CONVECTIVE FLUX, FLUID MASS CONSERVATION

1. CONVECTIVE FLUX, FLUID MASS CONSERVATION The flux of any quantity in any direction is the rate per unit time per unit area that the quantity crosses a face normal to that direction. Quantities that can be fluxed in fluids include fluid mass, mass of a dissolved contaminant such as salt, mass of a suspended contaminant such as sediment, fluid momentum, fluid energy, heat, etc. There are three fundamental mechanisms of flux of interest in fluid mechanics: Convective flux, by which the quantity is carried with the flow; Diffusive flux, by which the quantity migrates from zones of high concentration to zones of low concentration by random molecular motion, and Radiative flux, by which the quantity (e.g. heat) is carried by waves such as electromagnetic waves (in the infrared spectrum in the case of heat). Here we are concerned only with the first two kinds.

2. A u1 u1tA x1 u1t CONVECTIVE FLUX, FLUID MASS CONSERVATION A flowing fluid carries any associated quantity with it convectively. We first consider the convective flux of fluid volume and mass. The tube shown below has rectangular cross-section with area A. The fluid velocity u1 in the tube is taken to be constant on the cross-section. At time t = 0 we mark a parcel of fluid, the downstream end of which is bounded by an orange face. In time t the leading edge of the marked parcel moves downstream a distance u1t, so that volume u1tA and mass u1tA has crossed the face in time t. Convective flux of fluid volume in x1 direction FC,vol,1 = the volume that moves across a face per unit face area per unit time = u1tA/(tA) = u1

3. A u1 u1tA x1 u1t CONVECTIVE FLUX, FLUID MASS CONSERVATION The mass that crosses normal to the section in time t is density x volume crossed = u1tA The convectivemass flux in the x1 direction across the section = u1tA/(tA) = u1. The heat in a fluid is characterized by the heat capacity cp = heat needed to raise one kg of the fluid 1 degree. In the SI system, [cp] = joules/kg/°K. Where  denotes the temperature of the fluid in °K, then, the amount of heat per unit volume in the fluid is cp ~ joules/m3. The convective flux of heat in the x1 direction = FC,heat,1 heat that moves across a face per unit face area per unit time = cpu1tA/(tA) = cpu1

4. A u1 u1tA x1 u1t CONVECTIVE FLUX, FLUID MASS CONSERVATION The momentum in the x1 direction that crosses normal to the section in time t = velocity x mass = u1 x u1tA = (u1)2 tA The convective fluxin the x1 direction of momentum in the x1 direction across the section = (u1)2tA/(tA) = (u1)2 Summary: the convective flux of a quantity in some direction = the quantity per unit volume x the flow velocity in the direction it is being fluxed. Volume flux in x1 direction = volume/volume x u1 = FC,vol,1 u1 Mass flux in x1 direction = mass/volume x u1 = FC,mass,1 = u1 Heat flux in x1 direction = heat/volume x u1 = FC,heat,1 = cpu1 Flux in x1 direction of momentum in x1 direction = momentum/volume x u1 = u1 x u1 = FC,mom,11 = (u1)2

5. CONVECTIVE FLUX, FLUID MASS CONSERVATION A scalar quantity such as volume, mass and heat can be fluxed three-dimensional space (x1, x2, x3), so that the flux of a scalar quantity is a vector. The vectorial convective flux of a scalar quantity = the quantity per unit volume x the velocity vector. Thus the flux vectors of volume, fluid mass and heat are: FC,vol,i = ui, FC,mass,i = ui and FC,heat,i = cpui. The flux of a scalar quantity across a face of specified direction ni is a scalar quantity, given as the dot produce of the flux vector and the unit normal of the face. Consider the illustrated face directed in the ni direction. The flux of fluid mass across the face is given as uini, i.e uini i.e. the component of the vector uini that is oriented normal to the face (and thus crosses it). In the diagram, the pink vector denotes ni, the red vector denotes uini, the blue vector denotes the component of uini that is parallel to the face (and thus does not cross it) and the green vector denotes the component uini of ui that is perpendicular to the face and thus is the mass flux across it. ui ni

6. CONVECTIVE FLUX, FLUID MASS CONSERVATION Let Fi denote the flux vector of some quantity and ni denote the unit normal vector of a face of area A. The discharge of the quantity across the face Qquan is a scalar given as The discharges of volume, mass and heat across the face are thus given as Note that no subscript is used in the case of volume discharge If ui is oriented normal to the face and has the value U which is constant across the face, the (volume) discharge is given as uini ui ni

7. CONVECTIVE FLUX, FLUID MASS CONSERVATION A vector quantity, such as momentum, has three components, each of which can be fluxed in each of the three directions (x1, x2, x3), so that the flux of a vector quantity is a matrix. The flux of momentum ui in the xj direction is given as FC,mom,ij = uiuj. For example, the flux in the x2 direction of momentum in the x1 direction is given as u1u2. Now how can momentum in the x1 direction be fluxed in the x2 direction? Consider a velocity vector that crosses a face diagonally.

8. ni ui ui u1 u1 u2 u2 x1 x3 x2 CONVECTIVE FLUX, FLUID MASS CONSERVATION More specifically, we assume that ui = momemtum per unit volume = (u1, u2 ,0). The animation illustrates how momentum in the x1 direction can be fluxed in the x2 direction. Note that in this case ni = (0, 1, 0).

9. CONVECTIVE FLUX, FLUID MASS CONSERVATION So let’s compute the flux in the x2 direction of momentum in the x1 direction. How much x1 momentum crossed the face normal to the x2 direction in time t? u2t u1t Where A denotes the area of the face, the volume of fluid that crossed in time t = Au2t. u1 The total amount of x1 momentum contained in this volume = momentum/volume * volume = u1Au2t. Flux of x1 momentum in x2 direction = u1Au2t/(At) = u2 x3 x1 x2

10. ni dA CONVECTIVE FLUX, FLUID MASS CONSERVATION The flux uiuj of momentum in the i direction across a face in the nj direction is a vector given as where ujnj denotes the component of the velocity that is actually normal to the face. Let nj denote a unit outward normal to a control volume that is fixed in space (so that fluid can flow in and out). The net inflow rate of the scalar quantity fluid massinto the control volume is The net inflow rate Qmom, inflow,i of the vector quantity fluid momentuminto the control volume is

11. ni dA CONVECTIVE FLUX, FLUID MASS CONSERVATION Equation of fluid mass conservation: /t(fluid mass in control volume) = net inflow rate of mass Now the control volume is fixed in space, so the equation can be rewritten as: But So

12. ni dA CONVECTIVE FLUX, FLUID MASS CONSERVATION Now the condition must be true for any volume V. The only way this can hold is if the integrand vanishes everywhere: The above equation is the equation of mass conservation, or continuity equation of a fluid. For an incompressible fluid, i.e. one for which  can be locally approximated as constant, the relation reduces to

13. CONVECTIVE FLUX, FLUID MASS CONSERVATION But this result is predicated on the following theorem:, if A = A(xi) is a continuous function of xi and simply-connected volume in a given domain, then everywhere within that domain. The equivalent 1D theorem can be stated as follows. Let f(x) be a continuous function such that for every set of values x1 and x2 within a given domain Then everywhere within that domain

14. CONVECTIVE FLUX, FLUID MASS CONSERVATION Proof: if everywhere within some domain, then where x is a free variable within that domain, Now take the derivative with respect to x to obtain and the desired result is proved.

15. L x2 x1 L CONVECTIVE FLUX, FLUID MASS CONSERVATION 2D corner flow of an incompressible fluid ui = (0, - U) Thus Now ui = (U, 0) Thus continuity is satisfied. The physical interpretation is as follows. Since no mass can be stored in the control “volume” because the fluid is incompressible, the inflow into the control volume must be precisely balanced by the outflow.