Convective momentum flux

The last but one term on the RHS denotes kinetic pressure and the last term the convective momentum flux. 

The necessary correction factor is best found from the convective momentum flux, i.e., the momentum carried by the moving fluid through a unit cross-sectional area of the channel in a unit time. This is the product of the linear velocity normal to the cross section and the mass velocity (or mass flux). For a difierential cross-sectional area dS, then, the momentum flux is... 

If fluid motion is restricted to the x direction in rectangular coordinates as illustrated above, then the only nonvanishing scalar of convective momentum flux is pVxVx, which has units of momentum per time per area or force per unit area. Hence, pvv contains forces per unit area that are transmitted across the surfaces that bound fluid within the control volume, and terms of this nature due to convection motion of a fluid must be included in the force balance. One must construct the product of each of the nine scalars generated from this second-rank tensor with the surface area across which the force (or stress) is transmitted. Information about these smfaces and the coordinate direction in which the force acts is contained in the subscripts of the velocity components. For the most general type of fluid flow in rectangular coordinates, the nine scalars that one can generate from convective momentum flux are... 

As an illustrative example, focus on the element in the first row and second column, pVjcVy, for the matrix representation of the convective momentum flux tensor. The subscript x on the first velocity component indicates that pv Vy is a force per unit area acting across a simple surface oriented with a unit normal vector in the x direction. The subscript y on the second velocity component reveals that this force acts in the y direction. If we perform this analysis for all nine components in the matrix (8-6) the three entries in the first row represent X, y, and z components, respectively, of the vector force per unit area that is transmitted across the simple surface defined by a constant value of the x coordinate, which means that the unit normal vector to the surface is colinear with the X direction. In vector-tensor notation, this is... 

These vector forces per unit area are obtained by taking the dot product of a unit normal vector with the second-rank tensor due to convective momentum flux. Each force is transmitted across the surface in the direction of n. If n represents an outward-directed unit normal vector on each differential element of surface dS that surrounds fluid within the control volume, then the vector force acting on the system due to convective momentum flux is... 

Vector-Tensor Manipulation of the Accumulation Rate Process and Forces Due to Convective Momentum Flux... 

Next, a vector-tensor identity is employed to expand the convective momentum flux term in (8-23)... 

Add the tensorial rank of each cofactor and subtract 2 for the dot operation. Scalars are zeroth-rank tensors and vectors are first-rank tensors. Since the del operator is a vector and convective momentum flux is a second-rank tensor, V pvv is a vector. The fcth component of V pvv is... 

Even though this vector-tensor identity was verified using summation notation in rectangular coordinates, it is valid in any coordinate system. It is extremely tedious to verify vector-tensor identities that involve the del operator in curvilinear coordinate systems because the unit vectors exhibit spatial dependence. Now it is possible to combine terms in the equation of motion due to the accumulation rate process and convective momentum flux. Equations (8-24) and (8-25) yield ... 

This represents a balance between forces due to convective momentum flux, fluid pressure, and gravity. The vector-tensor identity presented in Problem 8-7 is used to re-express forces due to convective momentum flux ... 

In the limit of vanishingly small Reynolds numbers, forces due to convective momentum flux are negligible relative to viscous, pressure, and gravity forces. Equation (12-4) is simplified considerably by neglecting the left-hand side in the creeping flow regime. For fluids with constant /r and p, the dimensionless constitutive relation between viscons stress and symmetric linear combinations of velocity gradients is... 

It is necessary to introduce the following vector-tensor identity for convective momentum flux (i.e., pv Vv) in the equation of motion (see Problem 8-7) ... 

Now, one of the terms that must be considered when one constructs the scalar dot product of v with convective momentum flux in the equation of motion is... 

The net convective momentum flux across the annular surface at x and x + Ax is zero, since the flow is fully developed and the terms are independent of x. This is true since at X is equal to at x 4- Ax. 

The net convective momentum flux is the rate of momentum entering the area ts.xW at... 

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