Formulation of an Initial-Boundary-Value Problem

One generally uses the spatial description of motion for a fluid. Thus the balance of mass, and the balance of linear momentum for a compressible Navier-Stokes fluid are

$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho v_i)}{\partial x_i} = 0,$$ (6.2.1)

$$\rho\left(\frac{\partial v_i}{\partial t} + \frac{\partial v_i}{\... ...i\partial x_k} + \mu \frac{\partial^2v_i}{\partial x_k\partial x_k} + \rho g_i.$$ (6.2.2)

Equation (6.2.2) is obtained by substituting into (4.1.14) for the acceleration from (3.6.3) and for the Cauchy stress from (6.1.3). Because of the presence of the convective part of the acceleration on the left-hand side of (6.2.2), equations governing the motion of a fluid are nonlinear. In the referential description of motion used to describe the deformations of a solid, the acceleration is linear in displacements. Were we to use the referential description of motion for studying the deformations of a fluid, the expression for the strain-rate tensor $\mathbf{D}$ in terms of the spatial gradients of the velocity field will be more involved. The nonlinearity in (6.2.2) implies that we can no longer use the principle of superposition. Whereas for solids, the deformations caused by gravity are generally negligible and hence are ignored, gravity is the main driving force for a fluid. A familiar example is the flow of a fluid in a river. Note that equations governing the deformations of a solid were expressed in terms of displacements, those for a fluid are written in terms of velocities.

For an incompressible Navier-Stokes fluid, equations expressing the balance of mass and the balance of linear momentum are

$$\frac{\partial v_i}{\partial x_i} = 0,$$ (6.2.3)

$$\rho\left(\frac{\partial v_i}{\partial t} + \frac{\partial v_i}{\... ...{\partial x_i} + \mu \frac{\partial^2v_i}{\partial x_k\partial x_k} + \rho g_i.$$ (6.2.4)

The mass density $\rho$ is a constant, and the pressure field $p$ is an arbitrary function of $\mathbf{x}$ and time $t$.

Equations (6.2.1) and (6.2.2) for a compressible Navier-Stokes fluid or (6.2.3) and (6.2.4) for an incompressible Navier-Stokes fluid are supplemented by the following initial and boundary conditions.

$$\rho (\mathbf{x},0) =\ \rho_0 (\mathbf{x})\ {\rm in}\ \Omega ,$$

$$v_i(\mathbf{x},0)=\ v^0_i(\mathbf{x})\ {\rm in}\ \Omega,$$ (6.2.5)

$$[-p\delta_{ij} + \lambda v_{k,k} \delta_{ij} + \mu (v_{i,j} + v_{j,i})]n_j =\ f_i(\mathbf{x},t)\ {\rm on}\ \partial_1\Omega\times (0,T),$$

$$v_i(\mathbf{x},t) =\ v^p_i (\mathbf{x},t)\ {\rm on}\ \partial_2\Omega\times (0,T).$$ (6.2.6)

Here $\Omega$ is the domain of study which may equal only a part of the region occupied by the fluid. Equations (6.2.5) specifying the mass density and the velocity at time $t = 0$ are the initial conditions, and (6.2.6) prescribing surface tractions on a part of the boundary and velocities on the remainder of the boundary are boundary conditions. If the velocity field is prescribed on the entire boundary, then for an incompressible fluid, the pressure field is indeterminate.

For a steady flow, the field variables $\rho,\ p$ and $\mathbf{v}$ in the spatial description are independent of time $t$. Thus $$\frac{\partial\rho}{\partial t} = 0 = \frac{\partial\mathbf{v}}{\partial t}$$ in equations (6.2.1) through (6.2.4), and initial conditions (6.2.5) are not required.

A flow is called irrotational if the spin tensor or the vorticity vanishes everywhere in the domain. Governing equations for the steady irrotational flow of an incompressible Navier-Stokes fluid are

$$\frac{\partial v_i}{\partial x_i} =\ 0,$$ (6.2.7)

$$\frac{\partial v_i}{\partial x_j}v_j =\ - \frac{\partial p}{\partial x_i} + g_i.$$ (6.2.8)

In writing (6.2.8) we have absorbed the constant mass density in $p$. Thus the viscosity of a fluid does not influence the steady irrotational flow of an incompressible Navier-Stokes fluid.