Introduction to Fluid Mechanics Chapter 5 Introduction to Differential Analysis of Fluid Motion

Main Topics قانون بقای جرم تابع جریان برای جریان دو بعدی تراکم ناپذیر سینماتیک معادله مومنتوم

قانون بقای جرم: Basic Law for a System

Conservation of Mass Differential CV and Taylor series Infinitesimal control volume of dimensions dx, dy, dz Area of right face = dy dz Mass flow rate through the right face of the control volume

Net mass flow rate into CV: Net mass flow rate out of CV:

Or, if we apply the definition of the divergence of a vector After substitution, Dividing through by volume dxdydz Or, if we apply the definition of the divergence of a vector

Conservation of Mass Rectangular Coordinate System

Conservation of Mass Rectangular Coordinate System “Del” Operator

Conservation of Mass Rectangular Coordinate System

Conservation of Mass Cylindrical coordinates

Conservation of Mass Special Cases Steady compressible flow Cartesian Cylindrical

Conservation of Mass Rectangular Coordinate System Incompressible Fluid: Steady Flow:

EXE :A two-dimensional converging duct is being designed for a high-speed wind tunnel. The bottom wall of the duct is to be flat and horizontal, and the top wall is to be curved in such a way that the axial wind speed u increases approximately linearly from u1 " 100 m/s at section (1) to u2 " 300 m/s at section (2) .Meanwhile, the air density ρ is to decrease approximately linearly from ρ1 " 1.2 kg/m3 TO ρ2 " 0.85 .The converging duct is 2.0 m long and is 2.0 m high at section (1). (a) Predict the y-component of velocity, v(x, y), in the duct. (b) Plot the approximate shape of the duct, ignoring friction on the walls. (c) How high should the duct be at section (2), the exit of the duct?

EXE 2:

Conservation of Mass Cylindrical Coordinate System

Conservation of Mass Cylindrical Coordinate System

Conservation of Mass Cylindrical Coordinate System “Del” Operator

Conservation of Mass Cylindrical Coordinate System

Conservation of Mass Cylindrical Coordinate System Incompressible Fluid: Steady Flow:

تابع جریان برای جریان تراکم ناپذیر 2 بعدی Two-Dimensional Flow یاد آوری خطوط جریان دو مولفه سرعت را ببریم در دل مفهوم ψ Stream Function y  

به درد به دست آوردن دبی حجمی   در طول خط جریان تابع ψ در طول خط جریان ثابت است Hence we can specify individual streamlines by their stream function values: ψ 5 0, 1, 2, etc. What is the significance of the ψ values? به درد به دست آوردن دبی حجمی

the flow rate across AB is

Stream Function for Two-Dimensional Incompressible Flow Cylindrical Coordinates Stream Function y(r,q)

Motion of a Fluid Particle (Kinematics) Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field Fluid Rotation Fluid Deformation Angular Deformation Linear Deformation

کدام یک از حرکت ها تولید تنش برشی میکند؟

y t1 t0 particle path x

Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field مشتق اصلی یا مشتق مادی

شتاب جابه جایی شتاب محلی

Motion of a Fluid Particle (Kinematics) Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field

Fluid Translation: Acceleration of a Fluid Particle in a Velocity Field (Cylindrical)

چرخش سیال Fluid Rotation کی چرخش داریم؟ u3(x1) u1(x3) کی چرخش داریم؟ باید گشتاور داشته باشیم. نیروهای عمود بر سطح باعث شتاب و تغییر شکل طولی میشن و گشتاور ایجاد نمیکنند. نیروها وتنشهای برشی لازمه که فقط برای جریان ویسکوز معنی میده x1

Rotation Definition of rotation b Time = t + dt Time=t y Dy a Dx x Concept of rotation: Because of shear in the fluid, during flow, an element may not only get translated, but also ‘rotated’. The rotation is given by the definition in this page. Dx x Assume Vy|x < Vy|x+Dx and Vx|y > Vx|y+Dy

Fluid Rotation

Ψ=0 if x=0 , 180 or r=R

Motion of a Fluid Particle (Kinematics) Fluid Deformation: Angular Deformation

Motion of a Fluid Particle (Kinematics) Fluid Deformation: Angular Deformation

Motion of a Fluid Particle (Kinematics) Fluid Deformation: Linear Deformation

Momentum Equation Newton’s Second Law

Momentum Equation Forces Acting on a Fluid Particle

Momentum Equation Forces Acting on a Fluid Particle

Momentum Equation Differential Momentum Equation

Momentum Equation Newtonian Fluid: Navier-Stokes Equations

Momentum Equation Special Case: Euler’s Equation

Computational Fluid Dynamics Some Applications

Computational Fluid Dynamics Discretization