**Principal Investigator**
Denis J Evans **Project **r61

Research School of Chemistry **Machine **VP

**Co-Investigator** Billy
D Todd

Research School of Chemistry

**Strain Rate Dependence of Heat Transfer as Applied
to Planar Poiseuille Flow**

Previous work done in the group demonstrated that molecular dynamics simulations of a fluid under the influence of a sinusoidal transverse force could generate heat flow even in the absence of a temperature gradient, the direction of flow being from high to low shear regions of the fluid. A simple phenomenological explanation for this unexpected property was proposed by postulating an additional coupling term to the familiar Fourier heat equation, one which predicts that in addition to the heat flow generated by a temperature gradient, an extra component of heat flux exists which is proportional to the gradient in the square of the strain rate tensor.

The purpose of this project is to consider the case of planar Poiseuille flow through thin porous solid walls. This has involved writing a molecular dynamics program to perform nonequilibrium simulations in order to measure the temperature and heat flux profile across the channel, and hence to determine the standard classical thermal conductivity as well as the newly proposed strain rate coupling constant.

In addition, the algorithm has been extended to include an investigation of the dependence of the thermal conductivity and viscosity on distance within the pore. In order to calculate viscosity as a function of cross-channel distance one needs to know accurately the Pxy (shear stress) component of the pressure tensor, which for atomic fluids is often calculated by the Irving-Kirkwood method. This method relies upon an expansion of the difference of delta function terms associated with pairs of atoms. If the density of the fluid is uniform in space then the expansion reduces to unity and an accurate and simple expression may be used to determine the pressure tensor of the fluid. However, if the fluid density is not uniform in space, as is the case of a fluid sandwiched in a narrow channel, then additional terms in the expansion begin to dominate and the expression for the pressure tensor is no longer simple.

In an attempt to overcome this difficulty we have developed a simple technique in which the potential and kinetic contributions to the pressure tensor are treated independently of each other. This is done by creating a set of imaginary infinitesimal planes parallel to the atomic walls that extend into the fluid. Potential contributions to the pressure tensor at a particular plane are then calculated as the force per unit area exerted on the plane by a pair of atoms whose line of force intersects it. Kinetic components of the pressure tensor are calculated from instantaneous impulses that the imaginary plane experiences when an atom transfers momentum upon intersecting it.

A similar method has been developed to calculate the heat flux vector for inhomogeneous fluids. The heat flux vector needs to be calculated before the thermal conductivity and cross coupling dependence may be determined. These techniques have led to several recent publications and have proven to be more efficient and reliable than the Irving-Kirkwood method for inhomogeneous liquids.

**What are the basic questions addressed?**

In cases where fluid flow involves non-uniform strains, is there a coupling of the strain rate tensor to the heat flow? How do the viscosity and thermal conductivity across a narrow pore behave in regions where the fluid density oscillates rapidly as a function of distance across the channel?

**What are the results to date and future of the
work?**

Accurate techniques to calculate the pressure tensor and heat flux vector have been developed. We are now close to answering the question: is there a contribution to the heat flux vector from the strain rate tensor. The above techniques have been developed for simple atomic fluids. We will in time extend these methods to consider the case of more complex molecular fluids.

**What computational techniques are used and why
is a supercomputer required?**

Molecular dynamics techniques are used to solve Newtonian equations of motion under nonequilibrium planar Poiseuille flow. The simulations involve numerical solutions to systems of equations which involve up to several thousand particles. Each run involves about 100 000 timesteps, and a number of runs are required for a sufficient set of results to be compiled. It is thus essential to have powerful and fast computing to perform the many calculations involved.

**Publications**

*Heat flux vector in highly inhomogeneous nonequilibrium
fluids,* B.D. Todd, P.J. Daivis, and D.J.
Evans, Physical Review E, **51**, 4362-4368 (1995).

*The heat flux vector for highly inhomogeneous
nonequilibrium fluids in very narrow pores*,
B.D Todd. and D.J. Evans, Journal of Chemical Physics, **103**,
9804-9809 (1995).

*The pressure tensor for inhomogeneous fluids*,
B.D.Todd, D.J. Evans, and P.J. Daivis, Physical Review E,** 52,
**1627-1638 (1995).

*A technique for the calculation of mass, energy
and momentum densities at planes in molecular dynamics simulations*,
P. J. Daivis, K. P. Travis and B. D. Todd, Journal of Chemical
Physics, Accepted for publication.