tailieunhanh - Fundamentals of Finite Element Analysis phần 7

Ứng dụng trong Cơ học chất lỏng được cân bằng với tốc độ dòng chảy ròng khối lượng vào khối lượng. Tổng khối lượng bên trong khối lượng là dV, và kể từ khi dV là hằng số, chúng ta phải có ∂ dV = ∂ t rong khi các điều khoản tương ứng với kết quả từ dòng chảy trong y và z | I Text Hutton Fundamentals of Finite Element Analysis 8. Applications in Fluid Mechanics The McGraw-Hill Companies 2004 296 CHAPTER 8 Applications in Fluid Mechanics be in balance with the net mass flow rate into the volume. Total mass inside the volume is p dV and since dV is constant we must have 9p dV mass flow in mass flow out and the partial derivative is used because density may vary in space as well as time. Using the velocity components shown the rate of change of mass in the control volume resulting from flow in the x direction is m x p u d y dz d p u p u 7 - dx d y dz d x while the corresponding terms resulting from flow in the y and z directions are d pV p V m y p V dx dz dx dz m z p w dx d y d p w d l Ay Ua - dz dx d y d z - d y d y p w The rate of change of mass then becomes 9p -f- dV mx lily mz d t u ĩỉg 21 dx dy dz dx dy dz Noting that dV dx dy dz Equation can be written as udr Á 1 0 d t d x d y d z l_d x d y d z_ Equation is the continuity equation for a general three-dimensional flow expressed in Cartesian coordinates. Restricting the discussion to steady flow with respect to time of an incompressible fluid density is independent of time and spatial coordinates so Equation becomes u Ề IT 0 d x d y d z Equation is the continuity equation for three-dimensional incompressible steady flow expressed in Cartesian coordinates. As this is one of the most fundamental equations in fluid flow we use it extensively in developing the finite element approach to fluid mechanics. Rotational and Irrotational Flow Similar to rigid body dynamics consideration must be given in fluid dynamics as to whether the flow motion represents translation rotation or a combination of the two types of motion. Generally in fluid mechanics pure rotation . I Text Hutton Fundamentals of 8. Applications in Fluid Finite Element Analysis Mechanics The McGraw-Hill Companies 2004 297 Figure Fluid element in a rotational flow and b .