0.5 Equations of motion and energy in cartesian coordinates  (Page 7/12)

Under the first heading we have any of the specializations of the velocity as a vector field. These are essentially kinematic restrictions.

• (ia) Isochoric motion. (i.e., constant density) The velocity field is solenoidal
  $$-\frac{1}{\rho }\frac{D\rho }{Dt}=\nabla •\mathbf{v}=\Theta =0$$
  The equation of continuity now gives
  $$\frac{D\rho }{Dt}=0$$
  that is, the density does not change following the motion. This does not mean that it is uniform, though, if the fluid is incompressible, the motion is isochoric. The other equations simplify in this case for we have $\alpha$ , $\beta$ , and $\gamma$ of the constitutive equations functions of only $\Phi$ and $\Psi$ of the invariants of the rate of deformation tensor. In particular for a Newtonian fluid
  $$\begin{array}{c}\hfill T_{ij}=-p{\delta }_{ij}+2\mu e_{ij}\\ \hfill \rho \frac{D\mathbf{v}}{Dt}=\rho \mathbf{f}-\nabla p+\mu {\nabla }^2\mathbf{v}\end{array}$$
  The energy equation is
  $$\rho \frac{DU}{Dt}=\nabla •\left(k\nabla T\right)+Y,$$
  and for a Newtonian fluid
  $$Y=-4\mu \Phi .$$
  Because the velocity field is solenoidal, the velocity can be expressed as the curl of a vector potential.
  $$\mathbf{v}=\nabla \times \mathbf{A}.$$
  The Laplacian of the vector potential can be expressed in terms of the vorticity.
  $$\begin{array}{ccc}\hfill \mathbf{w}& =& \nabla \times \mathbf{v}\hfill \\ & =& \nabla \times (\nabla \times \mathbf{A})\hfill \\ & =& \nabla (\nabla •\mathbf{A})-{\nabla }^2\mathbf{A}\hfill \\ & =& -{\nabla }^2\mathbf{A},\text{if}\nabla •\mathbf{A}=0\hfill \end{array}$$
  If the body force is conservative, i.e., gradient of a scalar, the body force and pressure can be eliminated from the Navier-Stokes equation by taking the curl of the equation.
  $$\frac{D\mathbf{w}}{Dt}=\mathbf{w}•\nabla \mathbf{v}+v{\nabla }^2\mathbf{w}$$
  where $\nu$ is the kinematic viscosity.
  Isochoric motion is a restriction that has to be justified. Because it is justified in so many cases, it is easier to identify the cases when it does not apply. We showed during the discussion of the effects of compressibility that compressibility or non-isochoric is important in the cases of significant Mach number, high frequency oscillations such as in acoustics, large dimensions such as in meteorology, and motions with significant viscous or compressive heating.
• (ib) Irrotational motion. The velocity field is irrotational
  $$\mathbf{w}=\nabla \times \mathbf{v}=0$$
  It follows that there exists a velocity potential (x,t) from which the velocity can be derived as
  $$\mathbf{v}=\nabla \varphi$$
  and in place of the three components of velocity we seek only one scalar function. (Note that some authors express the velocity as the gradient of a scalar and others as the negative of a gradient of a scalar. We will use either to conform to the book from which it was extracted.) The continuity equation becomes
  $$\frac{D\rho }{Dt}+\rho {\nabla }^2\varphi =0$$
  so that for an isochoric (or incompressible), irrotational motion, is a potential function satisfying
  $${\nabla }^2\varphi =0$$
  The Navier-Stokes equations become
  $$\nabla \left[\frac{\partial \varphi }{\partial t}+\frac{1}{2}{\left(\nabla \varphi \right)}^2\right]=\mathbf{f}-\frac{1}{\rho }\nabla p+({\lambda }^{\text{'}}+2\nu )\nabla \left({\nabla }^2\varphi \right)$$
  In the case of an irrotational body force $\mathbf{f}=-\nabla \Omega$ and when p is a function only of , this has an immediate first integral since every term is a gradient. Thus if $P\left(\rho \right)=\int dp/\rho$ ,
  $$\frac{\partial \varphi }{\partial t}+\frac{1}{2}{\left(\nabla \varphi \right)}^2+\Omega +P\left(\rho \right)-({\lambda }^{\text{'}}+2\nu ){\nabla }^2\varphi =g\left(t\right)$$
  is a function of time only.
  Irrotational motions with finite viscosity are only very special motions because the no-slip boundary conditions on solid surfaces usually will cause generation of rotation. Usually irrotational motion is associated with inviscid fluids because the no-slip boundary condition then will not apply and initially irrotational motion will remain irrotational.
• (ic) Complex lamellar motions, Betrami motions, ect. These names can be applied when the velocity field is of this type. Various simplifications are possible by expressing the velocity in terms of scalar fields. We shall not discuss them further here.
• (id) Plane flow. Here the motion is restricted to two dimensions which may be taken to be the 012 plane. Then v3 = 0 and x3 does not occur in the equations. Also, the vector potential and the vorticity have only one nonzero component.
  Incompressible plane flow. Since the flow is solenoidal, the velocity can be expressed as the curl of the vector potential. The nonzero component of the vector potential is the stream function.
  $$\begin{array}{ccc}\hfill \mathbf{v}& =& \nabla \times \mathbf{A}\hfill \\ \hfill v_i& =& {\epsilon }_{ij3}{A}_{3,j}={\epsilon }_{ij}{\psi }_{,j}\hfill \\ \hfill (v_1,v_2,v_3)& =& \left(\frac{\partial \psi }{\partial x_2},-\frac{\partial \psi }{\partial x_1},0\right)\hfill \end{array}$$
  The vorticity has only a single component, that in the 03 direction, which we will write without suffix
  $$\begin{array}{ccc}\mathbf{w}\hfill & =\hfill & \nabla \times \mathbf{v}\hfill \\ w_i\hfill & =\hfill & {\epsilon }_{ijk}{v}_{k,j},j,k=1,2\hfill \\ w\hfill & =\hfill & \frac{\partial v_2}{\partial x_1}-\frac{\partial v_1}{\partial x_2}\hfill \\ & =\hfill & -{\nabla }^2\psi \hfill \end{array}$$
  If the body force is conservative, i.e., gradient of a scalar, then the body force and pressure disappear from the Navier-Stokes equation upon taking the curl of the equations. In plane flow
  $$\frac{Dw}{Dt}=\nu {\nabla }^{2}w$$
  Thus for incompressible, plane flow with conservative body forces, the continuity equation and equations of motion reduce to two scalar equations.
  Incompressible, irrotational plane motion. A vector field that is irrotational can be expressed as the gradient of a scalar. Since the flow is incompressible, the velocity vector field is solenoidal and the Laplacian of the scalar is zero, i.e., it is harmonic or an analytical function.
  $$\begin{array}{ccc}\hfill \mathbf{v}& =& \nabla \varphi \hfill \\ \hfill \nabla •\mathbf{v}& =& 0\hfill \\ & =& {\nabla }^2\varphi \hfill \end{array}$$
  Since the flow is incompressible, it also can be expressed as the curl of the vector potential, or in plane flow as derivatives of the stream function as above. Since the flow is irrotational, the vorticity is zero and the stream function is also an analytical function, i.e., $\mathbf{v}=\nabla \times \mathbf{A},0=\mathbf{w}=\nabla \times \mathbf{v}=-{\nabla }^2\mathbf{A}+\nabla \left(\nabla •\mathbf{A}\right),\Rightarrow {\nabla }^2\psi =0$ . Thus
  $$\begin{array}{ccccc}{v}_1\hfill & =\hfill & \frac{\partial \varphi }{\partial x_1}\hfill & =\hfill & \frac{\partial \psi }{\partial x_2}\hfill \\ v_2\hfill & =\hfill & \frac{\partial \varphi }{\partial x_2}\hfill & =\hfill & -\frac{\partial \psi }{\partial x_1}\hfill \end{array}$$
  These relations are the Cauchy-Riemann relations show that the complex function $f=\varphi +i\psi$ is an analytical function of $z=x_1+i{x}_2$ . The whole resources of the theory of functions of a complex variable are thus available and many solutions are known.
  Steady, plane flow. If the fluid is compressible but the flow is steady (i.e., no quantity depends on t) the equation of continuity becomes
  $$\frac{\partial \left(\rho v_1\right)}{\partial x_1}+\frac{\partial \left(\rho v_2\right)}{\partial x_2}=0$$
  A stream function can again be introduced, this time in the form
  $$v_1=\frac{1}{\rho }\frac{\partial \psi }{\partial x_2},v_2=-\frac{1}{\rho }\frac{\partial \psi }{\partial x_1}$$
  The vorticity is now given by
  $$\rho w=-{\nabla }^2\psi +\frac{\nabla \psi •\nabla \rho }{\rho }$$
• (ie) Axisymmetric flows. Here the flow has an axis of symmetry such that the flow field can be expressed as a function of only two coordinates by using curvilinear coordinates. The curl of the vector potential and velocity has only one non-zero component and a stream function can be found.
• (if) Parallel-flow perpendicular to velocity gradient. If the flow is parallel, i.e., the streamlines are parallel and are perpendicular to the velocity gradient, then the equations of motion become linear in velocity if the fluid is Newtonian.
  $$\begin{array}{c}\mathrm{If}\mathbf{v}•\nabla \mathbf{v}=0,\mathrm{then}\frac{D\mathbf{v}}{Dt}=\frac{\partial \mathbf{v}}{\partial t},\mathrm{and}\hfill \\ \rho \frac{\partial \mathbf{v}}{\partial t}=\mathbf{f}+\nabla •\mathbf{T}\hfill \end{array}$$
  The second type of specializations are limiting cases of the equations of motion.