UVA PHYS 3040 - Fluid mechanics and living organisms

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Fluid mechanics and living organismsIn this chapter we discuss the basic laws of fluidflow as they apply to life processes at various sizescales. For example, fluid dynamics at lowReynolds’ number dominates the universe of uni-cellular organisms: bacteria and protozoa. Fluidflow—both uniform and pulsatile—is the domi-nant process by which both air and blood circulatein land animals, including humans.1. Euler’s equationWe consider the form of Newton’s second lawappropriate to a fluid. The mass of a small volumeof fluid is∆m = ρ ∆V ;its acceleration is therefore determined by theforces acting upon it,∆m dv→dt ≡ ρ ∆V ∂v→∂t + (v→ ⋅∇)v→  = ∆F→ .The forces are generally of two kinds:1. external, long-range forces—gravity, elec-tromagnetism;2. internal forces—especially pressure andviscosity.Thus, e.g., if the fluid is in a gravitational field oflocal acceleration g→ ,∆F→ = ρ∆V g→ .On the other hand, consider a pressure gradientin—say—the x-direction, as shown to the right.Clearly the net x-component of force is∆Fx = p(x) − p(x+dx)A ≈ − ∂p∂x ∆V ;In the absence of viscous forces the equation ofmotion is thereforeρ ∂v→∂t + (v→ ⋅∇)v→  = − ∇p + ρ g→ .This is sometimes known as Euler’s equation(L. Euler, 1755).ExampleWe first apply Euler’s equation to two problems inhydrostatics. In hydrostatics the fluid is not mov-ing, hence v→ = 0 and we obtain−∇p + ρ g→ = 0 .Defining g→ = −g z∧ we easily see that p(x,y,z;t) ≡ p(z)so thatdpdz = −ρ g .We now have two cases to consider:1. The fluid is incompressible (ρ. = 0).2. The fluid is a perfect gas:ρ = µRT p ,p(x) p(x+dx)Fx = A p(x) − p(x+dx)Physics of the Human Body 37Chapter 4: Fluid mechanics and living organismswhere R = 8.3 J oK−1 gm−mol−1 is the gas con-stant, T the absolute temperature, and µ the gram-molecular mass.In Case 1 we may easily integrate the equation togetp(z) = p(0) − ρgz= p0 1 − zz0where, if p0 is 1 atmosphere (= 0.76 m ρHg g ) andthe fluid is water, the height z0 in which the pres-sure changes by 1 atm isz0 = ρHgρH2O × 0.76 m ≈ 10.3 m ≈ 34 ft .In Case 2, we may write1p dpdz = − µ gRTorp(z) = p0 e−µ g z ⁄ RT .The scale height of the atmosphere is thenz0 = 8.3 × 2730.029 × 9.8 ≈ 8 km .End of example2. Conservation of fluidA volume dV = dx dy dz of fluid containsdN = n dVmolecules (where n is the number density) anddm = ρ dVmass (where ρ is the mass density). If a fluid isflowing with a velocity vx in the x-direction (say),then in time dt we expectdN = n vx dt dy dzparticles to be transported across an imaginarysurface of area dA = dy dz . This leads to theconcept of the flux vectorj→ = n v→that describes the transport of particles across thesurface. The dimensionality of j→ is L−2 T−1, that isnumber per unit area per unit time. Correspond-ingly we can define the flux vector of mass trans-port (dimensionality M L−2 T−1)J→ = ρ v→ .Now imagine a boxwith edges dx, dyand dz in the re-spective direc-tions. We erect avector of unitlength pointingoutward from each face. The surface integral of anyvector over the surface of this box is defined as thesum, over all six faces, of the component of thevector along the outward normal. Thus for the fluxj→ we have1∫∫box j→ ⋅ dS→ = −jx(x) + jx(x + dx) dy dz+ −jy(y) + jy(y + dy) dx dz+ −jz(z) + jz(z + dz) dx dy .This represents the rate at which the particles flowout of the box, that is,38 Chapter 4: Fluid mechanics and living organisms1. The notation jx(x) stands for jx(x,y,z,t); similarly, jx(x+dx) means jx(x+dx,y,z,t).∫∫box j→ ⋅ dS→ = − ddt (n dx dy dz) = − ∂n∂t dV .If we then compare both sides (by expanding thesmall differences −jx(x) + jx(x + dx) to first order)we have− ∂n∂t = ∂∂x jx + ∂∂y jy + ∂∂z jz =df ∇ ⋅ j→ .Note we have defined a new differential operator,the divergence of a vector, asd iv j→ ≡ ∇ ⋅ j→ =df ∂∂x jx + ∂∂y jy + ∂∂z jz .The equation of continuity, or equation of numberconservation is therefore∂n∂t + ∇ ⋅ j→ = 0 .Suppose the quantity being conserved is some-thing other than particle number—for example,energy, electric charge, mass, or even probability.The conservation laws for such quantities are iden-tical in form, with the number density and numberflux replaced with the appropriate density (energy,charge, mass, probability) and the flux vector j→replaced by the corresponding current density.3. The Navier-Stokes equationWe now consider viscous fluids. Viscosity arisesfrom the transfer of momentum between a fluidand a solid in relative motion, or between adjacentrelatively moving layers of fluid. The derivation ofthe expression for internal forces resulting fromviscosity in the most general case is somewhatadvanced and would take us too far afield. Thuswe give only the end result in the specialized (andsimplified) case of an isotropic, incompressible liq-uid. In real life there is no such thing as an incompress-ible fluid, or solid, for that matter. If one squeezeshard enough, anything is guaranteed to compress.However, when atoms or molecules are touching(or nearly so) as in the case of liquids and solids,the energy needed to decrease even slightly thevolume of a given number of molecules is enor-mous. Hence for practical purposes, over a consid-erable range of pressure, we can neglect anychange in density.When we assume the density is constant, we seethat conservation implies that the divergence ofthe velocity vector vanishes,∇⋅ v→ = 0.The additional force per unit volume of fluid thentakes the form η∇2 v→ where η is the viscosity. Itsdimensionality is ML−1T−1 . The Euler equationthen takes the form (ignoring body forces such asgravitation)ρ ∂v→∂t + (v→ ⋅∇)v→  = −∇p + η∇2 v→ .This is known as the Navier-Stokes equation.The physical meaning of viscosity can be under-stood in terms of the force exerted on a fixed plateby a parallel moving plate, if there is a layer of fluidin between, as shown below:The velocity of the fluid has a uniform gradient inthe vertical (y) direction, and the force acting onthe fixed plate isFxAplate =df η ∂vx∂y . LiquidvForceon platePhysics of the Human Body 39Chapter 4: Fluid mechanics


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