## 1 Introduction  
Recently, there has been a systematic increase in interest in the use of magnetorheological (MR) fluids with viscosity controlled by magnetic or electric signals; these fluids can be found in a wide variety of devices (e.g., robots [1], testing stands [2], lifts [3] and two DoF brakes for investigating linear-rotary motors [4]) and seem to be particularly useful in automotive and motorcycle drive systems such as brakes and clutches [5, 6]. In cars and motorcycles, there is a need for the frequent coupling and decoupling of internal combustion or electric drive motors with a vehicle's wheel-drive system, and during such transient states, the clutches operate with varying slips, which result in friction (MR fluid-solid bodies), slip heat generation and, in consequence, temperature increases in the MR fluids having a direct influence on the clutch life. In recent years, this problem has been widely discussed by many authors [6, 7, 8, 9]. A power flow diagram for MR clutches in transient states (acting with slippage) is depicted in Figure 0(a),0. Magnetorheological (MR) clutches can also be used in industrial drive systems because of the ease of the coupling and decoupling process and the possibility of acting as an overload protection system [5]. Industrial drive systems work for long periods in a steady continuous-duty state with no slippage; hence, the generation of heat in the MR fluid associated with transient states does not play a significant role. In such cases electrical power consumption necessary for the production of a magnetic field and maintenance of a coupling state becomes the key issue for MR clutches. Electric power in an excitation coil connected to the flow of the excitation current is released in the form of Joule heat and becomes--next to the heat sources associated with friction in bearings--an additional heat source, increasing the temperature of the working clutch. For MR devices with a disc structure, commonly met in clutches, the possibility of a different number of discs is indicated, and each number of discs is associated with different properties. In article [10], the authors considered in detail the problem of the optimal selection of the number of discs N in MR clutches (N is equal to the number of discs of a clutch primary (driving) member) in the industrial drive systems from the viewpoint of various criteria and limitations. In particular, the authors not only took into account the external dimensions of the clutches: their lengths and diameters, masses and volumes but also characteristic factors: the maximum torque per mass ratio \(T_{c}/m\) and the maximum torque per volume ratio \(T_{c}/V\). The result of a detailed comparative analysis was the conclusion that the optimal (recommended) number of discs in a multi-disc magnetorheological clutch, taking into account the above-mentioned criteria, is equal to N = 2. In drawing this conclusion, the authors did not take into account the thermal operating conditions of the clutches. To their knowledge, in the available technical literature, the problem of power consumption by magnetorheological clutches in a steady continuous-duty states with no slippage, as well as copper losses in the excitation coil and spatial temperature distribution in such clutches (particularly, the maximum values of coils and MR fluid temperatures), has not been considered thus far. Hence, the main goal of this article is to consider the above-mentioned problems for MR clutches applied in the industrial continuous-duty drive systems.