1. Fluid Dynamics and the Applications of Differential Equations By Shi-Jeisun Xie, Summer 2011

2. 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 equilibrium o 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.

3. 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

4. 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 degree o 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

5. 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 viscosity o 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

6. Viscous Flow vs. Inviscid Flow Inviscid (Ideal)Viscous Less viscous More viscous

7. 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 field o 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 property o Turbulent flow is by definition unsteady o 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

8. Laminar Flow vs. Turbulent Flow Laminar Turbulent

9. Characteristics and Models of Fluid Flow - Stress and Strain Relationship Newtonian fluids show a linear relationship between stress and rate of strain o For Newtonian fluids, viscosity is the coefficient of proportionality and is constant for a particular fluid o In fluid dynamics, viscosity is commonly used to characterize shear properties o Most fluids of mid/low viscosity are Newtonian. e.g. water, air Non-Newtonian fluids show a nonlinear relationship between stress and rate of strain o For non-Newtonian fluids, stress and strain rate are dependent on numerous factors, and a constant viscosity cannot be defined o Many highly viscous fluids and polymer solutions are non- Newtonian

10. 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

11. 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

12. 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

13. 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:

14. Conservation of Mass - Incompressible Fluids For incompressible fluids, ρ = constant, thus becomes in which case

15. 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

16. 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

17. 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

18. 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 vector o For a cubic control volume, the momentum = (momentum per unit volume)(volume element ΔxΔyΔz) We can write the above equation as or

19. 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

20. 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

21. 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

22. Conservation of Linear Momentum - Surface Forces Surface forces are those that act on the surfaces of the control volume, such as pressure and viscous stress o Pressure act normal to the surface and is the stress on the fluid at rest o 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

23. Conservation of Linear Momentum - Stress Tensors Pressure is isotropic: it is the same in all directions for a given point o 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 point o Viscous stress has nine components, with three directional stresses on each constant surface o e.g. for a surface of constant x, the fluid stresses acting in the x, y, z directions are τ xx τ xy τ xz, respectively o e.g. the sum of surface forces in the x direction are τ xx, τ yz, τ zx, - p Expressing the total stress tensor in rectangular coordinates thus gives

24. 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 direction o 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

25. Conservation of Linear Momentum - Derivation of Surface Forces (cont.) Following F sx 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 F sy and F sz and rearranging the terms, we obtain

26. Conservation of Linear Momentum - Derivation of Surface Forces (cont.) Because τ x =τ xx +τ yx +τ zx (and similarly for τ y, τ z ), using the definition of the gradient operator in Cartesian coordinates (see previous slides), we obtain since and we obtain

27. 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 F b, 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

28. The Navier-Stokes Equations The Navier-Stokes equations are differential equations that describe the motion of fluid o 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 F b Application of the Navier-Stokes equation requires information on the stress tensor term τ, which depends on the specific type of fluid flow

29. 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

30. 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

31. Incompressible Newtonian Fluids - Derivation of the Viscous Stress Term Using 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

32. The Navier-Stokes Equations - Compressible Newtonian Fluids For compressible Newtonian fluids, the Navier-Stokes equation is similar to that of incompressible fluids, with some exceptions o 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 τ ij ) o The term for the bulk viscosity is where μ v is the bulk for second coefficient of viscosity, δ ij is the Kronecker delta ( = 1 when i = j, = 0 when i ≠ j)

33. Because δ ij = 1 only for τ xx, τ yy, τ zz, and δ ij = 0 for all other τ ij 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

34. 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 equations o 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

35. Examples of CFD Some examples of CFD for turbulence modeling include o 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 Re 3 ) 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

36. 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

37. 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.