Quick Recapitulation of Fluid Mechanics Amey Joshi 07-Feb-018 1 Equations of ideal fluids onsider a volume element of a fluid of density ρ. If there are no sources or sinks in, the mass in it will change only if there is a flux across its surface. The flux of the fluid across a surface element ds is ρv ds, where v is the velocity of the fluid. The flux across the entire surface is entire surface is ρv ds S This quantity ought to be equal to the rate of change of mass inside. That is, d ρd = ρv ds S or ( ) ρ + (ρv) d = 0 Since this equation is true for any volume element, we have the equation of continuity ρ + (ρv) = 0, (1) which is equivalent to 1 dρ = v () ρ If a fluid element has mass M and volume then the above equation becomes 1 d = v (3) The quantity on the right hand side is the rate of change of volume of a moving fluid element per unit volume. If the fluid is incompressible then it will be zero, in which case the velocity field is solenoidal. Thus, the two equivalent definitions of an incompressible fluid are 1 d = 0 (4) v = 0 (5) 1
Liquids are usually considered to be incompressible. But it is also possible to consider a gas to be incompressible. onsider a solid moving through a gas. The front side of the solid pushes against the gas, compressing it. But that compression is readily relaxed by a flow of gas outside the compressed zone. This relaxation happens as the speed of sound. Therefore, as long as the solid is moving at a speed much lesser than the speed of sound, the gas can be thought as an incompressible fluid. Now consider a volume element of the fluid so small that the density is uniform throughout it. Let is denote it by δ so that the mass in it is ρδ. Newton s second law applied to it gives ρδ dv = F body + F surface, where F body is the force acting on all portions of the volume and F surface is the force acting only on its surface. If is convenient to write F body = ρδ f so that f is thebody force per unit mass. The surface force acting on a surface element ds is written as Fi surface = σ ij δs j, where σ ij is the ij-th component of the stress tensor. Thus we have, ρδ dv i = ρδ f i + σ ij δs j The left hand side of the above equation is the rate of change of momentum of the volume element δ. If we were to extend it to the rate of change of momentum of a volume of the fluid, we would write ρ dv ( i d = ρf i + σ ) ij d x j Since this equation is true for any arbitrary volume of the fluid, we have ρ dv i = ρf i + σ ij x j, (6) also called the auchy equation. Note that we have not used any property specific to the fluids to arrive at equation (6). Therefore it is applicable to all continuous materials. Fluids are defined as materials in which even the slightest shear forces induce a motion in them. On the other hand, a solid material can remain in equilibrium under shear forces. Therefore, for a static fluid, σ ij cannot have diagnoal terms and we can write it as σ ij = pδ ij (7) An ideal fluid is the one for which equation (7) is true even for fluids in motion. The auchy equation (6) in this case becomes + v v = 1 ρ p + f (8)
Equation (8) is called the Euler equation. So far we have considered flow of mass and momentum across the surface of a fluid element. We now consider the flow of energy. The internal energy U of a volume of a fluid changes because heat is supplied to (or extracted from) it or work is done on (by) it. Thus, δu = δq pδ, which is also equivalent to the first law of thermodynamics. Let u, q and v be the specific (that is, per unit mass) internal energy, heat supply and volume respectively. learly, v = 1/ρ, so that the above equation gives, δq δt = δu δt + p δ δt ( 1 ρ In the limit δt 0 and using equation () we get ) ρ du + p v = ρdq The quantity ρdq/ is the rate of change of heat per unit volume. Let us denote it by Q r. Then the rate of change of heat in a volume is Q r d If G is the heat flux vector then Q r d = S G ds, or that Q r = G. The heat flux is related to temperature gradient as Q r = K T, where K is the thermal conductivity of the material. Therefore, Q r = (K T ) and hence equation (9) becomes (9) ρ du + p v (K T ) = 0 (10) Equations (1), (8) and (9) are the equations governing flow of ideal fluids with ρ, v, p, T and u. The equations of state relating ρ, p with T and u with T give the additional equations to match the number of equations with number of unknowns. The vorticity equation We can write (8) as + ( v ) v ω = 1 ρ p + f, 3
where ω = v is called the vorticity. above equation we get Applying the curl operation to the (v ω) = 1 ρ ρ p + f Usually the external body force is convervative so that f = 0 so that = (v ω) + 1 ρ p. (11) ρ Equation (10) is called the vorticity equation for ideal fluids. A barotropic fluid is the one for which density is a function of pressure alone. In the case of barotropic fluids, ρ = (dρ/dp) p and the vorticity equation simplifies to = (v ω). (1) A baroclinic fluid is the one for which density is not a function of pressure alone and baroclinicity is a measure of misalignment between p and ρ. Atmosphere in the tropics is usually barotropic while the mid-latitudes and the poles have a baroclinic atmosphere. Ex[anding the right hand side of equation (10), = v( ω) ω( v) v ω + ω v. Since ω = 0, dω = ω( v) + ω v If the fluid is also assumed to be incompressible, the vorticity equation simplifies to dω = ω( v). (13) Let us define circulation of a flow as Γ = v dx = ω ds, (14) where is a closed curve in the fluid enclosing an area S. Now, ( ) dγ = dv dx + dv v v dv = dx + d The second integral on the right hand side is zero. Using Euler s equation (8) and assuming that the body force is conservative, ( ) dγ 1 = ρ p Ψ dx, S 4
where f = Ψ. Since Ψ dx = Ψ ds = 0 we have ( dγ = 1 1 ρ p dx = ρ ρ x dx + 1 ρ ρ y dy + 1 ) ρ ρ z dz Since the functions ρ 1 ρ/ x, ρ 1 ρ/ y and ρ 1 ρ/ z are single valued, the integral on the right hand side is zero leading to Kelvin s circulation theorem Equation (8) for steady flows can be written as ( ) v v ω + 1 ρ p f = 0 dγ = 0 (15) Let us assume that the density is constant and that the body force is conservative, with a potential Ψ so that ( v + p ) ρ + Ψ v ω = 0 Let us now integrate the left hand side along a material curve, that is ( v + p ) ρ + Ψ dx dx (v ω) = 0 Now, dx (v ω) = v v ω) = 0 so that ( v + p ) ρ + Ψ dx = 0 Since this is true for any material curve, we conclude also called Bernoulli principle. v + p + Ψ = a constant, (16) ρ 3 Equations of Newtonian fluids Newton postulated that fluids have an internal friction. In the case of a fluid flowing in the XZ-plane, he considered that there is a velocity gradient along the Y -axis. For such a flow dv x /dy 0. The frictional force in the X-direction 5
was postulated to proportional to the velocity gradient and in a direction to nullify it. Thus, F x = µ dv x dy, where the constant of proportionality is called the viscosity of the fluid. We can therefore conjecture that the stress tensor of section?? will be propotional to the velocity gradient, that is σ ij = µ j x i This is almost true and we get more insights into it if we write it as σ ij = µ ( j + ) i µ ( j ) i x i x j x i x j and consider its value when the fluid rotates in a plane as a rigid body. In that case, v = Ω x or that v i = ɛ ijk Ω j x k so that so that i x l = ɛ ijk Ω j δ lk = ɛ ijl Ω i l x i = ɛ ljk Ω j δ ik = ɛ lji Ω i i x l + l x i = (ɛ ijl + ɛ lji )Ω i = 0 In a rigid rotation, there is no relative motion between layers of fluid. Therefore, we expect the internal friction to be zero. To ensure this, we need, ( j σ ij = pδ ij + a + ) i k + bδ ij, x i x j x k To ensure that for a fluid at rest we have σ ij = pδ ij, we require b = a/3. Further, we identify a as the viscosity of the fluid so that ( j σ ij = pδ ij + µ + i ) x i x j 3 δ k ij. (17) x k Fluids for which (17) is valid are called Newtonian fluids. Substituting (17) in the auchy equation (6), we get ρ dv i = p + [ ( j µ + i )] x i x i x i x j 3 δ k ij + ρf i. (18) x k It is called the Navier-Stokes equation. For incompressible fluids with constant viscosity it takes the familiar form i + v i j = 1 p + ν v i, (19) x j ρ x i 6
where ν = µ/ρ is called the kinematic viscosity of the fluid. arrying out manipulations similar to those leading to (1), we get the vorticity equation for viscous fluids, = (v ω) + ν ω. (0) The term ω ensure that Kelvin s circulation theorem does not hold good. If ω = 0 in an ideal fluid at one instant of time, (1) guarantees that it was zero in the past and that it will be zero in the future. Equation (0), on the other hand permits creation and dissipation of vorticity. For although the vorticity may be zero at an instant of time, its derivatives need not be zero at that time. 4 Sound waves In this section we will consider how sound waves propagate in ideal fluids. Let us consider sound waves in an ideal gas. The equation of state of an ideal gas is p = nrt, (1) where n is the number of moles and R is the universal gas constant. R = N a k B, where N a is the Avogadro number and k B the Boltzmann constant. Let m be the molecular weight of the gas. Then p = nn ak B T = nmn a k B m T = ρr gt, () where R g is a constant that depends on the nature of the gas. We will use () as the equation of state of an ideal gas. We will also use the fact that the internal energy of an ideal gas is u = c T, (3) where c is the specific heat at constant volume. onsider an ideal gas of density ρ 0 at pressure p 0 in equilibrium. Let the pressure by perturbed to p 0 + p 1 leading to a density fluctuation of ρ 0 + ρ 1. From equation () But R g T = dp/dρ so that where p 0 + p 1 = (ρ 0 + ρ 1 )R g T p 1 = ρ 1 R g T. p 1 = c sρ 1, (4) c s = dp dρ (5) We will interpret the meaning of c s later. If we substitute the perturbed quantities in the equation of continuity, we get ρ 1 + ρ 0 v 1 = 0, (6) 7
where we have ignored higher than linear order terms in perturbed quantities and have used the fact that in the unperturbed state ρ 0 does not depend on t and v 0 = 0. Doing the same for Euler equations, we get ρ 0 1 = p 1 Using equation (4), we get ρ 0 1 + c s ρ 1 = 0 (7) Taking time derivative of (6) and divergence of (7) we get ρ 1 + ρ 0 v 1 = 0 ρ 0 v 1 + c s ρ 1 = 0 Subtracting the second equation from the first we get ρ 1 = c s ρ 1, (8) which is a wave equation with c s being the speed of the wave. The waves of compression and rarefaction produced by perturbation in density are sound waves. 5 Hydrodynamics in rotating frames The time rate of change of a vector A in frame of reference rotating with angular velocity Ω with respect to a fixed frame of reference is related to the time rate of change of the same vector in the fixed frame of reference as ( da ) fixed = ( da ) rotating + Ω A (9) In particular, if A = x, the position vector, then ( ) ( ) dx dx = + Ω x (30) fixed rotating ( ) ( ) dv dv = + Ω v + Ω (Ω x) + dω x (31) fixed rotating Therefore, Euler equation (8) in rotating frame becomes + v v = 1 p + f Ω v Ω (Ω x) (3) ρ 8
and Navier-Stokes equation (19) becomes + v v = 1 ρ p + f + ν v Ω v Ω (Ω x) (33) where we have assumed Ω to be a constant. Now, ( ) Ω x = Ω (Ω x) (The fastest way to see this is to first find scalar function Ω x ) so that + v v = 1 ( ) Ω x ρ p + f + ν v Ω v + If f = Ψ then we get, + v v = 1 (Ψ ρ p ) Ω x + ν v Ω v (34) At the long length scales of geophysical or astronomical problems, the viscous term ν v is negligible and we ignore it. If we write then we are left with Ψ eff = Ψ Ω x + v v = 1 ρ p Ψ eff Ω v. (35) If and L are typical velocity and length scales then the advective term is of the order /L and the oriolis term if of the order Ω. Their ratio ɛ = ΩL, (36) is a dimensionless quantity called Rossby number. L/ is the typical time scale of the problem and 1/Ω is the period associated with rotation. If the Rossby number is small and the flow is steady, we can approximate equation (35) as 1 ρ p Ψ eff Ω v = 0. (37) If we write the second term as g eff e r then we have, 1 ρ p + g effe r + Ω v = 0. The magnitude of oriolis force is usually very small as compared to gravity. Therefore, whern we write the vertical and horizontal components of the above equation, we get 1 ρ ρ r = g eff (38) h p = ρ(ω v) h, (39) 9
where the subscript h denotes the horizontal component. Equations (38) and (39) are called geostrophic approximation. From (39) it is clear that the velocity in the horizontal plane is perpendicular to the pressure gradient. That is why when air rushes in a low pressure region, it does not rush in straight but enters tangentially creating cyclones. In the case of ideal fluids with constant density and under the influence of conservative body forces, equation (3) becomes, ( ) p v ω = ρ + v Ω x + Ψ Ω v Applying the curl operator to this equation we get Since Ω is constant, we can as well write it as = (v ω) + (v Ω) (40) (ω + Ω) = [v (ω + Ω)] (41) This equation has same form as (1) and we therefore conclude d (ω + Ω) ds = 0, (4) S which is a generalization of the Kelvin s circulation theorem for rotating frames of reference. onsider an area S in a fluid rotating at a constant angular velocity. Let there be no vorticity inside S. If S gets deformed in the course of time because of its motion in the fluid, in order to keep the flux of ω + Ω through it constant, it might experience an evolution of ω. Thus, it is quite natural to expect vorticity in rotating fluids. For steady flows with low vorticity, we can approximate equation (40) as (v Ω) = 0, which for an incompressible fluid rotating at a constant Ω mean This equation is called the Taylor-Proudman theorem. (Ω )v = 0 (43) 10