Fluid Mechanics (C.V. analysis) Dept. of Experimental Orthopaedics and Biomechanics Bioengineering Reza Abedian (M.Sc.)

Analysis of a Fluid Sys. Flowing through a C.V. We use this general representation of a flowing F.S. through a C.V.to develop a relationship between a F.S. and a C.V. known as: –The control volume approach, or –The Raynolds transport theorem Note: the C.V. approach is also called the Eulerian Approach in contrast to the Lagrangian approach which describes the motion of individual particle as a function of time

Analysis of a Fluid Sys. Flowing through a C.V. Let X = total amount of a fluid property (Mass, Energy, Momentum) –At time t, C.V. and C.S. coincide, (X s ) t = (X C.V. ) t –At time t + dt, F.S has moved and changed shape volume of fluid entered C.V. in dt carrying volume of fluid leaving C.V. in dt carrying Thus the amount of X s at time t + dt is Subtraction Derivation

Continuation of Derivation of Reynold’s Transport Theorem: Last result Take limit as Reynold’s Transport Theorem

Analyzing the Theorem 1: rate of change of total amount of extensive property X within the moving fluid system 2: rate of change of X contained within the fixed control volume 3: rate of outflow of X through the control surface downstream 4: rate of inflow of X through the control surface upstream 3 – 4  net rate of out flow of X passing through the C.S EQUATION STATES that the difference between the rate of change of X within the system and that within the control volume is equal to the net rate of outflow from the control volume

Why is the Reynold’s transport theorem Important? Laws of physics (Newtons 2 nd law or conservation of momentum, conservation of energy and mass) apply to fluid systems… It is difficult to follow a fluid system as if flows (Lagrangian approach) It is easier to observe fluid characteristics (density, velocity, pressure, momentum, etc at fixed points in space that is using a C.V. (Eulerian approach) Thus we need a way to relate the behavior of a fluid system (S) to the quantities that can be observed within a control volume Solution: the Reynold’s Transport Theorem

Equation of Continuity No flow across the stream tube sides Only flow through the ends of streamtube Let X=m in Reynold’s transport theorem Conservation of mass

Equation of Continuity

Differential Form of the Equation of Continuity Mass flux: mass flow rate = mass/area.time u,v and w are all positive and velocities in each directions are as follows: –In upstream face in x –In downstream face x

fluxes Reynold´s Transport Theprem: Differential form of the continuity equation for any kind of flow

Vector analysis notation In Cartesian cordinates (x,y,z) the unit vectors are We use scalar functions Q(x,y,z,t) such as density or temperature as well as vector functions F(x,y,z,t) such as velocity and momentum The „del“ or „nabla“ operator is defined as: Gradient of a scalar function: Divergence of a vector function:

Application to fluid dynamics: Let q be the velocity vector of a flow in Cartesian coordinates and ρ the density,consider the vector flux ρq: Then the divengence of ρq is: Then the differential form of the continuity equation is: Rule of derivatives of product:

Special form of the continuity equation: For an incompressible flow:

Vorticity Another operation related to the „del“ or „Nabla“ operator is the Curl or Rotational of a vector function F=(F x F y F z ) The curl operation is used to define the vorticity of a flow The vorticity is related to the angular velocity if a fluid particles An irrotational flow is one that has no angular velocity, for a 2D flow in xy plane: