## Abstract

In the current chapter, some applications of tensor analysis to fluid dynamics are presented. Governing equations of fluid motion and energy are obtained and analyzed. We shall discuss about continuity equation, equation of motion, and mechanical energy transport equation and four forms of energy equation. Finally, we shall talk about the divergence from transfer equations of different parameters of motion. The tensor form of equations has advantages over the component form: these are, first, compact writing of equations and, second, independency from reference frames, etc. Moreover, it allows to obtain new forms of equations on the basis of governing ones easily.

### Keywords

- stress tensor
- Navierâ€“stokes equation
- energy
- continuity
- vorticity
- divergence form

## 1. Introduction

The mathematical model of moving fluid includes a set of equations, which are usually written as transport equations of main physical parameters—density, velocity, energy, etc. These equations are conservation laws in fluid flows. Traditionally the component form of the equations is usually used, but at the same time, the componentless form (Gibbs approach) could be applied to obtain and transform these equations. In this chapter several main conservation laws are discussed and represented in tensor form, which has many advantages against usually used component form, like simplicity and compactness, independence on reference frames, less errors in transformations, etc. Below we obtain and analyze continuity and momentum equations and vorticity and energy transport equations, and we discuss also about the divergent form of transport Eqs.

## 2. Continuity equation

Continuity equation is the mass conservation law for a fluid flow and is presented as a scalar equation, which connects density *ρ* and velocity of fluid particles

where

Hereinafter, the Einstein summation convention is used by default.

It also can be written in two other equivalent forms [1, 2]:

In the case of incompressible fluid, we could obtain its simplified form:

Let us apply gradient operator to continuity equation (Eq. (1)):

As a result, we obtain vector equation:

which could be written in a more compact form:

or by a little bit different way:

These equations contain gradient of vector

For incompressible fluid the left part of this relation is equal to zero; therefore for rotation of velocity vector, we can write:

In the case of compressible fluid in accordance with Eq. (6), we have additional terms in the right part of the equation:

Continuity equation can be also written in tensor form:

The tensor

Finally, continuity equation can be written in form

Convective derivatives of density and pressure (and any another scalar quantitatives) also can be written in tensor form:

i.e., convective derivative is equal to trace of corresponding tensor.

In addition, for the divergence of the product of scalar and vector functions, we can obtain the following relation:

## 3. Equations of motion of fluid with constant and variable properties

The equation of a motion in terms of stress [4, 5] is

where

where *p* is the pressure; *μ* is the fluid shear (dynamic) viscosity; and

Then equation of motion of incompressible fluid (Navier–Stokes equation) at

Additional term

In case of compressible fluid with variable viscosity, the equation will contain a term with divergence

Let us introduce the denotation:

then we can write Eq. (16) as

Divergence of this tensor at

As a result, equation of motion of compressible fluid with variable viscosity has the form:

in Cartesian coordinates

If we represent fluid particle acceleration as the sum of local and convective terms, then (Eq. (19)) will take the form:

considering viscosity variability is especially important for turbulent flow modeling using the Boussinesq hypothesis with turbulent viscosity

Let us apply divergence operation to the Navier–Stokes equation for compressible fluid with variable viscosity. With this purpose we shall apply operation

If the fluid motion occurs in gravity force field, then there is potential *z* is the vertical coordinate and the body force per unit mass is

Function *U* is linear; therefore

As a result of applied divergence operation to Navier–Stokes equation at

In the case of incompressible fluid, we have

if also

Now we consider the general case of fluid motion, taking into account its compressibility.

The set of equations of motion of an incompressible fluid contains two—Navier–Stokes and continuity (one vector equation and one scalar equation) [2, 3]:

This set of two equations is closed: it contains two unknown quantities—velocity vector

In case of compressible flows at

where *T*. To determine temperature *T*, state equation is used; usually in fluid dynamics, it is the Mendeleev-Clapeyron equation. Energy equation could be written as the equation of specific internal energy transport:

where *u* is the specific internal energy (for ideal gas it could be expressed with the help of the isochore heat capacity, *λ* is the thermal conductivity of the material; and

Mendeleev-Clapeyron equation has the form

where *R* is the universal gas constant. In case of real gas or fluid, the state equation becomes more complicated.

For liquids it usually supposes *ρ = const*. This condition is applicable for gas motions and also in case the velocities of gas particles are less than 1/3 of sound velocity.

Eqs. (1), (25), (26), and (27) are valid for laminar regime of motion. In case of turbulent regime in these equations, correlations will appear, caused by velocity, density, and temperature pulsations. For closure of the set of equations of turbulent motion, additional relations are required.

## 4. Vorticity vector and its associated tensor

Vorticity

in component form

where *εijk* is the Levi-Civita tensor in component form. Components of

Vorticity vector

And vice versa, vector

In Eq. (30) spin tensor

Let us prove expression Eq. (30):

The same for Eq. (31):

Let us descry components of this second-rank tensor: when

values for all

The matrix of components of this tensor is

It can be seen that it is a matrix of components of the antisymmetric tensor

It is easy to see also that

## 5. Vorticity transport equation

Rotation of convective acceleration of a fluid particle could be written as

Evidence of this equation can be performed by writing convective acceleration according to the formula:

Gradient of vorticity is the pseudo tensor of rank 2:

Trace of this tensor is

Let us assume that the fluid is incompressible, *μ = const*, and its motion occurs in the field of potential mass forces. In this case with the help of Eq. (34), we can obtain the Navier–Stokes equation in the form:

now we apply curl operation (

We could rewrite the second term of the left part in this form:

but

As a result, we can obtain transport equation of vortices in an incompressible viscous fluid, which is named as the generalized Helmholtz equation:

or in more compact form:

It is necessary to note that in the case of compressible fluid and at

This transport equation can be written in another form, considering the equality

If we apply divergence operation to Eq. (36), then for incompressible fluid we obtain

because

On the other hand,

and, finally, we have

For the second power of vorticity, we can write

and now, if we scalar multiply transport equation of vortices by

then we obtain scalar transport equation of

For incompressible fluid

As we already know

Let us write one more equation:

Therefore,

This equation also can be written in the next form:

One more interesting relation is

but as we already mentioned,

The vector product of gradients of scalar functions gives us a vector; in terms of rotation of a vector function, we can write

Really, the left part of this equation is a vector:

and the right part is.

Therefore, Eq. (44) is valid.

The vector product of gradients of scalar functions also can be written as

Here we have in component form:

It is easy to see that this is equal to expression for

## 6. Mechanical energy equation

Mechanical energy balance equation can be obtained as a scalar product of each member of Eq. (12) on velocity vector

Transformations of the left part lead us to the following results:

It is easy to see that in sum the left part is the material derivative of kinetic energy of a fluid particle—quantity

Usually stress tensor is defined as the sum

Now we can represent equation of mechanical energy balance (Eq. (46)) considering

The first member of the right part of Eq. (46) is power of stresses

It is easy to be proven if we rewrite this expression in component form in Cartesian coordinates. In this case the left part of Eq. (49) is

The first term of the right part of Eq. (49) in component form is

The second term of the right part of Eq. (49) in component form is

Finally Eq. (49) in component form is

Due to the symmetry of the stress tensor *i* by *k* and vice versa, takes the form

We could simplify the last term of the right part of Eq. (46) if we introduce potential *z* axis is positive upwards) as earlier

After substituting the above expressions into Eq. (46), we obtain the equation for mechanical energy of a fluid flow:

In the left part of this equation, we observe the total mechanical energy of a fluid flow as the sum of kinetic and potential energy of the flow [6]. Often the right part of Eq. (51) is written in another form, where stress tensor is written as the sum

The first member is

The third member is

When we substitute these terms in Eq. (52) and then in Eq. (51), the equation for mechanical energy of a fluid flow will take the form:

As a result, we could conclude that the rate of change of total mechanical energy of a flow is equal to the sum of the powers of the pressure forces and viscous friction.

Navier–Stokes equation for a steady flow of viscous incompressible fluid is

The Laplacian of velocity in the right part can be written in the form:

It can be obtained by consideration of operation

The member with the basis vector

The same can be written for the members with basis vectors

And therefore formula Eq. (55) is valid.

One more useful expression based on Eq. (55) is

Using Eqs. (34) and (55) and considering the mass force in field of gravity with the help of potential *U*, *z* axis is positive upwards), we obtain instead Eq. (54):

where

In incompressible fluid

The gradient of total mechanical energy of a fluid particle

It is possible to obtain the divergence form of Eq. (54) for incompressible fluid considering

Then Eq. (54) will take the form:

Using concepts of identity tensor

The last term of Eq. (59) can be considered in a more simple form due to relation for incompressible fluid

## 7. Energy equation for moving fluid

The first law of thermodynamics connects internal energy, heat, and work. In the case of moving fluid, it can be written as follows:

where *t* is the time; *u* is the specific internal energy;

The physical meaning of this equation is that the rate of change of internal energy per unit volume is equal to rate of energy supply due to heat conduction, due to dissipation of mechanical energy of the flow, and due to heat from external or internal sources. Since stress tensor

This is the energy equation in terms of transfer of specific internal energy *u*.

Vector

where *T* is the temperature and *λ* is the coefficient of thermal conductivity.

Fourier’s law of thermal conductivity can also be written in terms of enthalpy, which for an ideal gas is related to temperature by the formula *ср* is the isobar heat capacity. Then considering

where

In Cartesian coordinates Eq. (62) can be written as follows:

The terms

where *μ* is the fluid shear viscosity and

Now we could write the dissipative term

Thus, in component form Rayleigh function

or in usual notations

This function can also be written in the componentless form:

For perfect gases [6, 7] internal energy is connected with temperature by the relation

The energy equation (Eq. (62)) can be also written in terms of enthalpy

Here we also used continuity equation (Eq. (2)). Finally, we can obtain energy equation in the form of enthalpy transport as

This is the second form of energy equation for the perfect gas in which

One more form of the energy equation can be written if we introduce stagnation enthalpy

Here we used the relation:

which is easy to be proven if we write it down in the component form considering symmetry of stress tensor*τ*

As a result, the dissipative term in Eq. (73) can be written as follows:

where

In usual axis designations, the first term is

while second term is

Finally function

The fourth form of the energy equation can be written in terms of entropy *s* transport. According to the fundamental thermodynamic relation,

Hence we have

and then if we substitute quantity

All forms of the equation energy (in terms of internal energy, enthalpy, stagnation enthalpy, and entropy) are equivalent.

Equation for temperature field of an arbitrary gas in the form of equation of transport of temperature *T* can be obtained from Eq. (62) or Eq. (71). In these cases quantities

The subscripts in derivatives here fix the parameters, with the constancy of which the derivatives are calculated. From these formulas the expressions for derivatives can be obtained:

If we substitute them into Eq. (62) and Eq. (71), we obtain two forms of the equation energy in terms of temperature transport.

In heat transfer problems, boundary conditions are specified in three different kinds—the first, second, and third kind:

The boundary conditions of the first kind consist in setting the temperature on the surface of the body.

The boundary conditions of the second kind are setting of the distribution of the heat flux density q on the surface of the body.

The boundary conditions of the third kind consist in setting the temperature of the flow over the surface of the body and the heat transfer conditions on its surface.

## 8. Divergence form of transport equations

Material derivative of any physical quantity Θ multiplied by density *ρ* always can be written in the “divergent” form as

This directly follows from the continuity equation [Eq. (2)].

Let us consider in detail the following cases for three ranks of a certain physical quantity Θ.

### 8.1 Quantity Θ is a scalar

Let us assume that quantity Θ is temperature *T:*

We could prove this equality if we write the left and right parts in component form. For the left part, we have

For the right part, we have

Since the expression in parentheses is zero (due to continuity equation), the equality of the left and right parts is obvious.

### 8.2 Quantity Θ is a vector

Let us assume that quantity Θ is velocity

Here in the last term, we see tensor

We could prove this equality if we write the left and right parts in the component form and use continuity equation.

For the left part, we have

The first term of the right part is

The second term of the right part is

The right part as a whole is

Therefore, the expression (Eq. (83)) is valid in case Θ is a vector.

### 8.3 Quantity Θ is a tensor

Let us assume that Θ is a tensor, for instance, stress tensor

The left part of this equation in the component form can be written as follows:

The right part is

Therefore, the expression (Eq. (83)) is also valid in case Θ is a tensor.

It is necessary to note the derivative

There are different forms of deviatoric stress rate for an arbitrary second-rank tensor

Jaumann G.:

Rivlin R.:

Truesdell C.:

Oldroyd J., Sedov L.I., etc. [8, 10, 11]

At present, the question of which derivative is more appropriate to use when constructing rheological equations is unclear. The most common is the rotational derivative by Gustav Jaumann. The corresponding material derivative in the form by Jaumann, for an arbitrary tensor of the second rank, has the form

Material derivative in the form by Rivlin is written as follows:

It is easy to see that Rivlin’s derivative differs from Jaumann’s one by the additional term

Rotational derivative of a symmetric tensor is also a symmetric tensor. As an example, let us consider the rotational derivative of strain-rate tensor and spin tensor:

As a result, we have obtained the symmetrical second-rank tensor.

## 9. Conclusions

In this chapter, some applications of tensor calculus in fluid dynamics and heat transfer are presented. Typical transformations of equations and governing relations are discussed. Main conservation equations are given and analyzed. The governing equations of fluid motion and energy were obtained.