Flow of mechanically incompressible, but thermally expansible viscous fluids A. Mikelic, A. Fasano, A. Farina Montecatini, Sept

LECTURE 1. Basic mathemathical modelling LECTURE 2. Mathematical problem LECTURE 3. Stability

Definitions and basic equations We start by recalling the mass balance, the momentum and energy equations, as well as the Clausius-Duhem inequality in the Eulerian formalism. Following an approach similar to the one presented in [1], pages 51-85, we denote by: the density, velocity, specific internal energy, absolute temperature and specific entropy, satisfying the following system of equations: [1]. H. Schlichting, K. Gersten, Boundary-Layer Theory, 8th edition, Springer, Heidelberg, 2000.

where: denotes the material derivative. is the rate of strain tensor. T is the Cauchy stress tensor. Further, is the heat flux vector. is the gravity acceleration. Indeed is the unit vector relative to the x 3 axis directed upward. In the energy equation the internal heat sources are disregarded.

Introducing the specific (i.e. per unit mass) Helmholtz free energy and using the energy equation, Clausius-Duhem inequality becomes

Constitutive equations Key point of the model is to select suitable constitutive equations for in terms of the dependent variables The constitutive equations selection has to be operated considering the physical properties of the fluids we dial with.

Main physical properties The fluid is mechanically incompressible but thermally dilatable (i.e. the fluid can sustain only isochoric motion in isothermal conditions) T 1 T 2 > T 1 The fluid behaves as a linear viscous fluid (Newtonian fluid) Shear stress proportional to shear rate

V in, T in V fin, T fin thermal expansion coefficient We assume that  may depend on temperature, but not on pressure, i.e.

So, if, for instance,  is constant, in general we have  T   function of temperature ^^

V in, T in V fin, T fin refernce configuration actual configuration i.e. J is a function of T, and can be expressed in terms of the thermal expansion coeff. 

Since J=J(T), continity equation reads as a constraint linking temperature variations with the divergence of the velocity field.

Remark If T is uniform and constant the flow is isochoric First consequence of our modelling: We have introduced a CONSTRAINT. The admissible flows are those that fulfill the condition force that maintains the constraint

The constraint  T  models the fluid as being incompressible under isothermal conditions, but whose density changes in response to changes in temperature. In other words, given a temperature T the fluid is capable to exert any force for reaching the corresponding density. Physical remark rigid box fluid  C According to the model According to the reality The box does not breake  C The box breakes

Let us now turn to a discussion of the constraint. We start considering the usual incompressibility constraint We proceed applying classical procedure. We modify the forces by adding, as in classical mechanics, a term (the so called constraint response) due to the constraint itself. We thus consider Where the subscript r indicates the portion of the stress given by constitutive equations and subscript c refers to the constraint response. The contraint response is required: (i)to have no dependendance on the state variables (ii)to produce no entropy in any motion that satisfies the constarint, i.e.

Hece, the above conditions demand that: There is no constitutive equation for p, i.e. p does not depend on any variable defining the state of the system. p, usually referred to as mechanical pressure, is a new state variable of the system.

Remark FcFc xx Smooth constraints in classical mechanics (D'Alembert ) FcFc Constarint response xx Displacement compatible with the constraint

In the general case (density-temperature constraint, for instance) it is assumed that the dependent quantities are determined by constitutive functions only up to an additive constraint response (see [2], [3])  Adkins, 1958 [3] Green, Naghdi, Trapp, 1970 r constitutive, c constraint The contraint responses are required to do not dissipate energy Recalling

we have no entropy production, i.e. no dissipation Considering various special subsets of the set of all allowable thermomechanical processes, the above equation leads to (1) when T and are such that

Equation (1) reduces to

Hence There is no Where p is a function of position and time. Recall: There is no constitutive quation for p constitutive quation for p. The function p defined above is usually referred to as mechanical pressure and is not the thermodynamic pressure that is defined through an equation of state.

Remark 1 There is also another formulation of the theory. We briefly summarize it We start considering the theory of a compressible (i.e. unconstrained) fluid, introducing the Helmholtz free energy and we develop the standard theory considering which gives. We deduce that: There is no equation of state defining the pressure p. The distinctions of the two constraint theory approaches are as assumes follows: The first approach assumes the existence of an additive constraint response, postulated to produce no entropy, whereas the deduces above approach deduces the additive constraint response.

Remark 2 Constraints that are nonlinear cannot, in general, be considered by the Procedure prevuously illustrated. For example, consider this constraint (1) Now the conditions to be imposed are: 1. whenever 2. does not depend on the state variables We can no longer demand that condition 2 is fulfilled !! (1) Such a constraint is for illustrative purposes only and does not necessarily correspond to a physically meaningful situation.

Deduction of the constitutive equations Assumption 1. Assumption 2. Linear viscous fluid Setting so that we have

So, we are left with and assume OK !!

Summarizing ASSUMTION

Remark In formulating constitutive models, such as, for instance,  we must have in mind some inferential method for quantifying them. Concerning  ) we shall return to this point later.