Thermal energy equation incompressible

For high-viscosity fluids (e.g., oils), high rod velocities, or small gaps, the thermal energy generation by viscous dissipation may be important. In this case the steady-state, incompressible, thermal energy equation reduces to... 

Neglect the effects of viscous work (i.e., viscous dissipation), and compare with the steady-state form of the thermal-energy equation for an incompressible fluid. Are they the same equation If so, why If not, why ... 

Derive the nondimensional thermal-energy equation for an axisymmetric, semi-infinite stagnation flow of a constant-property incompressible fluid. 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

The approach to calculation of the temperature field and heat transfer follows closely the hydrodynamic calculation outlined above. For incompressible flow of a fluid with constant and uniform properties, neglecting the input to the thermal field by viscous dissipation, the thermal-energy equation (obtained by a combination of the energy and momentum equations) is... 

Let us now return to the equations of motion for a Newtonian fluid. With the constitutive equation, (2-80) [or (2-81) if the fluid is incompressible], the continuity equation, (2-5) [or (2-20) if the fluid is incompressible], and the Cauchy equation of motion, (2-32), we have achieved a balance between the number of independent variables and the number of equations for an isothermal fluid. If the fluid is not isothermal, we can add the thermal energy equation, (2-52), and the thermal constitutive equation, (2-67), and the system is still fully specified insofar as the balance between independent variables and governing equations is concerned. 

We begin with the dimensional form of the continuity, momentum, and thermal energy equations for an incompressible Newtonian fluid, simplified only to the extent that we neglect viscous dissipation ... 

The temperature of an incompressible fluid element in a deforming medium is governed by the equation of thermal energy, Eq. 2.9-14. This excluding the reversible compression term, and in terms of specific heat, is... 

The boundary layer equations may be obtained from the equations provided in Tables 6.1-6.3, with simplification and by an order-of-magnitude study of each term in the equations. It is assumed that the main flow is in the x direction. The terms that are too small are neglected. Consider the momentum and energy equations for the two-dimensional, steady flow of an incompressible fluid with constant properties. The dimensionless equations are given by Eqs. (6.46) to (6.48). The principal assumption made in the boundary layer is that the hydrodynamic boundary layer thickness 8 and the thermal boundaiy layer thickness 8t are small compared to a characteristic dimension L of the body. In mathematical terms,... 

In the next section, incompressible flow with constant properties and no body forces is discussed. Under such conditions, the governing momentum equations are decoupled from the governing energy equation. Once the flow field is known, different temperature distributions may be computed with different types of thermal boundary conditions. 

Now consider the flat plate shown in Fig. 12-3. The plate surface is maintained at the constant temperature Tw, the free-stream temperature is 7U, and the thermal-boundary-layer thickness is designated by the conventional symbol 5,. To simplify the analysis, we consider low-speed incompressible flow so that the viscous-heating effects are negligible. The integral energy equation then becomes... 

The temperature form of the energy equation for incompressible pure substances is yielded from (3.85) under the assumption of constant thermal conductivity... 

We are concerned in this book with the motion and transfer of heat in incompressible, Newtonian fluids. For this case, the equations of motion, continuity, and thermal energy,... 

We are now in a position to begin to consider the solution of heat transfer and fluid mechanics problems by using the equations of motion, continuity, and thermal energy, plus the boundary conditions that were given in the preceding chapter. Before embarking on this task, it is worthwhile to examine the nature of the mathematical problems that are inherent in these equations. For this purpose, it is sufficient to consider the case of an incompressible Newtonian fluid, in which the equations simplify to the forms (2 20), (2-88) with the last term set equal to zero, and (2-93). 

If the colunm were an adiabatic system, the increase in temperature would only. be a function of pressure. In practice, the problem becomes significant only when short columns packed with small particles (<5/on) are used at hig pressures (>20MPa). We can estimate the effect to a first approximation by calculating the mechanical work under adiabatic conditions and incompressibility of the fluid (5) and equating it to the thermal energy Q ... 

Kamiadakis and Beskok [6] developed a code H Flow with implementation of spectral element methods. They employed both the Navier-Stokes (incompressible and compressible) and energy equations in order to compute the relative effects of compressibility and rarefaction in gas microflow simulations. In addition, they also considered the velocity slip, temperature jump, and thermal creeping boundary conditions in the code Flow. The spatial discretization of fi Flow was based on spectral element methods, which are similar to the hp version of finite-element methods. A typical mesh for simulation of flow in a rough micro-channel with different types of roughness is shown in Fig. 1. The two-dimensional domain is broken up into elements, similar to finite elements, but each element employs high-order interpolants based on... 

Table 2.5. Energy equation for an incompressible fluid with constant thermal conductivity DT...

The equation of motion and the equation of energy balance can also be time averaged according to the procedure indicated above (SI, pp. 336 et seq. G7, pp. 191 et seq. pp. 646 et seq.). In this averaging process there arises in the equation of motion an additional component to the stress tensor t(,) which may be written formally in terms of a turbulent (eddy) coefficient of viscosity m(I) and in the equation of energy balance there appears an additional contribution to the energy flux q(1), which may be written formally in terms of the turbulent (eddy) coefficient of thermal conductivity Hence for an incompressible fluid, the x components of the fluxes may be written... 

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