Equations of Continuity

Outline Time Derivatives & Vector Notation Differential Equations of Continuity Momentum Transfer Equations

Infinitesimal pieces of fluid Introduction In order to calculate forces exerted by a moving fluid as well as the consequent transport effects, the dynamics of flow must be described mathematically (kinematics). FLUID Fluid is considered to be a continuous medium (or continuum) made up of very small individual pieces of the fluid substance that experiences deformation and displacement (by translation and rotation) during the flow. Each infinitesimal piece of fluid material is considered as a moving “particle”, whose position and/or velocity must be mathematically defined using the principles of kinematics. However, because there are an infinite number of “particles” present in a flowing fluid, a system must be developed to describe the fluid motion. Continuous medium Infinitesimal pieces of fluid

Perspectives of Fluid Motion Eulerian Perspective – the flow as seen at fixed locations in space, or over fixed volumes of space (the perspective of most analysis) Lagrangian Perspective – the flow as seen by the fluid material (the perspective of the laws of motion) Fluid system: finite piece of the fluid material (Lagrangian) Control volume: finite fixed region of space (Eulerian) Fluid particle: differentially small finite piece of the fluid material (Lagrangian) Coordinate: fixed point in space (Eulerian)

Lagrangian Perspective The motion of a fluid particle is relative to a specific initial position in space at an initial time. In the Lagrangian description of fluid flow, individual fluid particles are "marked," and their positions, velocities, etc. are described as a function of time. In the example shown, particles A and B have been identified. Position vectors and velocity vectors are shown at one instant of time for each of these marked particles. As the particles move in the flow field, their positions and velocities change with time, as seen in the animated diagram. The physical laws, such as Newton's laws and conservation of mass and energy, apply directly to each particle. 

Lagrangian Perspective z Lagrangian coordinate system pathline position vector The position of each particle (a finite parcel of the fluid) is defined (hence the subscript label p) and tracked. y partial (local) time derivatives x

Lagrangian Perspective Consider a small fluid element with a mass concentration  moving through Cartesian space: y y t = t1 t = t2 x x z z Because the particle is a “fluid”, it is subject to deformation during displacement.

Lagrangian Perspective Consider a small fluid element with a mass concentration  moving through Cartesian space: y y t = t1 t = t2 x x z z

Lagrangian Perspective Total change in the mass concentration with respect to time: If the timeframe is infinitesimally small:

Lagrangian Perspective Total Time Derivative Substantial Time Derivative Also called the Lagrangian derivative, material derivative, or particle derivative, the velocity terms follow the path of motion of the fluid particle. local derivative convective derivative

Lagrangian Perspective stream velocity vector notation  gradient

Lagrangian Perspective Problem with the Lagrangian Perspective The concept is pretty straightforward but very difficult to implement (since to describe the whole fluid motion, kinematics must be applied to ALL of the moving particles), often would produce more information than necessary, and is not often applicable to systems defined in fluid mechanics.

Eulerian Perspective flow Motion of a fluid as a continuum z Motion of a fluid as a continuum flow Fixed spatial position is being observed rather than the position of a moving fluid particle (x,y,z). In the Eulerian perspective, the system is a fixed volume in space where the fluid material flows, rather than following the property changes in a small piece of the flowing fluid. y x

Eulerian Perspective flow Eulerian coordinate system z Motion of a fluid as a continuum flow Velocity expressed as a function of time t and spatial position (x, y, z) y Eulerian coordinate system x

Eulerian Perspective Difference from the Lagrangian approach: Eulerian In the Eulerian perspective, since the infinitesimal control volume is at a fixed location, the analysis is no longer dependent on just one specific fluid element; instead, any changes in the property defining the control volume would be attributed to the difference between the properties of the incoming and outgoing fluid material, as well as other external factors that may interact with the system. Also, since the control volume is fixed, deformation occurs without a net change in volume.

Eulerian Perspective Difference from the Lagrangian approach: Eulerian

Outline Time Derivatives & Vector Notation Differential Equations of Continuity Momentum Transfer Equations

Equation of Continuity differential control volume:

Differential Mass Balance

Differential Mass Balance Substituting: Rearranging:

Differential Equation of Continuity Dividing everything by ΔV: Taking the limit as ∆x, ∆y and ∆z  0:

Differential Equation of Continuity divergence of mass velocity vector (v) Partial differentiation:

Differential Equation of Continuity Rearranging: substantial time derivative If fluid is incompressible:

Differential Equation of Continuity In cylindrical coordinates: If fluid is incompressible:

Outline Time Derivatives & Vector Notation Differential Equations of Continuity Momentum Transfer Equations

Differential Equations of Motion

Control Volume Fluid is flowing in 3 directions For 1D fluid flow, momentum transport occurs in 3 directions Recall: a unidirectional flow induces momentum transport in 3 directions (2 normal to the direction of the flow + 1 parallel to the direction of the flow) Momentum transport is fully defined by 3 equations of motion

Momentum Balance Consider the x-component of the momentum transport:

Momentum Balance Due to convective transport:

Momentum Balance Due to molecular transport:

Momentum Balance Consider the x-component of the momentum transport:

Momentum Balance Consider the x-component of the momentum transport:

Differential Momentum Balance Substituting:

Differential Momentum Balance Dividing everything by ΔV:

Differential Equation of Motion Taking the limit as ∆x, ∆y and ∆z  0: Rearranging:

Differential Momentum Balance For the convective terms: For the accumulation term:

Differential Equation of Motion Substituting:

Differential Equation of Motion Substituting: EQUATION OF MOTION FOR THE x-COMPONENT

Differential Equation of Motion Substituting: EQUATION OF MOTION FOR THE y-COMPONENT

Differential Equation of Motion Substituting: EQUATION OF MOTION FOR THE z-COMPONENT

Differential Equation of Motion Substantial time derivatives:

Differential Equation of Motion In vector-matrix notation:

Differential Equation of Motion Cauchy momentum equation Equation of motion for a pure fluid Valid for any continuous medium (Eulerian) In order to determine velocity distributions, shear stress must be expressed in terms of velocity gradients and fluid properties (e.g. Newton’s law)

Cauchy Stress Tensor Stress distribution:

Cauchy Stress Tensor Stokes relations (based on Stokes’ hypothesis)

Navier-Stokes Equations

Assumptions Newtonian fluid Obeys Stokes’ hypothesis Continuum Isotropic viscosity Constant density Divergence of the stream velocity is zero

Navier-Stokes Equations Applying the Stokes relations per component:

Navier-Stokes Equations Navier-Stokes equations in rectangular coordinates

Cylindrical Coordinates

Applications of Navier-Stokes Equations

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Steady state flow

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Unidirectional flow

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Unidirectional flow

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Constant fluid properties Isotropy (independent of position/direction) Independent with temperature and pressure

Application Euler’s Equation The Navier-Stokes equations may be reduced using the following simplifying assumptions: No viscous dissipation (INVISCID FLOW) Euler’s Equation

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: No external forces acting on the system

Application The Navier-Stokes equations may be reduced using the following simplifying assumptions: Laminar flow

Example Derive the equation giving the velocity distribution at steady state for laminar flow of a constant-density fluid with constant viscosity which is flowing between two flat and parallel plates. The velocity profile desired is at a point far from the inlet or outlet of the channel. The two plates will be considered to be fixed and of infinite width, with flow driven by the pressure gradient in the x-direction. Then afterwards, the solution may be applied to other configurations previously discussed during shell balance calculations.

Example Derive the equation giving the velocity distribution at steady state for laminar, downward flow in a vertical pipe of a constant-density fluid with constant viscosity which is flowing between two flat and parallel plates. Then afterwards, the solution may be applied to other configurations previously discussed during shell balance calculations.