Transport Phenomena – Fluid Mechanics Theory

Balance in Rectangular Cartesian Coordinates

Preparation Just what you need to know !

A shell momentum balance is used below to derive a general differential equation that can be then employed to solve several fluid flow problems in rectangular Cartesian coordinates. For this purpose, consider an incompressible fluid in laminar flow under the effects of both pressure and gravity in a system of length L and width W, which is at an angle β to the vertical. End effects are neglected assuming the dimension of the system in the x-direction is relatively very small compared to those in the y-direction (W) and the z-direction (L).

figure : differential shell momentum balance in rectangular Cartesian coordinatesFigure. Differential rectangular slab (shell) of fluid of thickness Δx used in z-momentum balance for flow in rectangular Cartesian coordinates. The y-axis is pointing outward from the plane of the computer screen.

Since the fluid flow is in the z-direction, vx = 0, vy = 0, and only vz exists. For small flow rates, the viscous forces prevent continual acceleration of the fluid. So, vz is independent of z and it is meaningful to postulate that velocity vz = vz(x) and pressure p = p(z). The only nonvanishing components of the stress tensor are τxz = τzx, which depend only on x.

Consider now a thin rectangular slab (shell) perpendicular to the x-direction extending a distance W in the y-direction and a distance L in the z-direction. A ‘rate of z-momentum’ balance over this thin shell of thickness Δx in the fluid is of the form:

Rate of z-momentum           In − Out + Generation = Accumulation
At steady-state, the accumulation term is zero. Momentum can go ‘in’ and ‘out’ of the shell by both the convective and molecular mechanisms. Since vz(x) is the same at both ends of the system, the convective terms cancel out because (ρ vz vz W Δx)|z = 0 = (ρ vz vz W Δx)|z = L. Only the molecular term (L W τxz ) remains to be considered, whose ‘in’ and ‘out’ directions are taken in the positive direction of the x-axis. Generation of z-momentum occurs by the pressure force acting on the surface [p W Δx] and gravity force acting on the volume [(ρ g cos β) L W Δx].

The different contributions may be listed as follows:

  • rate of z-momentum in by viscous transfer across surface at x is (L W τxz )| x
  • rate of z-momentum out by viscous transfer across surface at x + Δx is (L W τxz )| x + Δx
  • rate of z-momentum in by overall bulk fluid motion across surface at z = 0 is (ρ vz vz W Δx )| z = 0
  • rate of z-momentum out by overall bulk fluid motion across surface at z = L is (ρ vz vz W Δx )| z = L
  • pressure force acting on surface at z = 0 is p0 W Δx
  • pressure force acting on surface at z = L is − pL W Δx
  • gravity force acting in z-direction on volume of rectangular slab is (ρ g cos β) L W Δx

On substituting these contributions into the z-momentum balance, we get

(L W τxz ) | x − (L W τxz ) | x+Δx+ ( p 0 − p L ) W Δx + (ρ g cos β) L W Δx = 0
(1)


Dividing the equation by L W Δx yields

τxz | x+Δx − τxz | x

Δx

 = p 0 − p L + ρ g L cos β

L
(2)


On taking the limit as Δx → 0, the left-hand side of the above equation is exactly the definition of the derivative. The right-hand side may be written in a compact and convenient way by introducing the modified pressure P, which is the sum of the pressure and gravitational terms. The general definition of the modified pressure is P  =  p + ρ g h , where h is the distance upward (in the direction opposed to gravity) from a reference plane of choice. The advantages of using the modified pressure P are that (i) the components of the gravity vector g need not be calculated; (ii) the solution holds for any flow orientation; and (iii) the fluid may flow as a result of a pressure difference, gravity or both. Here, h is negative since the z-axis points downward, giving h = − z cos β and therefore P  =  p − ρ g z cos β. Thus, P0  =  p0 at z = 0 and PL  =  pL − ρ g L cos β at z = L giving p0 − pL + ρ g L cos β  =  P0 − PL  ≡  ΔP. Thus, equation (2) yields

dτxz

dx

 = ΔP

L
(3)


The first-order differential equation may be simply integrated to give

τxz  = ΔP

L

 x + C1
(4)


Here, C1 is an integration constant, which is determined using an appropriate boundary condition based on the flow problem. Equation (4) shows that the momentum flux (or shear stress) distribution is linear in systems in rectangular Cartesian coordinates.Since equations (3) and (4) have been derived without making any assumption about the type of fluid, they are applicable to both Newtonian and non-Newtonian fluids. Some of the axial flow problems in rectangular Cartesian coordinates where these equations may be used as starting points are given below.

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