1Physiology 472/572 - 2010 - Quantitative modeling of biological systems Lecture 14: Stresses in a fluid, Navier-Stokes equations Example of normal stress • a fluid at rest – hydrostatic equilibrium - no shear stress • stress vector T = -pn • all surfaces feel a pressure pushing in on them (imagine swimming deep under water) • p is the hydrostatic pressure • stress tensor is σij = -pδij, i.e., ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−=ppp000000σ • δij is the Kronecker delta: δij = 1 if i = j, δij = 0 if i ≠ j Incompressible fluid • water and other fluids have little density change under forces encountered in the body • can assume constant density Pressure in an incompressible fluid under gravity • consider a force balance • force on top surface = - p2 A • force on bottom surface = + p1 A • force of gravity = - ρ A(z2 - z1) g • total force is zero so p1 - p2 = ρ g (z2 - z1) • still holds even in a more complicated container2Example -mercury manometer • used in measuring blood pressure • from above, pc - pa = ρ g h • for mercury, ρ = 1.35 x 104 kg/m3, g = 9.81 m/s2 • consider h = 1 mm, then pc - pa = 133.3 Pa = 1333 dyn/cm2 = 1 mmHg • mmHg is often used as a unit of pressure Shear flow of a fluid • two parallel plates, one fixed (u = 0), one moving (u = U) • fluid in space between • fluid resists motion, force τΔS acts on area ΔS of fluid surface • produces shear stress in fluid • by force balance on a control volume, τ must be constant with y Newtonian fluids • for many fluids, shear stress is proportional to velocity gradient • constant of proportionality is viscosity μ No-slip condition • fluid adjacent to a solid surface moves with the surface • true for any continuum Velocity profile in shear flow • suppose u = u(y), τ = μ du/dy for a Newtonian fluid • when y = 0, u = 0 - no slip condition • when y = h, u = U3• τ is constant so u = Uy/h, τ = μU/h • viscous drag is proportional to viscosity • viscous drag is inversely proportional to spacing h Viscosity • units are stress/(velocity/distance) • in SI, Pa⋅s, in cgs, dyn⋅s/cm2 = P (Poise) • 1 P = 0.1 Pa⋅s, 1 cP = 0.001 Pa⋅s ~ viscosity of water • viscosity depends on temperature • for Newtonian fluids (air, water, etc) does not depend on velocity gradient • for many biological fluids, viscosity varies with velocity gradient - "Non-Newtonian" Incompressible fluid • for an incompressible fluid, ∇.u = ∂ui/∂xi = 0 • prove by considering a small control volume δx1 × δx2 × δx3 Stress tensor in a viscous incompressible fluid • recall τ = μ du/dy for a Newtonian fluid in shear flow • stress tensor must be symmetric (conservation of angular momentum) • in tensor form, the stress in simple shear flow is ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡ττ=0000000σ where τ = μ ∂u1/∂x2 using index notation • in a general flow, represent velocity field as u(x,t) • all the components of ∂ui/∂xj may be non-zero • viscous stress is σij (viscous) = μ (∂ui/∂xj + ∂uj/∂xi)4• total stress is (hydrostatic pressure) + (viscosity) × (velocity gradients) • σij = -pδij + μ (∂ui/∂xj + ∂uj/∂xi), i.e., σ = -pI + μ (∂u/∂x + (∂u/∂x)T) Fluid velocity and acceleration • to find acceleration of fluid, must take derivative of velocity traveling with fluid • if fluid moves from a low-speed to a high speed area, it speeds up, even if the velocity stays constant at each point • e.g. in one dimension, acceleration of fluid = ∂u/∂t + u ∂u/∂x • second term called advective acceleration • in the general case, fluid acceleration = ∂u/∂t + u . ∇u Navier-Stokes equations • general idea is to apply conservation of momentum • fluid may have a body force F (per unit volume) acting on it (e.g., gravity) • resultant force of stress in fluid (per unit volume) = ∇ . σ • conservation of momentum per unit volume gives ρ (∂u/∂t + u . ∇u) = ∇ . σ + F • (∇ . σ)j = (∂/∂xi) (-pδij + μ (∂ui/∂xj + ∂uj/∂xi)) = - ∂p/∂xj + μ ∇2uj using incompressibility condition • or ∇ . σ = -∇p + μ ∇2u • ρ (∂u/∂t + u . ∇u) = -∇p + μ ∇2u + F • Navier-Stokes equations in terms of (u,v,w) and (x,y,z) : 0 = zw + yv + xuF + w + zp- = ]zw w+ yw v+ xw[u + tw F + v + yp- = ]zv w+ yv v+ xv[u + tv F +u + xp- = ]zu w+ yu v+ xu[u + tu