An Internet Book on Fluid Dynamics Kinematics of Deformation and Motion In any homogeneous and isotropic continuum, whether fluid or solid, the deformation or motion of any incremental Lagrangian element can be decomposed into four fundamental deformations or motions. In these sections we will identify these fundamental deformations or motions and relate them to the displace- ments or velocities of the continuum. These relations are purely kinematic in that they do not involve any mechanical or physical properties of the continuum; consequently they are identical for a solid, fluid or any continuum. First we address the case of a solid where it is convenient to focus on the vector of displacements denoted by Xi that must be measured relative to some initial location or unstressed state of the substance. The four fundamental displacements are translation, rotation, shear (or extension) and dilation. Translation is obviously described by the displacement vector, Xi, and needs little further comment. At the next level of detail we can identify the deformation that is clearly related to the spatial differences in the displacements and therefore to the spatial gradient of the displacement, ∂Xi/∂xj. Thistensoriscalledthedisplacement gradient tensor. Note that without any displacement gradient there would be no deformation. Moreover, this displacement gradient tensor combines both the deformation and the rotation of the substance. Thus we separate these two displacements into the strain tensor, Eij, and the rotation tensor, Ωij ,whichare respectively the symmetric and antisymmetric parts of that displacement gradient tensor. The strain tensor is 1 ∂Xi ∂Xj Eij = + (Bba1) 2 ∂xj ∂xi and the rotation tensor is 1 ∂Xi ∂Xj Ωij = − (Bba2) 2 ∂xj ∂xi Note that Eij = Eji and Ωij = −Ωji (Bba3) We refer to the diagonal components in this tensor as the normal strains and the off-diagonal components as shear strains. Moreover the sum of the normal strains ∗ Φ = E11 + E22 + E33 (Bba4) is known as the dilation and is a measure of the volumetric expansion of the substance. In a precisely parallel way the motions of a fluid are characterized by the rates of change with time of the above measures. Thus the velocities, ui(xi,t), are just the Lagrangian rates of change with time of the displacements: DX u i i = Dt (Bba5) and the rate of deformation of the fluid is clearly related to the spatial gradients of the velocities, ∂ui/∂xj, a tensor that is called the velocity gradient tensor. Note that without any velocity gradient there would be no rate of deformation. Moreover, this velocity gradient tensor combines both the rate of deformation and the rate of rotation of the fluid. Thus we separate these by defining the strain rate tensor, eij,andthe ∗ rate of rotation tensor, ωij, that are respectively the symmetric and antisymmetric parts of that velocity gradient tensor, ∂ui/∂xj.Thustherate of strain tensor is 1 ∂ui ∂uj eij = + (Bba6) 2 ∂xj ∂xi and the rate of rotation tensor is ∗ 1 ∂ui ∂uj ωij = − (Bba7) 2 ∂xj ∂xi Note that ∗ ∗ eij = eji and ωij = −ωji (Bba8) We refer to the diagonal components in this matrix as the normal strain rates and the off-diagonal com- ponents as shear strain rates. Moreover the sum of the normal strain rates Φ=e11 + e22 + e33 (Bba9) is known as the dilation rate and is a measure of the rate of volumetric expansion of the fluid. Written out in full the rate of strain tensor (or matrix) has Cartesian components ∂u ∂v ∂w e e e xx = ∂x ; yy = ∂y ; zz = ∂z (Bba10) 1 ∂u ∂v exy = eyx = + (Bba11) 2 ∂y ∂x 1 ∂v ∂w eyz = ezy = + (Bba12) 2 ∂z ∂y 1 ∂w ∂u ezx = exz = + (Bba13) 2 ∂x ∂z In cylindrical coordinates, (r, θ, z), with velocities ur, uθ, uz in the r, θ, z directions, the strain rate tensor has components: ∂u ∂u u ∂u e r e 1 θ r e z rr = ∂r ; θθ = r ∂θ + r ; zz = ∂z (Bba14) 1 1 ∂ur ∂uθ uθ erθ = eθr = + − (Bba15) 2 r ∂θ ∂r r 1 1 ∂uz ∂uθ eθz = ezθ = + (Bba16) 2 r ∂θ ∂z 1 ∂ur ∂uz ezr = erz = + (Bba17) 2 ∂z ∂r In spherical coordinates, (r, θ, φ), with velocities ur, uθ, uφ in the r, θ, φ directions, the strain rate tensor has components: ∂ur 1 ∂uθ ur 1 ∂uφ ur uθ cot θ err = ; eθθ = + ; eφφ = + + (Bba18) ∂r r ∂θ r r sin θ ∂φ r r 1 ∂ uθ 1 ∂ur erθ = eθr = r + (Bba19) 2 ∂r r r ∂θ 1 sin θ ∂ uφ 1 ∂uθ eθφ = eφθ = + (Bba20) 2 r ∂θ sin θ r sin θ ∂φ 1 1 ∂ur ∂ uφ eφr = erφ = + r (Bba21) 2 r sin θ ∂φ ∂r r ∗ The other key kinematic property is the rate of rotation tensor, ωij given by ∗ ∗ 1 ∂ui ∂uj ωij = −ωji = − (Bba22) 2 ∂xj ∂xi ∗ ∗ Since ωij = −ωji this also allows us to define a very important kinematic property of a flow namely the ∗ vorticity, denoted by ωk which is just twice the rate of rotation tensor ωij so that ∗ ∂uj ∂ui ωk = −2ωij = − (Bba23) ∂xi ∂xj The vorticity is a key characteristic of a flow and will be addressed in many sections of this book. Written out in terms of its Cartesian components, the vorticity components are ∂w ∂v ω ω∗ − ω∗ − x =2zy = 2 yz = ∂y ∂z (Bba24) ∂u ∂w ω ω∗ − ω∗ − y =2xz = 2 zx = ∂z ∂x (Bba25) ∂v ∂u ω ω∗ − ω∗ − z =2yx = 2 xy = ∂x ∂y (Bba26) In cylindrical coordinates, (r, θ, z), with velocities ur, uθ, uz in the r, θ, z directions, the rotation tensor and vorticity have components: ∂u ∂u ω ω∗ − ω∗ 1 z − θ r =2zθ = 2 θz = r ∂θ ∂z (Bba27) ∂u ∂u ω ω∗ − ω∗ r − z θ =2rz = 2 zr = ∂z ∂r (Bba28) ∂ ru ∂u ω ω∗ − ω∗ 1 ( θ) − 1 r z =2θr = 2 rθ = r ∂r r ∂θ (Bba29) In spherical coordinates, (r, θ, φ), with velocities ur, uθ, uφ in the r, θ, φ directions, the rotation tensor and vorticity have components: ∗ ∗ 1 ∂(ruφ sin θ) ∂(ruθ) ωr =2ω = −2ω = − (Bba30) φθ θφ r2 sin θ ∂θ ∂φ ∗ ∗ 1 ∂ur ∂(ruφ sin θ) ωθ =2ω = −2ω = − (Bba31) rφ φr r sin θ ∂φ ∂r ∂ ru ∂u ω ω∗ − ω∗ 1 ( θ) − r φ =2θr = 2 rθ = r ∂r ∂θ (Bba32).