part of the pressure drop, i.e. about 80%, is considered to be pressure losses inside the cyclone due to the energy dissipation by the viscous stress of the turbulent rotational flow [9], The remaining 20% of the pressure drop are caused by the contraction of the fluid flow at the outlet, expansion at the inlet and by fluid friction on the cyclone wall surface. 

Thus, we have uz = uz(r), ur = ug = 0 and p = p(z). With this type of velocity field, the only non-vanishing component of the rate-of-deformation tensor is the zr-component. It follows that for the generalized Newtonian flow, rzr is the only nonzero component of the viscous stress, and that Tzr = rZT r). The -momentum equation is then reduced to,... 

Relative Importance of the Various Terms in the Analysis of the Isothermal Fiber Spinning of a Newtonian Melt Use the data and result of Problem 14.1(b) to evaluate the importance in the isothermal fiber spinning of a Newtonian melt analysis (nylon 6-6 at 285°C) of the inertial terms and gravity relative to the viscous stress terms. Using Eq. E14.1-2 for FD, evaluate the importance of the air-drag force term. 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu 2, -pv 2, and -pw 2 are always non-zero because they contain squared velocity fluctuations. The shear stresses -puV, -pu w, -pv w and are associated with correlations between different velocity components. If, for instance, u and v were statistically independent fluctuations, the time average of their product uV would be zero. However, the turbulent stresses are also non-zero and are usually large compared to the viscous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

From the viscous stress-strain curve using Equations (4.1), (4.2), and (8.2) we can calculate the collagen fibril length. The collagen fibril lengths in tendon range from about 20 pm for during tendon development to in excess... 

Figure 7.7. Total, elastic, and viscous stress-strain curves for uncrosslinked self-assembled type I collagen fibers.Total (open squares), elastic (filled diamonds), and viscous (filled squares) stress-strain curves for self-assembled uncrosslinked collagen fibers obtained from incremental stress-strain measurements at a strain rate of 10%/min. The fibers were tested immediately after manufacture and were not aged at room temperature. Error bars represent one standard deviation of the mean value for total and viscous stress components. Standard deviations for the elastic stress components are similar to those shown for the total stress but are omitted to present a clearer plot. The straight line for the elastic stress-strain curve closely overlaps the line for the viscous stress-strain curve. Note that the viscous stress-strain curve is above the elastic curve suggesting that viscous sliding is the predominant energy absorbing mechanism for uncrosslinked collagen fibers.

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