tailieunhanh - Introduction to Continuum Mechanics 3 Episode 12

Tham khảo tài liệu 'introduction to continuum mechanics 3 episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 426 Newtonian Viscous Fluid . Show that for a one-dimensional steady adiabatic flow of an ideal gas the ratio of temperature 0ị Õ2 at sections 1 and 2 is given by Ớ1 _ 1 ỳr - 1 Wi 2 1 y - 1 MỈ where y is the ratio of specific heat Mỵ and Af2 are local Mach numbers at section 1 and 2 respectively. . Show that for a compressible fluid in isothermal flow with no external work dM2 dv M2 V where M is the Mach number. Assume perfect gas. . Show that for a perfect gas flowing through a constant area duct at constant temperature conditions. dp l 2 p 2 M2 . For the flow of a compressible inviscid fluid around a thin body in a uniform stream of speed vm in the 1 - direction we let the velocity potential be ÍP -VO 1 Pl where P1 is assumed to be very small. Show that for steady flow the equation governing v iis with Mo vo co .2 r 2. n2 - mV 1 4. a 1 4. 1 n dxị dxị dxị 7 Integral Formulation of General Principles In Sections the field equations expressing the principles of conservation of mass of linear momentum of moment of momentum and of energy were derived by the consideration of differential elements in the continuum. In the form of differential equations the principles are sometimes referred to as local principles. In this chapter we shall formulate the principles in terms of an arbitrary fixed part of the continuum. The principles are then in integral form which is sometimes referred to as the global principles. Under the assumption of smoothness of functions involved the two forms are completely equivalent and in fact the requirement that the global theorem be valid for each and every part of the continuum results in the differential form of the balance equations. The purpose of the present chapter is twofold l to provide an alternate approach to the formulation of field equations expressing the general principles and 2 to apply the global theorems to obtain approximate solutions of some engineering problems using the concept of control