Equations of a Viscous, Heat-Conducting Fluid

Conservation Equations of a Viscous, Heat-Conducting Fluid.4-59... 

In an ideal fluid, the stresses are isotropic. There is no strength, so there are no shear stresses the normal stress and lateral stresses are equal and are identical to the pressure. On the other hand, a solid with strength can support shear stresses. However, when the applied stress greatly exceeds the yield stress of a solid, its behavior can be approximated by that of a fluid because the fractional deviations from stress isotropy are small. Under these conditions, the solid is considered to be hydrodynamic. In the absence of rate-dependent behavior such as viscous relaxation or heat conduction, the equation of state of an isotropic fluid or hydrodynamic solid can be expressed in terms of specific internal energy as a function of pressure and specific volume E(P, V). A familiar equation of state is that for an ideal gas... 

The mechanism of thermal conduction is analogous to that of viscous resistance to fluid flow. In the case of fluid flow, where a velocity gradient exists, momentum is transported from point to point by gas molecules in thermal conduction, where a temperature gradient exists, it is kinetic energy that is transported from point to point by gas molecules. If, in the mean-free-path treatment of viscosity, the momentum is replaced by the average kinetic energy s of a molecule, we obtain for one-dimensional heat flow in the x direction an equation analogous to Eq. (8) ... 

Equation 5 for / , the depth of the lubricating fluid, is based on melting due to viscous dissipation (term 1 on rhs), offset by energy losses due to heat conduction into the ice, both fast (term 3) and slow (term 4), and loss of lubricating fluid through a lateral squeeze flow (term 2) from under the blade. We have ignored heat transfer with the skate blade here, but it is considered in Penny et al. The lubrication equation is ... 

It is important to notice the similarity between Eqs. 1.1,1.6, and 1.20. The heat conduction equation, Eq. 1.1, describes the transport of energy the diffusion law, Eq. 1.6, describes the transport of mass and the viscous shear equation, Eq. 1.20, describes the transport of momentum across fluid layers. We note also that the kinematic viscosity v, the thermal diffusivity a, and the diffusion coefficient D all have the same dimensions L2/f. As shown in Table 1.10, a dimensionless number can be formed from the ratio of any two of these quantities, which will give relative speeds at which momentum, energy, and mass diffuse through the medium. 

Given the importance of low-Re, viscous flow on microscale aerodynamics, it is possible to take advantage of the dominant heat transfer effects to enhance microrotorcraft flight. These heat effects can be characterized using the standard transport equations and a Navier-Stokes solver. In order to accurately apply the physical properties, it is important to include the effect of temperature on the viscosity (using, e.g., Sutherland s, Wilke s, or Keyes laws), thermal conductivity, and specific heat of the surrounding fluid (air). 

In heat transfer in a fluid in laminar flow, the mechanism is one of primarily conduction. However, for low flow rates and low viscosities, natural convection effects can be present. Since many non-Newtonian fluids are quite viscous, natural convection effects are reduced substantially. For laminar flow inside circular tubes of power-law fluids, the equation of Metzner and Gluck (M2) can be used with highly viscous non-Newtonian fluids with negligible natural convection for horizontal or vertical tubes for the Graetz number Nq, > 20 and n > 0.10. 

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. 

In verbal form, Eq. (4-1) shows that the temperature of a fluid element in motion is affected by heat conduction, the q terms in the first bracket expansion effects, the second term on the right multiplied by T dp dt)p and viscous heating or viscous dissipation, the remainder of the terms on the right-hand side of the equation excepting Aq. Aq is for all other types of heat generation, such as phase changes, chemical sources, and electrical sources. 

Consider a simple nonuniform diameter natural circulation loop as shown in Fig. 1 with a horizontal heat source at the bottom and a horizontal heat sink at the top. The heat sink is maintained by providing cooling water to the secondary side of the cooler at a specified inlet temperature of Tj. In this analysis, the secondary side temperature is assumed to remain constant. The heat flux at the heat source is maintained constant. Assuming the loop to be filled with an incompressible fluid of constant properties except density (Boussinesq approximation where density is assumed to vary as p=pr[l-P(T-Tr)]) with negligible heat losses, axial conduction and viscous heating effects, the governing differential equations can be written as... 

Liquid metals constitute a class of heat-transfer media having Prandtl numbers generally below 0.01. Heat-transfer coefficients for liquid metals cannot be predicted by the usual design equations applicable to gases, water, and more viscous fluids with Prandtl numbers greater than 0.6. Relationships for predicting heat-transfer coefficients for liquid metals have been derived from solution of Eqs. (5-38a) and (5-38b). By the momentum-transfer-heat-transfer analogy, the eddy conductivity of heat is = k for small IVp,. Thus in the solu-... 

The situation is analogous to momentum flux, where the relative Importance of turbulent shear to viscous shear follows the same general pattern. Under certain ideal conditions, the correspondence between heat flow and momentum flow is exact, and at any specific value of rjr the ratio of heat transfer by conduction to that by turbulence equals the ratio of momentum flux by viscous forces to that by Reynolds stresses. In the general case, however, the correspondence is only approximate and may be greatly in error. The study of the relationship between heat and momentum flux for the entire spectrum of fluids leads to the so-called analogy theory, and the equations so derived are called analogy equations. A detailed treatment of the theory is beyond the scope of this book, but some of the more elementary relationships are considered. 

Equation 2.22 is a bit difficult to interpret physically because it is an equation for the change in temperature, not the change in energy, from which it is derived. Roughly, the left side represents the accumulation of the internal energy in a fluid element. The term with the thermal conductivity k represents the heat flow to and from the fluid element because of thermal conduction, while the viscous dissipation term reflects the rate of increase in internal energy because of work done on... 

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