ii (4)Grad 1 s thirteen moment method is applied to the problem of the shear flow and heat conduction between two concentric, rotating cylinders of infinite length

The solutions of the linearis"€

One of the simplest situations involving curvature is the problem of shear flow and heat conduction between two concentric, rotating cylinders of infinite length (cylindrical Couette flow). Moreover, this flow is one of the few that has been experimentally studied by several researchers over the entire range of gas density.

Table of Contents List of Figures List of Symbols Introduction

EQUATIONS OF MOTION AND BOUNDARY CONDITIONS

In the heat flow equations, the results of the stress equations are already used; therefore -(2/3 p~ ) qi are the only terms introduced by the collision integral. Since we are dealing with a cylindrical coordinate system that is rectangular, we have a local Cartesian coordinate system; therefore. Using this result, the remaining differential equations are extremely simplified and we can immediately integrate the conservation equations (Eq. s. 19). Equation (18) states that the torque is constant in the ring, while Eq. 19) states that the flow of thermal energy plus the rate at which the shear stress does work on the fluid is a constant.

An examination of the equations shows that solutions in reasonably simple form are difficult to find. To bring out the effect of curvature as simply as possible, linearized equations and boundary conditions will be used instead.

CYLINDRICAL COUETTE FLOW

The whole problem will be solved after evaluating the remaining two constants c and c2, which can be done by replacing q/, prr 1 and. The temperature of the gas at the surface of the inner cylinder is given by. As expected, there is no temperature spike at normal density. we have the surface of the outer cylinder in the limiting case when Re/M-+ 0.

However, when Re/M-+- 0, we encounter the problem of CHM approaching zero instead of the constant value given by the free molecule calculation. Thus, in the free molecular boundary, the temperature distribution and heat transfer are physically unrealistic, (see section IV.) Graph CHM vs.

On the other hand, Grade 1's distribution function [ Eq. wedge-like domains of influence of the two cylinders at the point P,. as well as the angular dependence of ti [Eq. When the width of the annulus is small compared to the inner cylinder radius, the solutions of the linearized Grad equations for p rQ. For example, in the limit Re/M __._ 0 the effect of the inner cylinder rotation on the gas dies off with radial distance like the solid.

We conclude that excellent agreement is obtained between the solutions of the linearized Grad equations for steady, planar Couette flow3. Similar conclusions can be drawn from a study of Goldberg's solution 11 of the linearized Grad equations for slow 11 flow11 over a sphere. Even for small (but finite) values of (b-a)/a, the heat transfer rate given by the solution of the linearized Grad equations approaches zero faster than the density at the boundary Re/M--?- 0 [Eq. for linearized plane Couette flow, and the heat transfer rate approaches the exact free molecule.

Because the geometry of the present problem is so simple, this dilemma can be solved with a slight modification of Grad's method. Goldberg's solution 11 of the linearized Grad equations for slow 11" flow over a sphere exhibits the same contradictions in the rate of heat transfer in the limit Re/M ~ 0 • Here p and. These remarks are applicable for any f that is a simple extension of the Chapman-Enskog polynomial.

Perhaps these problems can be avoided by using a two-tailed polynomial distribution of the form. Of course, no integral method is unique, but the choice of weighting function f to use seems to be. When calculating flow problems, we are often primarily interested in certain lower moments of the velocity distribution function rather than the function itself.

The temperature of the wire was kept constant and the heat loss at different pressures as measured. Here Tw is the temperature of the wire and Tb is the temperature of the gas at the wall of the bell jar.

FIG. I - SCHEMATIC PICTURE OF THE PROBLEM '

PART II

INTRODUCTION

If we examine the existing solutions of the Grad 1 equations, we are not surprised to find that most of the cases considered so far. So far, all solutions of Grad's equations have been obtained for the case of a stationary state, except for the Rayleigh problem 11. In this particular problem, the equations 11 are of an acoustic nature and the solutions, at least in the limiting cases, are obtained for a thermally insulated plate.

The features show an initial linear growth in time, and the solutions show interesting features that are quite different in nature from those of Navier-Stokes8. The present work deals with the one-dimensional unsteady problem, which can be considered as an extension.

LINEARIZED GRAD'S EQUATIONS

A total of fifteen unknowns are involved in Eq. 1) to (5); therefore, in addition, we need the equation of state, which is also obtained from a certain moment relation 4. Furthermore, from the definition of the moments and also from Eq. Therefore, in general, only five stresses need to be solved, and the total number of moment equations is reduced to thirteen. We can use the above relations to linearize Eq. 1) to (6) by dropping all products and squares of disturbances. One would obtain the steady state Oseen1 s type of equations by applying a Galilean transformation to the above equations 7.

The Laplace transform with respect to t of any quantity Q = Q(x, t) is. and the inverse transform is defined as. where

In principle, after finding the fundamental solutions of the problem, solutions corresponding to any other given functions can be generated. One would expect that to obtain all the transformed quantities, the integral. of the Green's function G~2) (x-J ) with respect to J is continuous;. However, the exact evaluation of these transformations involves a lot of difficulty due to the complicated expressions we encounter.

Characteristics (x, t) = constant of the linearized system are found by vanishing the determinant:. where dx/dt is the slope of the various characteristic curves. This behavior is to be expected, since no transverse quantity appears in the one-dimensional problem. linearization, all "characteristic curves" are straight lines and are known in advance. The second term contributes to the integral .. which is a damping term such that all induced disturbances decay exponentially.

The transformations, after being extended to power series of l/A, can be represented in the following form: To evaluate a contour integral, the singularities and branch points of the integrand must first be located in order to fully understand the behavior of the integral to understand. As you can see, they are all located to the left of the imaginary axis in the complex .4.

DISCUSSION AND CONCLUSIONS

Apparently the application of the delta force function introduces a kind of heat dipole, or equal and opposite. Thus, in the case of the heat input function, the wavefront and wake disturbances are equally important; but in the case of the force impulse function, most of the disturbance is in the waves. For t/tf < < 1, Grad's equations yield a kind of average behavior, as observed in reference 11 for Rayleigh's problem.

It would be desirable to examine the present problem with the help of a somewhat more. It would also be instructive to study other non-steady flow problems, such as the disturbance caused by the sudden heating of an infinite stationary flat plate, or the piston problem, which has some close similarities to the present problem. Yao-Tsu: Small disturbances in the unsteady flow of a compressible, viscous and heat conducting fluid.

Read: Plane Couette flow at low Mach number according to the kinetic theory of gases.

APPENDIX I

In this integral ~ = 0 is a branch point; so we can consider the contour as the one given in Figure 2. o 1 is equivalent to the integral along path I, but we know that I= - III - IV since II and V vanish as R goes to infinity. Along the road z = iy: hence. o3 is integrated in the same way as o 1. SOME STATEMENTS ABOUT FUNDAMENTAL SOLUTIONS. In this Appendix we will state some simple theorems with proofs about fundamental solutions of some special linear differential equations. l) is the fundamental solution of.

J 00 means integrate over all components of the position vector. Then the basic solution is. III.

APPENDIX IV SOLUTIONS OF

The above equations can be replaced by an equivalent set of first-order partial differential equations similar to the Grado scheme in distorted x and t coordinates introduced earlier. Equations (IV. 5), (IV. 6) and (IV. 7) are exactly the same as Grad's for 0 = 5/3, since they represent nothing more than conservation of mass, momentum and energy. If we use the Laplace transform method with zero initial conditions for Eqs. IV. 5) to (IV. 10), we obtain the following equations for the transformed quantities.

Therefore, it is important to understand the behavior of Q, or more precisely, the behavior of ~ s. Since we are interested here primarily in the value of the integrals over a wide range of time, a branch point approximation would serve the purpose. If we replace ;/ s in SF and expand it to powers of ,A., we get, keeping only the leading terms.