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Note that only we use column vectors (in cartesian coordinate system) above for clarity, but the equation is written using physical components (which are neither covariants ("column") nor contravariants ("row") ). However, if we chose a non-orthogonal curvilinear coordinate system, then we should calculate and write equations in covariant ("row vectors") or contravariant ("column vectors") form.
After an appropriate change of variables, it can also be written in conservation form:
where j is the momentum density at a given space-time point, F is the flux associated to the momentum density, and s contains all of the body forces per unit volume.
where is momentum in time t, is force averaged over . After dividing by and passing to the limit we get (derivative):
Let us analyse each side of the equation above.
The X component of the forces acting on walls of a cubic fluid element (green for top-bottom walls; red for left-right; black for front-back).
In the top graph we see approximation of function (blue line) using a finite difference (yellow line). In the bottom graph we see "infinitely many times enlarged neighborhood of point " (purple square from the upper graph). In the bottom graph, the yellow line is completely covered by the blue one, thus not visible. In the bottom figure, two equivalent derivative forms have been used: ], and the designation was used.
Surface forces act on walls of the cubic fluid element. For each wall, the X component of these forces was marked in the figure with a cubic element (in the form of a product of stress and surface area e.g. ).
Adding forces (their X components) acting on each of the cube walls, we get:
After ordering and performing similar reasoning for components (they have not been shown in the figure, but these would be vectors parallel to the Y and Z axes, respectively) we get:
We can then write it in the symbolic operational form:
There are mass forces acting on the inside of the control volume. We can write them using the acceleration field (e.g. gravitational acceleration):
Let us calculate momentum of the cube:
Because we assume that tested mass (cube) is constant in time, so
where ? represents the control volume. Since this equation must hold for any control volume, it must be true that the integrand is zero, from this the Cauchy momentum equation follows. The main step (not done above) in deriving this equation is establishing that the derivative of the stress tensor is one of the forces that constitutes Fi.
The Cauchy momentum equation can also be put in the following form:
Cauchy momentum equation(conservation form)
simply by defining:
where j is the momentum density at the point considered in the continuum (for which the continuity equation holds), F is the flux associated to the momentum density, and s contains all of the body forces per unit volume. u ? u is the dyad of the velocity.
Here j and s have same number of dimensions N as the flow speed and the body acceleration, while F, being a tensor, has N2.[note 1]
In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the Euler equations.
An example of convective acceleration. The flow is steady (time-independent), but the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow).
A significant feature of the Navier-Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a flow with respect to space. While individual continuum particles indeed experience time dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.
Regardless of what kind of continuum is being dealt with, convective acceleration is a nonlinear effect. Convective acceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamic effect is disregarded in creeping flow (also called Stokes flow). Convective acceleration is represented by the nonlinear quantity u · ?u, which may be interpreted either as (u · ?)u or as u · (?u), with ?u the tensor derivative of the velocity vector u. Both interpretations give the same result.
Advection operator vs tensor derivative
The convection term can be written as (u · ?)u, where u · ? is the advection operator. This representation can be contrasted to the one in terms of the tensor derivative.
The tensor derivative ?u is the component-by-component derivative of the velocity vector, defined by [?u]mi = ?m vi, so that
In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:
And by projecting the momentum equation on the flow direction, i.e. along a streamline, the cross product disappears due to a vector calculus identity of the triple scalar product:
If the stress tensor is isotropic, then only the pressure enters: (where I is the identity tensor), and the Euler momentum equation in the steady incompressible case becomes:
In the steady incompressible case the mass equation is simply:
that is, the mass conservation for a steady incompressible flow states that the density along a streamline is constant. This leads to a considerable simplification of the Euler momentum equation:
The convenience of defining the total head for an inviscid liquid flow is now apparent:
in fact, the above equation can be simply written as:
That is, the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant.
The Lamb form is also useful in irrotational flow, where the curl of the velocity (called vorticity) ? = ? × u is equal to zero. In that case, the convection term reduces to
The effect of stress in the continuum flow is represented by the ?p and ? · ? terms; these are gradients of surface forces, analogous to stresses in a solid. Here ?p is the pressure gradient and arises from the isotropic part of the Cauchy stress tensor. This part is given by the normal stresses that occur in almost all situations. The anisotropic part of the stress tensor gives rise to ? · ?, which usually describes viscous forces; for incompressible flow, this is only a shear effect. Thus, ? is the deviatoric stress tensor, and the stress tensor is equal to:[full ]
where I is the identity matrix in the space considered and ? the shear tensor.
The divergence of the stress tensor can be written as
The effect of the pressure gradient on the flow is to accelerate the flow in the direction from high pressure to low pressure.
As written in the Cauchy momentum equation, the stress terms p and ? are yet unknown, so this equation alone cannot be used to solve problems. Besides the equations of motion--Newton's second law--a force model is needed relating the stresses to the flow motion. For this reason, assumptions based on natural observations are often applied to specify the stresses in terms of the other flow variables, such as velocity and density.
The vector field f represents body forces per unit mass. Typically, these consist of only gravity acceleration, but may include others, such as electromagnetic forces. In non-inertial coordinate frames, other "inertial accelerations" associated with rotating coordinates may arise.
Often, these forces may be represented as the gradient of some scalar quantity ?, with f = ?? in which case they are called conservative forces. Gravity in the z direction, for example, is the gradient of -?gz. Because pressure from such gravitation arises only as a gradient, we may include it in the pressure term as a body force h = p - ?. The pressure and force terms on the right-hand side of the Navier-Stokes equation become
It is also possible to include external influences into the stress term rather than the body force term. This may even include antisymmetric stresses (inputs of angular momentum), in contrast to the usually symmetrical internal contributions to the stress tensor.
In order to make the equations dimensionless, a characteristic length r0 and a characteristic velocity u0 need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
Substitution of these inverted relations in the Euler momentum equations yields:
^In 3D for example, with respect to some coordinate system, the vector j has 3 components, while the tensors ? and F have 9 (3x3), so the explicit forms written as matrices would be:
Note, however, that if symmetrical, F will only contain 6 degrees of freedom. And F's symmetry is equivalent to ?'s symmetry (which will be present for the most common Cauchy stress tensors), since dyads of vectors with themselves are always symmetrical.