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Chapter 6 Conservation of Momentum: Fluids and Elastic Solids

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Chapter 6 Conservation of Momentum: Fluids and Elastic Solids Chapter 6 Conservation of Momentum: Fluids and Elastic Solids The description of the motion of fluids plays a fundamental role in applied mathematics. The basic model is a system of partial di↵erential equations of evolution type. 6.1 Equations of Fluid Motion Fluids consists of molecules; thus, on a microscopic level, a fluid is a discrete material. To obtain a useful approximation, we will describe fluid motion on the macroscopic level by taking into account forces that act on a parcel of fluid, which we assume to be a collection of sufficiently many molecules of the fluid so that the continuity assumption is valid. More precisely, we will assume the fluid to be a continuous medium contained in three-dimensional space R3 such that every parcel of the fluid, no matter how small in compar- ison with the whole body of fluid, can be viewed as a continuous material. Mathematically, a parcel of fluid (at every moment of time) is a bounded open subset A of the fluid that is assumed to be in an open set whose closure is the region containing the fluid. To ensure correctness of theR math- ematical operations that follow, we assume that each parcel A has a C1 boundary. At every given moment of time, a particle of fluid is identified with a point in . R Let ⇢ = ⇢(x, t)denotethedensityandu = u(x, t)thevelocityofthefluid at position x and time t R. The motion of the fluid is modeled by 2R 2 257 258 Fluids di↵erential equations for ⇢ and u determined by the fluid’s internal material properties, its container, and the external forces acting on the fluid. The components of u with respect to the usual coordinates of three- dimensional space are denoted by (u1,u2,u3). We assume that the function u is twice continuously di↵erentiable and satisfies other properties that are necessary for the correctness of the mathematical operations used in the following discussion. We will make remarks about the additional properties of u as needed. The changing position of the particle of fluid with initial position x 0 2R at time t =0isgivenbythecurvet γ(x0,t)in that is the solution of the initial value problem (IVP) 7! R ⇠˙ = u(⇠,t),⇠(0) = x0 (6.1) (see Appendix A.3 for a theorem on existence of solutions of ordinary dif- ferential equations). When γ is viewed as a function of two variables γ : R3 R R3 it is called the flow of u. Note that the notation γ(A, t), where ⇥ ! A is a subset of R3,isthesetofallpointsinR3 obtained by solving to time t the di↵erential equation with initial condition ⇠(0) = x0 for each x0 in A. For each such initial point, the value of the solution at time t (a point in R3) belongs to the set γ(A, t). 3 Let x1, x2,andx3 denote the Cartesian coordinates in R and e1, e2, e3 the usual unit direction vectors. Using this notation, the velocity vector u is u = u1e1 + u2e2 + u3e3. The gradient operator in Cartesian coordinates (also called nabla or del) is @ @ @ := e1 + e2 + e3 r @x1 @x2 @x3 or, equivalently, @ @x1 := @ . r 0 @x2 1 @ B @x3 C @ A This operator applied to a function f : R3 R gives its gradient in Cartesian coordinates ! @f @x1 f = 0 @f 1 . r @x2 @f B C B @x3 C @ A Fluids 259 Applied to a vector field u (with the notation u,where denotes the usual inner product in Euclidean space), the gradientr· operator gives· the divergence in Cartesian coordinates @u @u @u u = 1 + 2 + 3 . r· @x1 @x2 @x3 The expression (u )u,oftenwrittenu u,isthevector ·r ·r u @u1 + u @u1 + u @u1 1 @x1 2 @x2 3 @x3 @u2 @u2 @u2 u1 + u2 + u3 . 0 @x1 @x2 @x3 1 u @u3 + u @u3 + u @u3 1 @x1 2 @x2 3 @x3 @ A AparcelA of fluid identified at time zero is moved to γ(A, t) at time t. Reynolds’s transport theorem states that d ⇢(x, t) dx = ⇢ (x, t)+div(⇢u)(x, t) dx (6.2) dt t Zγ(A,t) Zγ(A,t) (see A.11). By conservation of mass, the rate of change of the total mass in A does not change as the parcel is transported by the flow; therefore, the left-hand side of equation (6.2) vanishes. Since A may be taken arbitrarily small (for example, a ball with arbitrarily small radius), it follows that ⇢t +div(⇢u)=0 (6.3) or, equivalently, ⇢ + (⇢u)=0. (6.4) t r· Equation (6.4) (or (6.3)) is called the equation of continuity; it states that the mass of a fluid is conserved by the fluid motion. Equation (6.2) is a general statement of the rate of change of total mass that holds as long as u is an arbitrary (smooth) vector field with flow γ. Adi↵erentialequationforthevelocityfieldu is obtained from the equa- tion for the conservation of momentum using Newton’s second law of motion. Note that a fluid has mass. It might also be charged. Thus, a fluid is sub- jected to body forces, which by definition are forces that act per unit of mass or per unit of charge. The most important body force is the gravitational force, which acts on every fluid simply because fluids have mass. Unlike the motion of particles or rigid bodies, fluids (by definition) have internal stress 260 Fluids (force per area) that is caused by the action of the fluid on itself. Stress is modeled by a function σ that assigns a vector in R3 to each pair consisting of a point (x, t) in space-time and a unit-length (space) vector ⌘ in R3 at this point. The value of the stress function at this pair is called the stress vector at the point x in the direction of the outer normal ⌘ at time t on imag- inary surfaces in space that contain this point and have this outer normal at time t. The stress vector has units of force per area. Using conservation of momentum and angular momentum, it is possible to prove that the stress function at each point in space-time is a symmetric linear transformation of space (see [17]). Let us simply incorporate these facts about the stress as assumptions. Alinearandsymmetrictransformationonvectorsmayberepresented by a (diagonalizable) matrix in the usual Cartesian coordinates. Thus, for each point (x, t)inspace-time,σ(x, t)maybeviewedasasymmetricmatrix, which is thus defined by six numbers (the elements on and above the main diagonal of the matrix). Total stress over the current position of parcel A at time t is given by TS := σ(x, t)⌘(x, t) d , S Z@(A,t) where ⌘ is the outer unit normal on the boundary of γ(A, t)andd is the element of surface area. Using the body force b per unit of mass, theS total body force on γ(A, t)is TB := ⇢(x, t)b(x, t) d , V Zγ(A,t) where d is the element of volume. TheV total momentum of γ(A, t)is ⇢(x, t)u(x, t) d . V Zγ(A,t) By Newton’s second law of motion (the time rate of change of momen- tum on a body is equal to the sum of the forces acting on the body), the mathematical expression for the conservation of momentum is d ⇢(x, t)u(x, t) d = σ(x, t)⌘(x) d + ⇢(x, t)b(x, t) d . dt V S V Zγ(A,t) Z@(A,t) Zγ(A,t) (6.5) Fluids 261 The region γ(A, t) in space is moving with time. Using the equation of continuity (6.3), the transport theorem (A.11), and some algebra, it follows that the left-hand side of the momentum balance (6.5) is given by d ⇢(x, t)u(x, t) d = ⇢(x, t)(u +(u )u)(x, t) d , (6.6) dt V t ·r V Zγ(A,t) Zγ(A,t) where ut denotes the partial derivative of u with respect to t. The expression ut +(u )u that appears in the right-hand side of equa- tion (6.6) is the material· derivativer of the velocity field u,whichisoften Du denoted by (x, t); its definition takes into account the motion of the fluid Dt particles with time: d u(γ(t, x ),t)=u (γ(t, x ),t)+Du(γ(t, x ),t)˙γ(t, x ) dt 0 t 0 0 0 = ut(γ(t, x0),t)+Du(γ(t, x0),t)u(γ(t, x0),t) =(u +(u )u)(γ(t, x ),t) t ·r 0 Du = (γ(t, x ),t). Dt 0 The total stress in the direction of an arbitrary constant vector field v in R3 is given by (σ(x, t)⌘(x)) vd . · S Z@(A,t) Using the symmetry of the stress tensor, we have (⌘) v =(σv) ⌘. · · This fact together with the divergence theorem implies (σ(x, t)⌘(x)) vd = div(σ(x, t)v) d . · S V Z@(A,t) Zγ(A,t) By an easy calculation in components with σ represented as a matrix and v a constant vector, we have the identity (σv)=( σ)v. (6.7) r r· 262 Fluids By our notation, div(σv)= (σv). Thus, we may view σ (the divergence of σ), as a linear operation onr vectors. In Cartesian coordinates,r· this operator is given by @11 + @12 + @13 @x1 @x2 @x3 σ = @21 + @22 + @23 . r· 0 @x1 @x2 @x3 1 @31 + @32 + @33 @x1 @x2 @x3 Using these facts, the total stress@ is A TS = σ(x, t)⌘(x) d = σ(x, t) d .
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