By Stein D.
During this engrossing biography, Dorothy Stein strips away the numerous layers of fantasy to bare a narrative way more dramatic and interesting than earlier debts have indicated
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14) , where the constant viscosity has been set equal to unity for convenience. 15). Other examples in process engineering include the following. 13) with h = 1 (Fig. 5). On the assumption that flow is taking place because the stresses are everywhere strong enough to overcome friction, we can find the simple `yield criterion', which the stress must satisfy, from an analysis using Coulomb's law of friction. This just says that we need to ensure that at each point of a flowing granular material there is a `slip plane', on an element of which the ratio of the shear (frictional) force to the normal force is equal to the coefficient of friction, say tan ¢, while on all other planes the ratio is less than tan 0.
The only common physical situation where higher-order derivatives occur naturally at this modelling stage is quantum mechanics, about which we will say more in Chapter 8. With this powerful motivation, we now describe a framework within which first-order systems with two independent variables may be considered. Alas, too few of the examples listed above (or indeed of any nonlinear system of partial differential equations) can be analysed as fully as those in Chapter 1, and those for which substantial progress can be made can often be most conveniently written as higher-order scalar equations.
Suppose that u(x, y) is such that Ou/8x = 0 with 1, Y<0' Now let the partial differential equation for u be replaced by 8u 82u 8x = C8y2' for small positive e. Verify that a solution of this equation is V/ 2 2 u(xy) _ f e: e' ds 9 (this will be derived in Chapter 6). Show that, as a -, 0, 1, u-> 1, Y<0' y > 0, for x > 0, and that this result is the same as that obtained by requiring u to be discontinuous only on a characteristic. 1 Motivation and models When we use vector systems of partial differential equations, we can model many more physical situations than when we are restricted to the scalar case.