Tham khảo tài liệu 'introduction to continuum mechanics 3 episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 386 Energy Equation For a Newtonian Fluid a20 a2e a2e 0 dx dy dz which in this problem reduces to d2e . 00 4y2 Thus d r and 0 C y C2 iii Using the boundary condition 0 0 at y 0 and 0 0U at y d the constants of integration are determined to be d C2 1 It is noted here that when the values of 0 are prescribed on the plates the values of on the plates are completely determined. In fact 0u-0 tZ. This serves to illustrate that in steady-state heat conduction problem governed by the Laplace equation it is in general not possible to prescribe both the values of 0 and the normal derivatives of 0 at the same points of the complete boundary unless they happen to be consistent with each other. Example The plane Couette flow is given by the following velocity distribution Vi ky v2 0 v3 0 i If the temperature at the lower plate is kept at 0 and that at the upper plate at 0ip find the steady- state temperature distribution. Solution. We seek a temperature distribution that depends only on y. From Eq. we have since 12 k 2 n . . 2q ii 0 pk k dy2 Newtonian Viscous Fluid 387 Thus d2e _ nk2 2 K iii which gives 0 22 _ C1 . c2 where Cl and c2 are constants of integration. Now at y 0 0 0 and at y d 0 0U therefore Ci and c2 0 1 d 2k d L 1 The temperature distribution is therefore given by 4À 2 2 2 0 iv d 2k r Vorticity Vector We recall from Chapter 3 Section and 14 that the antisymmetric part of the velocity gradient Vv is defined as the spin tensor w. Being antisymmetric the tensor w is equivalent to a vector at in the sense that Wx at X X see Sect. 2B16 . In fact at - 23el 31e2 12e3 - Since see Eq. dx Vv dx Ddx Wdx Ddx fi Xdx the vector at is the angular velocity vector of that part of the motion representing the rigid body rotation in the infinitesimal neighborhood of a material point. Further at is the angular velocity vector of the principal axes of D which we show below Let dx be a material element in the direction of the unit vector n at time .
