Fluid Dynamics and the Applications of Differential

Equations

By Shi-Jeisun Xie, Summer 2011

Fluid Mechanics

• Fluid mechanics is a discipline under continuum mechanics that studies fluids (liquids, gases, plasma) and the associated forces

• Fluid mechanics can be divided into:o Fluid statics or hydrostatics: the study of fluids at rest and

effects of forces on fluid equilibriumo Fluid dynamics: the study of fluids in motion and effects of

forces on fluid flow• Follows the continuum assumption:

o Fluids are considered to be continuous entities instead of discrete particles. Properties such as density, pressure, temperature, and velocity are well-defined at infinitesimally small points and vary continuously throughout.

Fluid Dynamics

• Fluid dynamics is the study of fluid flow and the effects of forces on fluids in motion

• This sub-discipline can be separated into hydrodynamics (liquids in motion) and aerodynamics (gases in motion)

• Based on the conservation laws of mass, linear momentum, and energy

• Expressed using the Reynolds transport theorem, which essentially states:o The final quantity inside a control volume is equal to the initial

quantity plus the amount that enters the control volume minus the amount that leaves the control volume

Characteristics and Models of Fluid Flow - Compressibility• Compressible fluids experience changes in density due to changes in

pressure or temperature; all fluids are compressible to some degreeo Compressible fluids use more general compressible flow

equations • In the case that density changes are insignificant and negligible, the

flow can be modeled as incompressible o Incompressible fluids do not experience changes in density as they

move in the flow fluid. That is,  o All single-component liquids at constant temperature and gases at

fluid velocity less than the speed of sound with insignificant temperature gradients behave as incompressible fluid

o For most biological systems, fluids are often treated as incompressible due to relatively constant pressure and temperature

o For acoustic problems, fluids are often treated as compressible to examine effects of sound compression waves

Characteristics and Models of Fluid Flow - Viscosity• Viscous fluids are significantly affected by fluid friction in respect to

fluid flow and have high resistance to shear stress• Inviscid fluids, or ideal fluids, experience no resistance to shear stress

and have no viscosityo Inviscid flow often use the Euler equations and Bernoulli's equation

• In regards to kinematics and dynamics, we are often concerned with the ratio of inertial forces to viscous forces, as represented by the Reynolds number Re o If Re is very low (>>1), inertial forces are negligible compared to

viscous forces, and the fluid is in Stokes flow (creeping flow)o If Re is very high, viscous forces are negligible compared to inertial

forces, and the flow can be modeled as inviscid• Near solid boundaries, viscosity generally cannot be ignored

o These regions utilize the boundary layer equations for computation

Viscous Flow vs. Inviscid Flow

Inviscid (Ideal) Viscous

Less viscous More viscous

Characteristics and Models of Fluid Flow - Laminar and Turbulent Flow• Fluid flow is laminar when there is no disruption among parallel

fluid layers in the flow fieldo Laminar flow has high momentum diffusion and low

momentum convection. That is, when the fluid flows past a solid surface, momentum diffuses across the boundary layer

o Flows with low Re (below 2100) are usually laminar • Fluid flow is turbulent when there are non-deterministic and

"random" changes in propertyo Turbulent flow is by definition unsteadyo Flows with high Re (above 4000) are usually turbulent

• Under steady flow, pressure, shear stress, velocity, and other fluid properties at a given point do not change with time. Otherwise, the flow is unsteady

Laminar Flow vs. Turbulent Flow

Laminar

Turbulent

Characteristics and Models of Fluid Flow - Stress and Strain Relationship• Newtonian fluids show a linear relationship between stress and

rate of straino For Newtonian fluids, viscosity is the coefficient of

proportionality and is constant for a particular fluido In fluid dynamics, viscosity is commonly used to characterize

shear propertieso Most fluids of mid/low viscosity are Newtonian. e.g. water, air

• Non-Newtonian fluids show a nonlinear relationship between stress and rate of straino For non-Newtonian fluids, stress and strain rate are dependent

on numerous factors, and a constant viscosity cannot be definedo Many highly viscous fluids and polymer solutions are non-

Newtonian

Newtonian Flow vs. Non-Newtonian Flow

Example of a Newtonian fluid - Water

Example of a non-Newtonian fluid - Paint

• Green: Newtonian • Red & Blue: Non-Newtonian

The Continuity Equations• Continuity equations are differential equations that describe

relations of some conserved quantity (mass, energy, momentum, etc.)

• They are based on the physical laws of conservation• The general form is

where φ is the a quantity, t is time, is the divergence operator or ∇gradient operator, f is flux (flow rate), s is generation or removal rate of the quantity. The equations of fluid dynamics can be expanded and expressed in rectangular (Cartesian), cylindrical, and spherical coordinates, but the Cartesian representation is the most commonly used. In Cartesian coordinates, is defined with ∇unit vectors u as

• For conserved quantities, s=0, and so

The Substantial Derivative

• Describes rate of change with respect to time of a quantity while in motion with velocity v

• In respect to fluid dynamics, it describes the rate of change with respect to time of a quantity moving along a path in accordance with fluid flow

• Also known as the material derivative, Stokes derivative, etc.• The operator for the substantial derivative is defined as:

         where x is a scalar or vector 

Conservation of Mass

• The mass continuity equation (mass is conserved, so s=0) is:         where ρ is mass density and v is velocity (ρv is "flux")• Because of the expansion

              the above equation can be written using the substantial derivative: 

Conservation of Mass - Incompressible Fluids

• For incompressible fluids, ρ = constant, thus            becomes         in which case

Derivation of the Mass Continuity Equation• [Rate of accumulation of mass in control volume] = [Flow rate of

mass into control volume] - [Flow rate of mass from control volume]o For a cubic control volume, the mass = (density)(volume

element ΔxΔyΔz)o Mass flow rate =(density)(cross-sectional area)(local velocity)

• Assuming that (for each direction) mass enters the control volume at x (for a surface of constant x and area ΔyΔz) and leaves at x+Δx, we can write the above equation as

Derivation of Mass Continuity Equation (cont.)

Dividing both sides by ΔxΔyΔz gives

Using the definition of the derivative

and taking the limit as Δx, Δy, Δz approach 0 gives

Derivation of Mass Continuity Equation (cont.)

Using the gradient operator for Cartesian coordinates

and replacing the unit vectors with ρv, we obtain the divergence of the mass flow rate per unit area

Thus, in accordance to the general continuity equation,

or

Conservation of Linear Momentum

• [Rate of momentum accumulation] = [rate of momentum flow in] - [Rate of momentum flow out] + Σ Forces

• The modeling for linear momentum is similar to that for mass, but whereas mass was a scalar, momentum is a vectoro For a cubic control volume, the momentum = (momentum per

unit volume)(volume element ΔxΔyΔz)• We can write the above equation as

                                                         or

Derivation of Linear Momentum Continuity Equation

Taking the limit as Δx, Δy, Δz approach 0 gives

where dV is the differential volume element dxdydz 

Again, using the gradient operator for Cartesian coordinates , we obtain∇

and

Derivation of Linear Momentum Continuity Equation (cont.)

Expanding the expression and derivative of momentum into , respectively∇

gives

where the third and fourth terms in the brackets both represent the product of velocity and the conservation of mass and thus are equal to zero, so

Conservation of Linear Momentum - Body Forces• Forces acting on the control volume are either body forces or

surface forces, thus• Body forces are those that act on the entire body (as oppose to

contact forces), such as gravity or electromagnetic forces. For fluid dynamics, we only consider gravity, thus

               and by taking the limit as Δx, Δy, Δz approach 0, we obtain

Conservation of Linear Momentum - Surface Forces • Surface forces are those that act on the surfaces of the control

volume, such as pressure and viscous stresso Pressure act normal to the surface and is the stress on the fluid

at resto Viscous stresses arise from the motion of the fluid and act both

normally and tangentially to the surface• The total stress tensor is the sum of the pressure and the viscous

stresses, or 

where σ is the stress tensor, τ is the viscous stresses, p is pressure and I is the identity matrix. The negatives sign arises because a compressive stress is considered to be negative and pressure is a positive quantity

Conservation of Linear Momentum - Stress Tensors • Pressure is isotropic: it is the same in all directions for a given

pointo Pressure is normal to every surface and directed inward

• Viscous stresses are deviatoric: they are not generally the same in all directions for a given pointo Viscous stress has nine components, with three directional

stresses on each constant surfaceo e.g. for a surface of constant x, the fluid stresses acting in the x,

y, z directions are  τxx τxy τxz, respectivelyo e.g. the sum of surface forces in the x direction are τxx, τyz, τzx, -p

• Expressing the total stress tensor in rectangular coordinates thus gives

Conservation of Linear Momentum - Derivation of Surface Forces•  The surface force arises from a gradient in the stress tensor ( ·∇ σ),

thus it is derived from the difference in pressure and the sum of the viscous forces for each respective directiono By convention, the fluid on the face with the greater algebraic

value of the defining space variable (e.g. x+Δx) exerts positive stress on the face that has the lesser value (e.g. x). This results in a (conventional) negative p factor. Thus, for the x direction

              Likewise

Conservation of Linear Momentum - Derivation of Surface Forces (cont.)Following Fsx as the example

we can write the right side expression as a product of the volume element and force per unit volume, or

Taking the limit as Δx, Δy, Δz approach 0, we obtain

Applying this form form to Fsy  and Fsz and rearranging the terms, we obtain

Conservation of Linear Momentum - Derivation of Surface Forces (cont.)

Because τx=τxx+τyx+τzx (and similarly for τy, τz), using the definition of the gradient operator in Cartesian coordinates (see previous slides), we obtain

since                                        and

we obtain

Conservation of Linear Momentum - General Form

Incorporating the force term into  we obtain

dividing both sides by the differential volume element and rearranging the left side expression yields

• This is the general equation for the conservation of linear momentum and is another form of Newton's second law of motion, expressed per unit of volume

• In some forms, ρg is replaced by Fb, a general term for body force• In the absence of fluid motion, v=0, τ=0, thus we obtain a form of

the equation of fluid statics

The Navier-Stokes Equations

• The Navier-Stokes equations are differential equations that describe the motion of fluido They state that the changes in momentum depends only

on external pressure and internal viscous stresses • The general form of the Navier-Stokes equation is

which is just another expression of the conservation of linear momentum using the substantial derivative for the left side expression and accounting ρg with Fb

• Application of the Navier-Stokes equation requires information on the stress tensor term τ, which depends on the specific type of fluid flow

Forms of the Navier-Stokes Equations• Different fluid types require analysis using different forms of the

Navier-Stokes equation• The stress tensor generally requires information on the viscosity of

fluid flow, and as such often deal with Newtonian fluids and fluids with predictable relationships between stress and strain rate

• Newtonian fluids follow general relationship        where μ is the viscosity constant• Understanding that viscous stress is symmetric for most fluids

• Using Newton's law of viscosity and the symmetric identity of the viscous stress tensor, we are able to develop the relationship

  • Noting that       is the same as the transpose of       and that in

vector form       can be written as the velocity gradient ∇v, we obtain

where (∇v)T is the transpose of ∇v.• Because the fluid is incompressible, it has been shown earlier that  ∇·v=0, and because the stress tensor is symmetric, ∇v=(∇v)T, so the viscous stress term becomes 

and the Navier-Stokes equation for incompressible Newtonian fluids becomes

The Navier-Stokes Equations - Incompressible Newtonian Fluids

Incompressible Newtonian Fluids - Derivation of the Viscous Stress TermUsing the constitutive relationship

we obtain for the viscous stress of the x momentum direction

Thus, the viscous stress term of the Navier-Stokes equation becomes

Given that for an incompressible fluid,

with similar results and derivation for the y and z momentum directions

we thus obtain

The Navier-Stokes Equations - Compressible Newtonian Fluids• For compressible Newtonian fluids, the Navier-Stokes equation is

similar to that of incompressible fluids, with some exceptionso The viscous stress tensor includes an additional term for the

bulk viscosity for the compressible particles of the fluid, which does exist for incompressible flow due to the nonexistence of flow divergence

• The bulk viscosity applies only when the viscous stress acts normally to the surface (i.e. when i = j for τij)o The term for the bulk viscosity is

where μv is the bulk for second coefficient of viscosity, δij is the Kronecker delta ( = 1 when i = j, = 0 when i ≠ j)

• Because δij = 1 only for τxx, τyy, τzz, and δij = 0 for all other τij

 

 

• By incorporating the bulk viscosity factor with the viscous stress tensor found for incompressible fluids, we obtain

and the Navier-Stokes equation for compressible fluids becomes

Compressible Newtonian Fluids - Derivation of the Bulk Viscosity Factor

Computational Fluid Dynamics - An Application of Navier-Stokes• Computational fluid dynamics (CFD) use algorithms, numerical

methods, and computer calculations to analyze fluid flow problems• In CFD follows a general basic procedure:

o Preprocessing: the physical boundaries and boundary conditions are defined, the control volume is divided into discrete cells, and the equations for physical modeling are defined

o Processing: the simulation runs and iteratively solves the defined equations as in or not in steady-state

o Postprocessing: the solutions are further analyzed and visualized • CFD can be used to model fluid flow, especially turbulent flow, by

finding or approximating solutions to the Navier-Stokes equationso Because turbulent flow is associated with a wide range of length

and time scales, resolution of these scales can be computationally costly depending on the finesse and accuracy of the model

Examples of CFD

• Some examples of CFD for turbulence modeling includeo Direct numerical simulation (DNS) - solves the Navier-Stokes

equations and resolves the entire range of length and time scales for turbulence  Allows for simulation of turbulent flow, but is extremely

expensive and memory-intensive at higher Reynolds numbers (computational cost is proportional to Re3)

o Reynolds-averaged Navier-Stokes (RANS) modeling - models fluid flow and Reynolds stresses using time-averaged equations incorporating Reynolds decomposition to approximate solutions to the Navier-Stokes equations Reynolds decomposition separates a quantity into its average

and fluctuating components

Visual Examples of CFD and DNS

DNS analysis of the turbulent heat flux

DNS analysis of turbulent kinetic energy

DNS analysis of turbulent mean velocity

Sources

• Truskey, George A., Fan Yuan, and David F. Katz. Transport Phenomena in Biological Systems. Upper Saddle River, NJ: Pearson Prentice Hall, 2009. Print.

• Batchelor, G. K. An Introduction to Fluid Mechanics. Cambridge: Cambridge Univ. Pr., 2000.

• Fox, Rodney O. Computational Models for Turbulent Reacting Flows. Cambridge, U.K.: Cambridge UP, 2003.

• Munson, Bruce Roy, Donald F. Young, and T. H. Okiishi. Fundamentals of Fluid Mechanics. Hoboken, NJ: J. Wiley & Sons, 2006.

• Segel, Lee A., and G. H. Handelman. Mathematics Applied to Continuum Mechanics. Philadelphia: Society for Industrial and Applied Mathematics, 2007.