Bruce A. Finlayson

Convective instability of ferromagnetic fluids

Convective instability of a ferromagnetic fluid is predicted for a fluid layer heated from below in the presence of a uniform vertical magnetic field. Convection is caused by a spatial variation in magnetization which is induced when the magnetization of the fluid is a function of temperature and a temperature gradient is established across the layer. A linearized convective instability analysis predicts the critical temperature gradient when only the magnetic mechanism is important, as well as when both the magnetic and buoyancy mechanisms are operative. The magnetic mechanism predominates over the buoyancy mechanism in fluid layers about 1 mm thick. For a fluid layer contained between two free boundaries which are constrained flat, the exact solution is derived for some parameter values and oscillatory instability cannot occur. For rigid boundaries, approximate solutions for stationary instability are derived by the Galerkin method for a wide range of parameter values. It is shown that in this case the Galerkin method yields an eigenvalue which is stationary to small changes in the trial functions, because the Galerkin method is equivalent to an adjoint variational principle.

Orthogonal collocation on finite elements

The effectiveness factor problem for heat and mass transfer with chemical reaction in a catalyst pellet is solved with a new technique especially suited to situations corresponding to high Thiele modulus when the solution is confined to a thin boundary region near the catalyst surface. The method of orthogonal collocation on finite elements combines the rapid convergence of the orthogonal collocation method with the convenience associated with finite difference methods of locating grid points or elements where the solution is important or has large gradients. The efficiency of the method results from block LU decompositions employed in the iterative schemes devised. The method is applied to two problems to illustrate the rate of convergence, the efficiency (as expressed by error versus computation time curves), and the use of the residual for optimum location of the finite elements. Comparisons are also made to usual orthogonal collocation and finite difference methods.

Heat transfer in packed beds—a reevaluation

Data for heat transfer from packed beds are reexamined in the light of new insights. Much of the data includes a length effect, resulting from a higher BidpϵR(1-ϵ)=0.27.

Combinatorial mixing of microfluidic streams

We have devised a microfluidic mixer design that produces all the mixture combinations of a given number of dilutions of the input compounds. As proof of the concept, we present a device that generates four titrations of two dye solutions, blue and yellow, and combinatorially mixes the blue titrations with the yellow titrations to deliver the sixteen mixture combinations in separate outlet microchannels. Our device features four different flow levels made by stacking nine laser-cut Mylar laminates. The fluidic network has a symmetric design that guarantees that the flow rates are the same at all the outlets, with deviations attributable to imperfections in the fabrication, assembly, or perfusion processes. Design rules for scaling up the number of compounds and/or dilutions are presented. The mixing scheme has broad applicability in high-throughput combinatorial testing applications such as drug screening, cell-based biochemical assays, lab-on-a-chip devices, and biosensors.

Packed bed reactor analysis by orthogonal collocation

The equations governing a packed bed reactor with radial temperature and concentration gradients are solved using the orthogonal collocation method. The method is shown to be faster and more accurate than finite difference calculations. Using the orthogonal collocation method it is straightforward to extend one-dimensional (lumped parameter) models to the two-dimensional models needed when radial temperature and concentration gradients are important. The two-dimensional model is necessary for large Biot numbers, hwR/ke, where hw is the wall heat transfer coefficient, R is the tube radius, and keis the tube radius, and ke is the effective thermal conductivity. For Biot number less than one, seventy-five per cent of the resistance to heat transfer is at the wall, and a one-dimensional (lumped parameter) treatment gives good results. Computations are illustrated for both plug flow and radially-varying velocity. In the latter case the velocity profile induces effective thermal conductivity and diffusivity profiles. Calculations made using the velocity profile predict a heat transfer coefficient which is used in the plug flow model. Good agreement is obtained between the models.

Finite element model of magnetoconvection of a ferrofluid

Combined natural and magnetic convective heat transfer through a ferrofluid in a cubic enclosure is simulated numerically. The momentum equation includes a magnetic term that arises when a magnetic fluid is in the presence of a magnetic field gradient and a temperature gradient. In order to validate the theory, the wall temperature isotherms and Nusselt numbers are compared to experimental work of Sawada et al. (Int. J. Appl. Electromagn. Mater. 4 (1994) 329). Results are obtained using standard computational fluid dynamics codes, with modifications to account for the Langevin factor when needed. The CFD code FIDAP uses the finite element method, sometimes with a user-defined subroutine. The CFD code FEMLAB uses the finite element method with a user-supplied body force.

Application of the Continuum Theory to Nematic Liquid Crystals

The apparent viscosity of nematic p‐azoxyanisole in Poiseuille flow at 122°C is calculated by numerically solving the differential equation derived by Atkin which is based on the continuum theory of Ericksen and Leslie for nematic liquid crystals. The theory requires knowledge of seven independent material properties, which are determined from experimental measurements of Miesowicz, Marinin and Zwetkoff, Zwetkoff, and the Orsay Liquid Crystal Group. Parodis relation is used to show that in cases where there are conflicting data, some data are inconsistent with Onsagers reciprocal relation. The calculations are compared with the Poiseuille flow experiments of Fisher and Fredrickson.

Heat transfer enhancement in ferrofluids subjected to steady magnetic fields

Finite element simulations of heat transfer to a ferrofluid in the presence of a magnetic field are presented for flow between flat plates and in a box.

Existence of Variational Principles for the Navier‐Stokes Equation

Frechet differentials are introduced to decide when a classical variational principle exists for a given nonlinear differential equation. The formalism is applied to the steady‐state Navier‐Stokes equation and the continuity equation, and no variational principle exists unless u × (∇ × u) = 0 or u· ∇u = 0. The concept of an adjoint equation is extended to nonlinear equations and a variational principle is derived for the Navier‐Stokes equation and its adjoint.

ORTHOGONAL COLLOCATION IN CHEMICAL REACTION ENGINEERING

Abstract The orthogonal collocation method is used to obtain approximate solutions to the differential equations modeling chemical reactors. There are two ways to view applications of the orthogonal collocation method. In one view it is a numerical method for which the convergence to the exact answer can be seen as the approximation is refined in successive calculations by using more collocation points, which are similar to grid points in a finite difference method. Another viewpoint considers only the first approximation, which can often be found analytically, and which gives valuable insight to the qualitative behavior of the solution. The answers, however, are of uncertain accuracy, so that the calculation must be refined to obtain useful numbers. However, with experience and appropriate caution, the first approximation is often sufficient and is easy to obtain. Thus it is very often useful in engineering work, where valid approximations are accepted. We present both viewpoints here: we use the first a...