Isoflux boundary

The mathematical model describing this problem was developed as follows. The potential obeys the Laplace equation in a domain limited by the cavity and the surrounding resist structure at the bottom, by lateral symmetry planes, and by an isoflux boundary far above the electrode surface ... 

Steady-state and transient constriction (spreading) resistances for a range of geometries for isothermal and isoflux boundary conditions are given. Analytical solutions for half-spaces and heat flux tubes and channels are reported. 

When Bi < 0.1, the values predicted by the expression approach the values corresponding to the isoflux boundary condition and when Bi > 100, the predicted values are within 0.1 percent of the values obtained for the isothermal boundary condition. The transition from the isoflux values to the isothermal values occurs in the range 0.1 < Bi < 100. 

The solution for the isoflux boundary condition and with external thermal resistance was recently reexamined by Song et al. [156] and Lee et al. [157]. These researchers nondimen-sionalized the constriction resistance based on the centroid and area-average temperatures using the square root of the contact area as recommended by Chow and Yovanovich [15] and Yovanovich [132,137,144-146,150], and compared the analytical results against the numerical results reported by Nelson and Sayers [158] over the full range of the independent parameters Bi, e, and x. Nelson and Sayers [158] also chose the square root of the contact area to report their numerical results. The analytical and numerical results were reported to be in excellent agreement. 

Solutions exist for a porous channel with two porous walls for laminar channel flow with heat transfer for asymmetric isothermal or isoflux boundaries (Prasad and Kulacki, 1984b Haji-Sheikh et al., 2006), with arbitrary uniform injection (Doughty and Perkins, 1973), arbitrary uniform suction or injection (Rao and Prasad, 1977 Nield et al., 1996), and for oscillating walls with small uniform suction or injection (Sharma and Yadav, 2005). 

Solutions exist for a porous channel with one porous wall for laminar channel flow with heat transfer for asymmetric isothermal or isoflux boundaries with arbitrary uniform suction (Rhee and Edwards, 1981). 

Nield, D.A., Junqueira, S.L.M., Lage, J.L., 1996. Eorced convection in a fluid-saturated porous-medium channel with isothermal or isoflux boundaries. J. Eluid Mech. 322, 201-214. 

The effect of single and multiple isotropic layers or coatings on the end of a circular flux tube has been determined by Antonetti [2] and Sridhar et al. [107]. The heat enters the end of the circular flux tube of radius b and thermal conductivity k3 through a coaxial, circular contact area that is in perfect thermal contact with an isotropic layer of thermal conductivity k, and thickness This layer is in perfect contact with a second layer of thermal conductivity k2 and thickness t2 that is in perfect contact with the flux tube having thermal conductivity k3 (Fig. 3.22). The lateral boundary of the flux tube is adiabatic and the contact plane outside the contact area is also adiabatic. The boundary condition over the contact area may be (1) isoflux or (2) isothermal. The dimensionless constriction resistance p2 layers = 4k3aRc is defined with respect to the thermal conductivity of the flux... 

For the UHF boundary condition, the local mixed convection Nusselt number for a horizontal, isoflux, continuous moving sheet can be expressed by the equation [71] ... 

The solution derived in Chapter 12 is for laminar channel flow with one porous wall with arbitrary, variahle suction or injection, with no heat transfer, electric or magnetic field. This appendix presents a comprehensive review of historical work on porous element flow solutions, analytically, numerically, and experimentally. Results are organized and classified by the type of porous element and number of porous walls (for channels), the nature of flow within the porous element (laminar or turbulent), the size (small, large, arbitrary) and nature (uniform or variable) of injection/suction Renumber into or out of the porous element, the type of transverse and axial boundary conditions at the porous surface (suction or injection), whether or not the flow has an electric or magnetic component or any other special feature, such as moving boundaries, non-Newtonian flow, and whether or not there is heat transfer. Heat transfer solutions are organized by boundary condition type, either isothermal or isoflux wall. 

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